]>
2017
10
10
ISSN 2008-1898
460
Homeomorphism metric space and the fixed point theorems
Homeomorphism metric space and the fixed point theorems
en
en
The purpose of this
paper is to introduce the concept of the homeomorphism metric
space and to prove the fixed point theorems and the best proximity
point theorems for generalized contractions in such spaces. The
multiplicative metric space is a special form of the homeomorphism
metric space. The results of this paper improve and extend the previously
known ones in the literature.
5132
5141
Yinglin
Luo
Department of Mathematics
Tianjin Polytechnic University
China
tjluoyinglin@sina.com
Yongfu
Su
Department of Mathematics
Tianjin Polytechnic University
China
tjsuyongfu@163.com
Wenbiao
Gao
Department of Mathematics
Tianjin Polytechnic University
China
15822752271@163.com
Homeomorphism metric space
multiplicative metric space
metric space
\(b\)-metric space
generalized contraction
fixed point
best proximity point
Article.1.pdf
[
[1]
R. P. Agarwal, E. Karapınar, B. Samet , An essential remark on fixed point results on multiplicative metric spaces, Fixed Point Theory Appl., 2016 (2016), 1-3
##[2]
A. E. Bashirov, E. M. Kurplnar, A. Ozyaplcl, Multiplicative calculus and its applications , J. Math. Anal. Appl., 337 (2008), 36-48
##[3]
A. E. Bashirov, E. Misirli, Y. Tandogdu, A. Ozyaplcl , On modeling with multiplicative differential equations, Appl. Math. J. Chinese Univ., 26 (2011), 425-438
##[4]
S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5-11
##[5]
K. Fan , Extensions of two fixed point theorems of F. E. Browder, Math. Z., 112 (1969), 234-240
##[6]
L. Florack, H. Van Assen , Multiplicative calculus in biomedical image analysis, J. Math. Imaging Vision, 42 (2012), 64-75
##[7]
W. A. Kirk, S. Reich, P. Veeramani, Proximinal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim., 24 (2003), 851-862
##[8]
Y. Su, J.-C. Yao, Further generalized contraction mapping principle and best proximity theorem in metric spaces, Fixed Point Theory Appl., 2015 (2015), 1-13
##[9]
J. Zhang, Y. Su, Q. Cheng, A note on ‘A best proximity point theorem for Geraghty-contractions’, Fixed Point Theory Appl., 2013 (2013), 1-4
##[10]
J. Zhang, Y. Su, Q. Cheng , Best proximity point theorems for generalized contractions in partially ordered metric spaces, Fixed Point Theory Appl., 2013 (2013), 1-7
]
Degenerate ordered Bell numbers and polynomials associated with umbral calculus
Degenerate ordered Bell numbers and polynomials associated with umbral calculus
en
en
In this paper, we study degenerate ordered Bell polynomials with the viewpoint of Carlitz's degenerate Bernoulli and Euler polynomials and derive by using umbral calculus some properties and new identities for the degenerate ordered Bell polynomials associated with special polynomials.
5142
5155
Taekyun
Kim
Department of Mathematics, College of Science
Department of Mathematics
Tianjin Polytechnic University
Kwangwoon University
China
Republic of Korea
tkkim@kw.ac.kr
Dae San
Kim
Department of Mathematics
Sogang University
Republic of Korea
dskim@sogang.ac.kr
Gwan-Woo
Jang
Department of Mathematics
Kwangwoon University
Republic of Korea
gwjang@kw.ac.kr
Lee-Chae
Jang
Graduate School of Education
Konkuk University
Republic of Korea
lcjang@konkuk.ac.kr
Degenerate ordered Bell polynomial
umbral calculus
Euler polynomials
Article.2.pdf
[
[1]
E. T. Bell , Postulational bases for the umbral calculus, Amer. J. Math., 62 (1940), 717-724
##[2]
L. Carlitz , Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math., 15 (1979), 51-88
##[3]
L. Comtet , Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel Publishing, Holland (1974)
##[4]
R. Dere, Y. Simsek, Applications of umbral algebra to some special polynomials, Adv. Stud. Contemp. Math., 22 (2012), 433-438
##[5]
A. Di Crescenzo, G.-C. Rota, On umbral calculus, Ricerche Mat., 43 (1994), 129-162
##[6]
D. V. Dolgiĭ, D. S. Kim, T. Kim, Korobov polynomials of the first kind, Sb. Math., 208 (2017), 60-74
##[7]
T. Kim , Identities involving Laguerre polynomials derived from umbral calculus, Russ. J. Math. Phys., 21 (2014), 36-45
##[8]
D. S. Kim, T. Kim, J. J. Seo, Higher-order Daehee polynomials of the first kind with umbral calculus, Adv. Stud. Contemp. Math., 24 (2014), 5-18
##[9]
S. Roman, The theory of the umbral calculus, J. Math. Anal. Appl., 95 (1983), 528-563
##[10]
S. Roman , The umbral calculus, Academic Press, New York (1984)
##[11]
G. C. Rota, B. D. Taylor, An introduction to the umbral calculus, Analysis, geometry and groups: a Riemann legacy volume, Hadronic Press, Palm Harbor (1993)
##[12]
G. C. Rota, B. D. Taylor , The classical umbral calculus, SIAM J. Math. Anal., 25 (1994), 694-711
]
Uniqueness and properties of positive solutions for infinite-point fractional differential equation with p-Laplacian and a parameter
Uniqueness and properties of positive solutions for infinite-point fractional differential equation with p-Laplacian and a parameter
en
en
Using new methods for dealing with an infinite-point fractional differential equation with p-Laplacian and a parameter, we study the existence of unique positive solution for any given positive parameter \(\lambda\), and then give some clear properties of positive solutions which depend on the parameter \(\lambda>0\), that is, the positive solution \(u_\lambda^{*}\) is continuous, strictly increasing in \(\lambda\) and \(\lim_{\lambda\rightarrow +\infty}\|u_\lambda^*\|=+\infty,\lim_{\lambda\rightarrow 0^+}\|u_\lambda^*\|=0.\) Our analysis relies on some new theorems for operator equations \(A(x,x)=x\) and \(A(x,x)=\lambda x\), where \(A\) is a mixed monotone operator.
5156
5164
Li
Wang
School of Mathematical Sciences
Shanxi University
P. R. China
985388806@qq.com
Chengbo
Zhai
School of Mathematical Sciences
Shanxi University
P. R. China
cbzhai@sxu.edu.cn
Uniqueness
positive solution
\(p\)-Laplacian
infinite-point fractional differential equation
mixed monotone operator
Article.3.pdf
[
[1]
D. Baleanu, S. D. Purohit, J. C. Prajapati, Integral inequalities involving generalized Erdélyi-Kober fractional integral operators, Open Math., 14 (2016), 89-99
##[2]
D. Baleanu, S. D. Purohit, F. Uçar, On Grüss type integral inequality involving the Saigo’s fractional integral operators, J. Comput. Anal. Appl., 19 (2015), 480-489
##[3]
H.-L. Gao, X.-L. Han, Existence of positive solutions for fractional differential equation with nonlocal boundary condition, Int. J. Differ. Equ., 2011 (2011), 1-10
##[4]
L.-M. Guo, L.-S. Liu, Y.-H. Wu, Existence of positive solutions for singular fractional differential equations with infinitepoint boundary conditions, Nonlinear Anal. Model. Control, 5 (2016), 635-650
##[5]
L.-M. Guo, L.-S. Liu, Y.-H. Wu, Existence of positive solutions for singular higher-order fractional differential equations with infinite-point boundary conditions , Bound. Value Probl., 2016 (2016), 1-22
##[6]
L. Hu , Existence of solutions to a coupled system of fractional differential equations with infinite-point boundary value conditions at resonance, Adv. Difference Equ., 2016 (2016), 1-13
##[7]
D. Kumar, S. D. Purohit, A. Secer, A. Atangana , On generalized fractional kinetic equations involving generalized Bessel function of the first kind, Math. Probl. Eng., 2015 (2015), 1-7
##[8]
D. Kumar, J. Singh, D. Baleanu, Numerical computation of a fractional model of differential-difference equation, J. Comput. Nonlinear Dyn., 11 (2016), 1-6
##[9]
B.-X. Li, S.-R. Sun, Y. Sun , Existence of solutions for fractional Langevin equation with infinite-point boundary conditions, J. Appl. Math. Comput., 53 (2017), 683-692
##[10]
X. Y. Lu, X. Q. Zhang, L. Wang, Existence of positive solutions for a class of fractional differential equations with m-point boundary value conditions, (Chinese) J. Systems Sci. Math. Sci., 34 (2014), 218-230
##[11]
I. Podlubny , Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)
##[12]
S. D. Purohit, Solution of fractional partial differential equations related to quantum mechanics, Adv. Appl. Math. Mech., 5 (2013), 639-651
##[13]
S. D. Purohit, S. L. Kalla, On fractional partial differential equations related to quantum mechanics, J. Phys. A, 44 (2011), 1-8
##[14]
H. M. Srivastava, D. Kumar, J. Singh, An efficient analytical technique for fractional model of vibration equation, Appl. Math. Model., 45 (2017), 192-204
##[15]
C.-B. Zhai, L.-L. Zhang, New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems, J. Math. Anal. Appl., 382 (2011), 594-614
##[16]
X.-Q. Zhang, Positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions, Appl. Math. Lett., 39 (2015), 22-27
##[17]
Q.-Y. Zhong, X.-Q Zhang, Positive solution for higher-order singular infinite-point fractional differential equation with p-Laplacian, Adv. Difference Equ., 2016 (2016), 1-11
]
Convergence theorems and stability results of pointwise asymptotically nonexpansive mapping in Banach space
Convergence theorems and stability results of pointwise asymptotically nonexpansive mapping in Banach space
en
en
The purpose of this paper is to approximate
the fixed point of pointwise asymptotically nonexpansive mapping
using the generalized Mann and generalized Ishikawa iterative scheme.
And under the condition that the pointwise asymptotically nonexpansive
mapping is compact, the stability results of the two iterative schemes
are studied.
The main results of this paper modify and improve many important recent
results in the literature.
5165
5173
Qiansheng
Feng
Department of Mathematics
Tianjin University
China
fengqiansheng-2008@163.com
Nan
Jiang
Department of Mathematics
Tianjin University
China
nanj@tju.edu.cn
Yongfu
Su
Department of Mathematics
Tianjin Polytechnic University
China
suyongfu@tjpu.edu.cn
Pointwise asymptotically nonexpansive mapping
generalized Mann iterative scheme
generalized Ishikawa iterative scheme
stability result
convergence theorem
Article.4.pdf
[
[1]
I. D. Arandjelović, Note on asymptotic contractions , Appl. Anal. Discrete Math., 1 (2007), 211-216
##[2]
J. Balooee, Weak and strong convergence theorems of modified Ishikawa iteration for an infinitely countable family of pointwise asymptotically nonexpansive mappings in Hilbert spaces, Arab J. Math. Sci., 17 (2011), 153-169
##[3]
A. O. Bosede, B. E. Rhoades, Stability of Picard and Mann iteration for a general class of functions, J. Adv. Math. Stud., 3 (2010), 23-25
##[4]
H. Dehghan, Demiclosed principle and convergence of a hybrid algorithm for multivalued *-nonexpansive mappings, Fixed Point Theory, 14 (2013), 107-115
##[5]
K. Goebel, W. A. Kirk , A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174
##[6]
A. M. Harder, T. L. Hicks, Stability results for fixed point iteration procedures, Math. Japon., 33 (1988), 693-706
##[7]
G. Khalilzadeh, R. Sarikhani, Fixed Point for pointwise asymptotically nonexpansive mapping in Banach Space which has Frechet Differential Norm, Int. J. Math. Anal., 7 (2013), 425-432
##[8]
W. A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl., 277 (2003), 645-650
##[9]
W. A. Kirk , Asymptotic pointwise contractions, Plenary Lecture, the 8th International Conference on Fixed Point Theory and Its Applications, Chiang Mai University, Thailand, (2007), 16-22
##[10]
W. A. Kirk, H.-K. Xu, Asymptotic pointwise contractions, Nonlinear Anal., 69 (2008), 4706-4712
##[11]
W. M. Kozlowski , Fixed point iteration processes for asymptotic pointwise nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 377 (2011), 43-52
##[12]
Z. Ma, L. Wang, Demiclosed principle and convergence theorems for asymptotically strictly pseudononspreading mappings and mixed equilibrium problems, Fixed Point Theory Appl., 2014 (2014), 1-20
##[13]
Z. Opial , Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597
##[14]
S. Rezapour, R. H. Haghi, B. E. Rhoades, Some results about T-stability and almost T-stability, Fixed Point Theory, 12 (2011), 179-186
##[15]
B. E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure Appl. Math., 21 (1990), 1-9
##[16]
J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43 (1991), 153-159
##[17]
K.-K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301-308
##[18]
R.Wangkeeree, H. Dehghan , Strong and-convergence of Moudafi’s iterative scheme in CAT(0) spaces, J. Nonlinear Conv. Anal., 16 (2015), 299-309
##[19]
H.-K. Xu, Asymptotic and weakly asymptotic contractions, Indian J. Pure Appl. Math., 36 (2005), 145-150
##[20]
Q. Yuan, B. E. Rhoades, T-Stability of Picard Iteration in Metric Spaces, Fixed Point Theory and Appl., 2008 (2008), 1-4
##[21]
T. Zamfirescu, Fix point theorems in metric spaces, Arch. Math., 23 (1972), 292-298
]
Estimates of higher order fractional derivatives at extreme points
Estimates of higher order fractional derivatives at extreme points
en
en
We extend the results concerning the fractional derivatives of a
function at its extreme points to fractional derivatives of
arbitrary order. We also give an estimate of the error and present two examples to illustrate the validity of the results.
The presented results are valid for both Caputo and Riemann-Liouville fractional derivatives.
5174
5181
Mohammed
Al-Refai
Department of Mathematical Sciences
United Arab Emirates University
UAE
m_alrefai@uaeu.ac.ae
Dumitru
Baleanu
Department of Mathematics and Computer Science
Institute of Space Sciences
Cankaya University
Turkey
Romania
dumitru@cankaya.edu.tr
Extreme points
higher order fractional derivatives
Caputo derivative
Riemann-Liouville derivative
Article.5.pdf
[
[1]
A. B. Abdulla, M. Al-Refai, A. Al-Rawashdeh, On the existence and uniqueness of solutions for a class of non-linear fractional boundary value problems, J. King Saud Univ. Sci., 28 (2016), 103-110
##[2]
R. P. Agarwal, M. Benchohra, S. Hamani, A survey on existing results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010), 973-1033
##[3]
M. Al-Refai , Basic results on nonlinear eigenvalue problems with fractional order, Electron. J. Differential Equations, 2012 (2012), 1-12
##[4]
M. Al-Refai, On the fractional derivative at extreme points, Electron. J. Qual. Theory Differ. Equ., 2012 (2012), 1-5
##[5]
M. Al-Refai, Yu. Luchko, Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications, Fract. Calc. Appl. Anal., 17 (2014), 483-498
##[6]
M. Al-Refai, Yu. Luchko , Maximum principle for the multi-term time-fractional diffusion equations with the Riemann- Liouville fractional derivatives, Appl. Math. Comput., 257 (2015), 40-51
##[7]
M. Al-Refai, Yu. Luchko, Analysis of fractional diffusion equations of distributed order: Maximum principles and its applications, Analysis, 36 (2016), 123-133
##[8]
D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, World Scientific Publishing, Hackensack (2012)
##[9]
D. Baleanu, O. Mustafa, Asymptotic Integration and Stability for Differential Equations of Fractional Order, World Scientific Publishing, Hackensack (2015)
##[10]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006)
##[11]
Y. Luchko, Fractional diffusion and wave propagation, Springer, 2014 (2014), 1-36
##[12]
J. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus , Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140-1153
##[13]
I. Podlubny , Fractional Differential Equations, Academic Press, San Diego (1999)
##[14]
J. Sabatier, O. P. Agarwal, J. A. Tenreiro Machado, Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering, Springer, Netherlands (2007)
##[15]
S. G. Samko, A. A. Kilbas, O. I. Marichev , Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Switzerland (1993)
##[16]
M. Stynes, J. L. Gracia, A finite difference method for a two-point boundary value problem with Caputo fractional derivative, IMA Journal of Numerical Analysis, 35 (2014), 698-721
##[17]
M. Syam, M. Al-Refai, Positive solutions and monotone iterative sequences for a class of higher order boundary value problems of fractional order, J. Fract. Calc. Appl., 4 (2013), 147-159
##[18]
W. Xie, J. Xiao, Z. Luo, Existence of solutions for Riemann-Liouville fractional boundary value problem, Abstract and Applied Analysis, Abstr. Appl. Anal., 2014 (2014), 1-9
##[19]
X.-J. Yang, Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems, Thermal Science, 2016 (2016), 1-13
##[20]
X. J. Yang , A new fractional operator of variablr order: application in the description of anomalous diffusion , Physica A: Statis. Mechanics Appl., 481 (2017), 276-283
##[21]
X. J. Yang, F. Gao, H. M. Srivastava , New rheological models within local fractional derivative, Rom. Rep. Phys., 2017 (2017), 1-12
##[22]
H. Ye, F. Liu, V. Anh, I. Turner, Maximum principle and numerical method for the multi-term time-space Riesz-Caputo fractional differential equations , Appl. Math. Comput., 227 (2014), 531-540
]
Nonlinear Mittag-Leffler stability of nonlinear fractional partial differential equations
Nonlinear Mittag-Leffler stability of nonlinear fractional partial differential equations
en
en
This paper focuses on the application of fractional backstepping control scheme for nonlinear fractional partial differential equation (FPDE). Two types of fractional derivatives are considered in this paper, Caputo and the Grünwald-Letnikov fractional derivatives. Therefore, obtaining highly accurate approximations for this derivative is of a great
importance. Here, the discretized approach for the space variable is used to transform the FPDE into a system of fractional differential equations. The convergence of the closed loop system is guaranteed in the sense of Mittag-Leffler stability. An illustrative example is given to demonstrate the effectiveness of the proposed control scheme.
5182
5200
Ibtisam Kamil
Hanan
Institute of Engineering Mathematics
Department of Mathematics and Computer Applications, College of Science
Universiti Malaysia Perlis
Al-Nahrain University
Malaysia
Iraq
ibtisamkamil83@gmail.com
Muhammad Zaini
Ahmad
Institute of Engineering Mathematics
Universiti Malaysia Perlis
Malaysia
mzaini@unimap.edu.my
Fadhel Subhi
Fadhel
Department of Mathematics and Computer Applications, College of Science
Al-Nahrain University
Iraq
dr_fadhel67@yahoo.com
Backstepping method
fractional Lyapunov function
fractional derivative
boundary control
fractional partial differential equation
Article.6.pdf
[
[1]
M. P. Aghababa , Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2670-2681
##[2]
N. Aguila-Camacho, M. A. Duarte-Mermoud, J. A. Gallegos, Lyapunov functions for fractional order systems, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2951-2957
##[3]
A. K. Alomari, F. Awawdeh, N. Tahat, F. Bani Ahmad, W. Shatanawi , Multiple solutions for fractional differential equations: Analytic approach, Appl. Math. Comput., 219 (2013), 8893-8903
##[4]
D. Baleanu, J. A. T. Machado, A. C. Luo, Fractional dynamics and control, Springer, New York (2011)
##[5]
T. A. Burton, Fractional differential equations and Lyapunov functionals, Nonlinear Anal., 74 (2011), 5648-5662
##[6]
S. Dadras, H. R. Momeni , Passivity-based fractional-order integral sliding-mode control design for uncertain fractionalorder nonlinear systems, Mechatronics, 23 (2013), 880-887
##[7]
D.-S. Ding, D.-L. Qi, Q. Wang, Non-linear Mittag-Leffler stabilisation of commensurate fractional-order non-linear systems, IET Control Theory Appl., 9 (2015), 681-690
##[8]
C. Farges, L. Fadiga, J. Sabatier, \(H_\infty\) analysis and control of commensurate fractional order systems, Mechatronics, 23 (2013), 772-780
##[9]
C. Farges, M. Moze, J. Sabatier, Pseudo-state feedback stabilization of commensurate fractional order systems, Automatica J. IFAC, 46 (2010), 1730-1734
##[10]
V. Lakshmikantham, S. Leela, M. Sambandham , Lyapunov theory for fractional differential equations, Commun. Appl. Anal., 12 (2008), 365-376
##[11]
Y. H. Lan, H. B. Gu, C. X. Chen, Y. Zhou, Y. P. Luo, An indirect Lyapunov approach to the observer-based robust control for fractional-order complex dynamic networks, Neurocomputing, 136 (2014), 235-242
##[12]
Y.-H. Lan, Y. Zhou, LMI-based robust control of fractional-order uncertain linear systems, Comput. Math. Appl., 62 (2011), 1460-1471
##[13]
Y. Li, Y. Chen, I. Podlubny , MittagLeffler stability of fractional order nonlinear dynamic systems, Automatica, 45 (2009), 1965-1969
##[14]
Y. Li, Y.-Q. Chen, I. Podlubny , Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl., 59 (2010), 1810-1821
##[15]
J.-G. Lu, Y.-Q. Chen, W. Chen, Robust asymptotical stability of fractional-order linear systems with structured perturbations, Comput. Math. Appl., 66 (2013), 873-882
##[16]
K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York (1974)
##[17]
F. Padula, S. Alcántara, R. Vilanova, A. Visioli, \(H_\infty\) control of fractional linear systems, Automatica, 45 (2013), 2276-2280
##[18]
I. Pan, S. Das, Intelligent fractional order systems and control: an introduction, Springer, New York (2012)
##[19]
I. Petráš, Fractional-order nonlinear systems: modeling, analysis and simulation, Springer, New York (2011)
##[20]
J. Sabatier, O. P. Agrawal, J. A. T. Machado, Advances in fractional calculus, Springer, Dordrecht (2007)
##[21]
H. Sheng, Y. Chen, T. Qiu, Fractional processes and fractional-order signal processing: techniques and applications, Springer, New York (2011)
##[22]
B. Shi, J. Yuan, C. Dong, Pseudo-state sliding mode control of fractional SISO nonlinear systems , Adv. Math. Phys., 2013 (2013), 1-7
##[23]
E. Sousa, How to approximate the fractional derivative of order \(1 < \alpha\leq 2\), Int. J. Bifurcation. Chaos, 2012 (2012), 1-13
##[24]
T. Takamatsu, H. Ohmori , Sliding Mode Controller Design Based on Backstepping Technique for Fractional Order System, SICE JCMSI, 9 (2016), 151-157
##[25]
Y. Tang, X. Zhang, D. Zhang, G. Zhao, X. Guan, Fractional order sliding mode controller design for antilock braking systems, Neurocomputing, 111 (2013), 122-130
##[26]
J. C. Trigeassou, N. Maamri, A. Oustaloup, Lyapunov Stability of Linear Fractional Systems: Part 1-Definition of Fractional Energy , ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2013 (2013), 1-10
##[27]
J. C. Trigeassou, N. Maamri, A. Oustaloup, Lyapunov Stability of Linear Fractional Systems: Part 2-Derivation of stability condition, ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2013 (2013), 1-10
##[28]
J.-C. Trigeassou, N. Maamri, J. Sabatier, A. Oustaloup , A Lyapunov approach to the stability of fractional differential equations , Signal Processing, 91 (2011), 437-445
##[29]
J. Wang, L. Lv, Y. Zhou , New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2530-2538
##[30]
X. Wen, Z. Wu, J. Lu, Stability analysis of a class of nonlinear fractional-order systems, IEEE Transactions on circuits and systems II: Express Briefs, 55 (2008), 1178-1182
##[31]
J. Yu, H. Hu, S. Zhou, X. Lin , Generalized Mittag-Leffler stability of multi-variables fractional order nonlinear systems , Automatica J. IFAC, 49 (2013), 1798-1803
##[32]
Y. H. Yuan, Q. S. Sun , Fractional-order embedding multiset canonical correlations with applications to multi-feature fusion and recognition, Neurocomputing, 122 (2013), 229-238
##[33]
X. F. Zhou, L. G. Hu, S. Liu, W. Jiang, Stability criterion for a class of nonlinear fractional differential systems, Appl. Math. Lett., 28 (2014), 25-29
]
Impact of non-separable incidence rates on global dynamics of virus model with cell-mediated, humoral immune responses
Impact of non-separable incidence rates on global dynamics of virus model with cell-mediated, humoral immune responses
en
en
In this paper, we study the dynamical behavior of a virus model into which cell-mediated and humoral immune responses are incorporated. The global stability of an infection-free equilibrium and four infected equilibria is established via a Lyapunov functional approach. The present construction methods are applicable to a wide range of incidence rates that are monotone increasing with respect to concentration of uninfected cells and concave with respect to the concentration of free virus particles. In addition, when the incidence rate is monotone increasing with respect to concentration of free virus particles, the functional approach plays an important role in determining the global stability of each of the four infected equilibria. This implies that the dynamical behavior of virus prevalence would be determined by basic reproduction numbers when the ``saturation effect" for free virus particles appears. We point out that the incidence rate includes not only separable incidence rate but also non-separable incidence rate such as standard incidence and Beddington-DeAngelis functional response.
5201
5218
Yoichi
Enatsu
Department of Applied Mathematics
Tokyo University of Science
Japan
yenatsu@rs.tus.ac.jp
Jinliang
Wang
School of Mathematical Science
Heilongjiang University
China
jinliangwang@aliyun.com
Toshikazu
Kuniya
Graduate School of System Informatics
Kobe University
Japan
tkuniya@port.kobe-u.ac.jp
Virus infection model
delay
global stability
incidence rate
Article.7.pdf
[
[1]
R. Arnaout, M. Nowak, D. Wodarz , HIV-1 dynamics revisited: Biphasic decay by cytotoxic lymphocyte killing?, Proc. Roy. Soc. Lond. B., 267 (2000), 1347-1354
##[2]
J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency , J. Animal Ecol., 44 (1975), 331-340
##[3]
S. Bonhoeffer, J. M. Coffin, M. A. Nowak, Human immunodeficiency virus drug therapy and virus load, J. Virol., 71 (1997), 3275-3278
##[4]
M. S. Ciupe, B. L. Bivort, D. M. Bortz, P. W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models, Math. Biosci., 200 (2006), 1-27
##[5]
D. L. DeAngelis, R. A. Goldstein, R. V. O’Neill, A model for trophic interaction , Ecology, 56 (1975), 881-892
##[6]
R. J. De Boer, A. S. Perelson , Towards a general function describing T cell proliferation, J. Theoret. Biol., 175 (1995), 567-576
##[7]
R. J. De Boer, A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison , J. Theoret. Biol., 190 (1998), 201-214
##[8]
K. Hattaf, N. Yousfi, A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Anal. Real World Appl., 13 (2012), 1866-1872
##[9]
G. Huang, Y. Takeuchi, W. Ma, Lyapunov functionals for delay differential equations model of viral infections , SIAM Journal on Appl. Math., 70 (2010), 2693-2708
##[10]
Y. Ji, M. Zheng , Dynamics analysis of a viral infection model with a general standard incidence rate, Abst. Appl. Anal., 2014 (2014), 1-6
##[11]
T. Kajiwara, T. Sasaki, Y. Takeuchi , Construction of Lyapunov functionals for delay differential equations in virology and epidemiology, Nonlinear Anal. Real World Appl., 13 (2012), 1802-1826
##[12]
A. Korobeinikov , Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883
##[13]
Y. Kuang , Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston (1993)
##[14]
C. C. McCluskey, Y. Yang , Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real World Appl., 25 (2015), 64-78
##[15]
Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 14-27
##[16]
M. A. Nowak, C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79
##[17]
R. Ouifki, G. Witten, Stability analysis of a model for HIV infection with RTI and three intracellular delays, BioSystems, 95 (2009), 1-6
##[18]
H. Peng, Z. Guo, Global stability for a viral infection model with saturated incidence rate, Abst. Appl. Anal., 2014 (2014), 1-9
##[19]
A. S. Perelson, P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44
##[20]
J. Prüss, R. Zacher, R. Schnaubelt , Global asymptotic stability of equilibria in models for virus dynamics, Math. Model. Nat. Phenom., 3 (2008), 126-142
##[21]
J.-L. Wang, M. Guo, X.-N. Liu, Z. Zhao, Threshold dynamics of HIV-1 virus model with cell-to-cell transmission, cellmediated immune responses and distributed delay, Appl. Math. Comput., 291 (2016), 149-161
##[22]
T. Wang, Z.-X. Hu., F. Liao, W. Ma , Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comp. Simulation, 89 (2013), 13-22
##[23]
J.-L. Wang, S.-Q. Liu, The stability analysis of a general viral infection model with distributed delays and multi-staged infected progression, Commun. Nonlinear Sci. Numer. Simul., 20 (2015), 263-272
##[24]
J.-L. Wang, J.-M. Pang, T. Kuniya, Y. Enatsu, Global threshold dynamics in a five-dimensional virus model with cellmediated, humoral immune responses and distributed delays, Appl. Math. Comput., 241 (2014), 298-316
##[25]
K. Wang, W. Wang, H. Pang, X.-N. Liu, Complex dynamic behavior in a viral model with delayed immune response, Phys. D, 226 (2007), 197-208
##[26]
Z.-P. Wang, R. Xu, Stability and Hopf bifurcation in a viral infection model with nonlinear incidence rate and delayed immune response, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 964-978
##[27]
S. Wang, D. Zou , Global stability of in-host viral models with humoral immunity and intracellular delays , Appl. Math. Model., 36 (2012), 1313-1322
##[28]
Y.-C. Yan, W. Wang, Global stability of a five-dimensional model with immune responses and delay, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 401-416
##[29]
Z.-H. Yuan, X.-F. Zou , Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays, Math. Biosci. Eng., 10 (2013), 483-498
##[30]
H. Zhu, Y. Luo, M. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay, Comput. Math. Appl., 62 (2011), 3091-3102
]
Existence of solution and Hyers-Ulam stability for a coupled system of fractional differential equations with \(p\)-Laplacian operator
Existence of solution and Hyers-Ulam stability for a coupled system of fractional differential equations with \(p\)-Laplacian operator
en
en
Models with \(p\)-Laplacian operator are common in different scientific fields including; plasma physics, chemical reactions design, physics, biophysics, and many others. In this paper, we investigate existence and uniqueness of solution and Hyers-Ulam stability for a coupled system of fractional differential equations with \(p\)-Laplacian operator. The Hyers-Ulam stability means that a differential equation
has a close exact solution which is generated by the approximate solution of the differential equation and the error in the approximation can be estimated. We use topological degree method and provide an expressive example as an application of the work.
5219
5229
Hasib
Khan
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering
Shaheed Benazir Bhutto University Sheringal
Hohai University
P. R. China
Pakistan
hasibkhan13@yahoo.com
Yongjin
Li
Department of Mathematics
Sun Yat-sen University
P. R. China
stslyj@mail.sysu.edu.cn
Hongguang
Sun
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering
Hohai University
P. R. China
shg@hhu.edu.cn
Aziz
Khan
Department of Mathematics
University of Peshawar
Pakistan
azizkhan927@yahoo.com
Existence and uniqueness of solution
Hyers-Ulam stability
topological degree method
\(p\)-Laplacian operator
Article.8.pdf
[
[1]
A. Ali, B. Samet, K. Shah, R. A. Khan , Existence and stability of solution to a toppled systems of differential equations of non-integer order, Bound. Value Probl., 2017 (2017), 1-13
##[2]
G. A. Anastassiou , On right fractional calculus, Chaos Solitons Fractals, 42 (2009), 365-376
##[3]
D. Baleanu, R. P. Agarwal, H. Mohammadi, S. Rezapour , Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces , Bound. Value Probl., 2013 (2013), 1-8
##[4]
D. Băleanu, O. G. Mustafa , On the global existence of solutions to a class of fractional differential equations, Comput. Math. Appl., 59 (2010), 1835-1841
##[5]
D. Băleanu, O. G. Mustafa, R. P. Agarwal , An existence result for a superlinear fractional differential equation, Appl. Math. Lett., 23 (2010), 1129-1132
##[6]
D. Băleanu, O. G. Mustafa, R. P. Agarwal , On the solution set for a class of sequential fractional differential equations, J. Phys. A, 43 (2010), 1-7
##[7]
J. Brzdęk, L. Cădariu, K. Ciepliński, A. Fošner, Z. Leśniak, Survey on recent Ulam stability results concerning derivations, J. Funct. Spaces, 2016 (2016), 1-9
##[8]
M. Caputo, Linear models of dissipation whose Q is almost frequency independent, II, Reprinted from Geophys. J. R. Astr. Soc., 13 (1967), 529–539, Fract. Calc. Appl. Anal., 11 (2008), 4-14
##[9]
L.-L. Cheng, W.-B. Liu, Q.-Q. Ye, Boundary value problem for a coupled system of fractional differential equations with p-Laplacian operator at resonance, Electron. J. Differential Equations, 2014 (2014), 1-12
##[10]
P. Găvruţa, S.-M. Jung, Y.-J. Li, Hyers-Ulam stability for second-order linear differential equations with boundary conditions, Electron. J. Differential Equations, 2011 (2011), 1-5
##[11]
A. Granas, J. Dugundji , Fixed point theory , Springer Monographs in Mathematics, Springer-Verlag, New York (2003)
##[12]
R. Hilfer (Ed.) , Applications of fractional calculus in physics , World Scientific Publishing Co., Inc., River Edge, NJ (2000)
##[13]
Z.-G. Hu, W.-B. Liu, J.-Y. Liu, Existence of solutions for a coupled system of fractional p-Laplacian equations at resonance, Adv. Difference Equ., 2013 (2013), 1-14
##[14]
F. Isaia , On a nonlinear integral equation without compactness, Acta Math. Univ. Comenian. (N.S.), 233–240. (2006)
##[15]
H. Jafari, D. Baleanu, H. Khan, R. A. Khan, A. Khan , Existence criterion for the solutions of fractional order p-Laplacian boundary value problems, Bound. Value Probl., 2015 (2015), 1-10
##[16]
R. A. Khan, A. Khan , Existence and uniqueness of solutions for p-Laplacian fractional order boundary value problems, Comput. Methods Differ. Equ., 205–215. (2014)
##[17]
R. A. Khan, A. Khan, A. Samad, H. Khan , On existence of solutions for fractional differential equation with p-Laplacian operator , J. Fract. Calc. Appl., 5 (2014), 28-37
##[18]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006)
##[19]
P. Kumam, A. Ali, K. Shah, R. A. Khan , Existence results and Hyers-Ulam stability to a class of nonlinear arbitrary order differential equations, J. Nonlinear Sci. Appl., 10 (2017), 2986-2997
##[20]
N. I. Mahmudov, S. Unul , Existence of solutions of \(\alpha\in (2, 3]\) order fractional three-point boundary value problems with integral conditions, Abstr. Appl. Anal., 2014 (2014), 1-12
##[21]
N. I. Mahmudov, S. Unul , Existence of solutions of fractional boundary value problems with p-Laplacian operator, Bound. Value Probl., 2015 (2015), 1-16
##[22]
N. I. Mahmudov, S. Unul , On existence of BVP’s for impulsive fractional differential equations, Adv. Difference Equ., 2017 (2017), 1-16
##[23]
K. S. Miller, B. Ross , An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1993)
##[24]
K. B. Oldham, J. Spainer, The fractional calculus, Theory and applications of differentiation and integration to arbitrary order, With an annotated chronological bibliography by Bertram Ross, Mathematics in Science and Engineering, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1974)
##[25]
I. Podlubny , Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)
##[26]
K. R. Prasad, B. M. B. Krushna, Multiple positive solutions for a coupled system of p-Laplacian fractional order two-point boundary value problems, Int. J. Differ. Equ., 2014 (2014), 1-10
##[27]
S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Theory and applications, Edited and with a foreword by S. M. Nikol’skiı, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon (1993)
##[28]
T.-F. Shen,W.-B. Liu, X.-H. Shen , Existence and uniqueness of solutions for several BVPs of fractional differential equations with p-Laplacian operator , Mediterr. J. Math., 13 (2016), 4623-4637
##[29]
X.-W. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 22 (2009), 64-69
##[30]
H.-G. Sun, Z.-P. Li, Y. Zhang, W. Chen , Fractional and fractal derivative models for transient anomalous diffusion: model comparison, Chaos Solitons Fractals, 102 (2017), 346-353
##[31]
H.-G. Sun, Y. Zhang, W. Chen, D. M. Reeves, Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media, J. Contam. Hydrol., 157 (2014), 47-58
##[32]
C. Urs, Coupled fixed point theorems and applications to periodic boundary value problems, Miskolc Math. Notes, 14 (2013), 323-333
]
Reduced differential transform method for solving time and space local fractional partial differential equations
Reduced differential transform method for solving time and space local fractional partial differential equations
en
en
We apply the new local fractional reduced differential transform method to obtain the solutions of some linear and nonlinear partial differential equations on Cantor set. The reported results are compared with the related solutions presented in the literature and the graphs are plotted to show their behaviors. The results prove that the presented method is faster and easy to apply.
5230
5238
Omer
Acan
Department of Mathematics, Art and Science Faculty
Siirt University
Turkey
omeracan@yahoo.com
Maysaa Mohamed
Al Qurashi
Department of Mathematics, Faculty of Art and Science
King Saud University
Saudi Arabia
maysaa@ksu.edu.sam
Dumitru
Baleanu
Department of Mathematics and Computer Sciences, Faculty of Art and Science
Institute of Space Sciences
Cankaya University
Turkey
Romania
dumitru@cankaya.edu.tr
Approximate solution
local fractional derivative
partial differential equations
reduced differential transform method
Article.9.pdf
[
[1]
O. Acan , N-dimensional and higher order partial differential equation for reduced differential transform method, Science Institute, Selcuk Univ., Turkey (2016)
##[2]
O. Acan, The existence and uniqueness of periodic solutions for a kind of forced rayleigh equation, Gazi Univ. J. Sci., 29 (2016), 645-650
##[3]
O. Acan, O. Firat, Y. Keskin, The use of conformable variational iteration method, conformable reduced differential transform method and conformable homotopy analysis method for solving different types of nonlinear partial differential equations, Proceedings of the 3rd International Conference on Recent Advances in Pure and Applied Mathematics, Bodrum, Turkey (2016)
##[4]
O. Acan, O. Firat, Y. Keskin, G. Oturanc, Solution of conformable fractional partial differential equations by reduced differential transform method, Selcuk J. Appl. Math., (2016), -
##[5]
O. Acan, O. Firat, Y. Keskin, G. Oturanc, Conformable variational iteration method, New Trends Math. Sci., 5 (2017), 172-178
##[6]
O. Acan, Y. Keskin , Approximate solution of Kuramoto-Sivashinsky equation using reduced differential transform method , AIP Conference Proceedings, 1648 (2015), 1-470003
##[7]
O. Acan, Y. Keskin, Reduced differential transform method for (2+1) dimensional type of the Zakharov-Kuznetsov ZK(n,n) equations, AIP Conference Proceedings, 1648 (2015), 1-370015
##[8]
O. Acan, Y. Keskin, A comparative study of numerical methods for solving (n + 1) dimensional and third-order partial differential equations, J. Comput. Theor. Nanosci., 13 (2016), 8800-8807
##[9]
O. Acan, Y. Keskin , A new technique of Laplace Pade reduced differential transform method for (1 + 3) dimensional wave equations, New Trends Math. Sci., 5 (2017), 164-171
##[10]
M. O. Al-Amr, New applications of reduced differential transform method, Alexandria Eng. J., 53 (2014), 243-247
##[11]
D. Baleanu, A. H. Bhrawy, R. A. Van Gorder, New trends on fractional and functional differential equations, Abstr. Appl. Anal., 2015 (2015), 1-2
##[12]
D. Baleanu, K. Sayevand, Performance evaluation of matched asymptotic expansions for fractional differential equations with multi-order, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 59 (2016), 3-12
##[13]
D. Baleanu, J. A. Tenreiro Machado, C. Cattani, M. C. Baleanu, X.-J. Yang, Local fractional variational iteration and decomposition methods for wave equation on Cantor sets within local fractional operators, Abstr. Appl. Anal., 2014 (2014), 1-6
##[14]
N. Baykus, M. Sezer , Solution of high-order linear Fredholm integro-differential equations with piecewise intervals, Numer. Methods Partial Differential Equations, 27 (2011), 1327-1339
##[15]
H. Beyer, S. Kempfle, Definition of physically consistent damping laws with fractional derivatives, Z. Angew. Math. Mech., 75 (1995), 623-635
##[16]
Z.-J. Cui, Z.-S. Mao, S.-J. Yang, P.-N. Yu, Approximate analytical solutions of fractional perturbed diffusion equation by reduced differential transform method and the homotopy perturbation method, Math. Probl. Eng., 2013 (2013), 1-7
##[17]
A. K. Golmankhaneh, X.-J. Yang, D. Baleanu, Einstein field equations within local fractional calculus, Rom. J. Phys., 60 (2015), 22-31
##[18]
Z.-H. Guo, O. Acan, S. Kumar , Sumudu transform series expansion method for solving the local fractional Laplace equation in fractal thermal problems, Therm. Sci., 20 (2016), 1-739
##[19]
P. K. Gupta, Approximate analytical solutions of fractional Benney-Lin equation by reduced differential transform method and the homotopy perturbation method, Comput. Math. Appl., 61 (2011), 2829-2842
##[20]
J.-H. He, A tutorial review on fractal spacetime and fractional calculus, Internat. J. Theoret. Phys., 53 (2014), 3698-3718
##[21]
M. A. E. Herzallah, D. Baleanu, On fractional order hybrid differential equations, Abstr. Appl. Anal., 2014 (2014), 1-7
##[22]
H. Jafari, H. K. Jassim, S. P. Moshokoa, V. M. Ariyan, F. Tchier, Reduced differential transform method for partial differential equations within local fractional derivative operators, Adv. Mech. Eng., 8 (2016), 1-6
##[23]
Y. Keskin , Partial differential equations by the reduced differential transform method, Science Institute, Selcuk Univ., Turkey (2010)
##[24]
Y. Keskin, G. Oturanc, Reduced differential transform method for partial differential equations, Int. J. Nonlinear Sci. Numer. Simul., 10 (2009), 741-750
##[25]
Y. Keskin, G. Oturanc , Reduced differential transform method for generalized KdV equations, Math. Comput. Appl., 15 (2010), 382-393
##[26]
Y. Keskin, G. Oturanc, The reduced differential transform method: A new approach to factional partial differential equations, Nonlinear Sci. Lett. A., 1 (2010), 207-217
##[27]
E. Kurul, N. B. Savasaneril, Solution of the two-dimensional heat equation for a rectangular plate, New Trends Math. Sci., 3 (2015), 76-82
##[28]
M. Ma, D. Baleanu, Y. Gasimov, X.-J. Yang, New results for multidimensional diffusion equations in fractal dimensional space , Rom. J. Phys., 61 (2016), 784-794
##[29]
F. Mainardi , Fractional relaxation-oscillation and fractional diffusion-wave phenomena , Chaos Solitons Fractals, 7 (1996), 1461-1477
##[30]
I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)
##[31]
A. Razminia, D. Baleanu, Fractional order models of industrial pneumatic controllers, Abstr. Appl. Anal., 2014 (2014), 1-9
##[32]
S. G. Samko, A. A. Kilbas, O. I. Marichev , Fractional integrals and derivatives, Theory and applications, Edited and with a foreword by S. M. Nikolski˘ı, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon (1993)
##[33]
A. Saravanan, N. Magesh, A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell-Whitehead-Segel equation, J. Egyptian Math. Soc., 21 (2013), 259-265
##[34]
M. Z. Sarikaya, H. Budak, Generalized Ostrowski type inequalities for local fractional integrals, Proc. Amer. Math. Soc., 145 (2017), 1527-1538
##[35]
W. R. Schneider, W. Wyss , Fractional diffusion and wave equations , J. Math. Phys., 30 (1989), 134-144
##[36]
B. K. Singh, Mahendra , A numerical computation of a system of linear and nonlinear time dependent partial differential equations using reduced differential transform method, Int. J. Differ. Equ., 2016 (2016), 1-8
##[37]
X.-J. Yang, Local fractional functional analysis and its applications, Asian Academic Publisher, Hong Kong (2011)
##[38]
X.-J. Yang, Advanced local fractional calculus and its applications, World Science, New York (2012)
##[39]
X.-J. Yang, D. Baleanu, H. M. Srivastava, Local fractional similarity solution for the diffusion equation defined on Cantor sets , Appl. Math. Lett., 47 (2015), 54-60
##[40]
X.-J. Yang, F. Gao, H. M. Srivastava, Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations, Comput. Math. Appl., 73 (2017), 203-210
##[41]
X.-J. Yang, L.-Q. Hua , Variational iteration transform method for fractional differential equations with local fractional derivative, Abstr. Appl. Anal., 2014 (2014), 1-9
##[42]
X.-J. Yang, H. M. Srivastava, C. Cattani, Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics, Rom. Rep. Phys., 67 (2015), 752-761
##[43]
X.-J. Yang, J. A. Tenreiro Machado, D. Baleanu, C. Cattani , On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, Chaos, 26 (2016), 1-5
##[44]
X.-J. Yang, J. A. Tenreiro Machado, C. Cattani, F. Gao, On a fractal LC-electric circuit modeled by local fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 47 (2017), 200-206
##[45]
X.-J. Yang, J. A. Tenreiro Machado, J. J. Nieto , A new family of the local fractional PDEs , Fund. Inform., 151 (2017), 63-75
##[46]
X.-J. Yang, J. A. Tenreiro Machado, H. M. Srivastava , A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach, Appl. Math. Comput., 274 (2016), 143-151
##[47]
Y. Zhang, C. Cattani, X.-J. Yang, Local fractional homotopy perturbation method for solving non-homogeneous heat conduction equations in fractal domains, Entropy, 17 (2015), 6753-6764
##[48]
Y. Zhang, X.-J. Yang, An efficient analytical method for solving local fractional nonlinear PDEs arising in mathematical physics, Appl. Math. Model., 40 (2016), 1793-1799
]
Sensitivity of non-autonomous discrete dynamical systems revisited
Sensitivity of non-autonomous discrete dynamical systems revisited
en
en
In this note, we construct a transitive non-autonomous discrete
system with strongly periodic density which is not sensitive.
Besides, we prove that every transitive non-autonomous discrete
system with almost periodic density is syndetically sensitive,
provided that it converges uniformly to a map, and that a product
system is multi-sensitive (resp., \(\mathcal{F}\)-sensitive) if and only
if there exists a factor system is multi-sensitive (resp., \(\mathcal{F}\)-sensitive),
where \(\mathcal{F}\) is a filterdual.
5239
5244
Xian-Feng
Ding
School of Sciences
Southwest Petroleum University
People’s Republic of China
dingxianfengswpu@163.com
Tian-Xiu
Lu
School of Mathematics and Statistics
Sichuan University of Science and Engineering
People’s Republic of China
lubeeltx@163.com
Jian-Jun
Wang
Department of Applied Mathematics
Sichuan Agricultural University
People’s Republic of China
jianjunw55@163.com
\(\mathcal{F}\)-sensitivity
Non-autonomous discrete system \(({\bf NADS})\)
sensitivity
transitivity
product system
Article.10.pdf
[
[1]
E. Akin , Recurrence in topological dynamics, Furstenberg families and Ellis actions, The University Series in Mathematics, Plenum Press, New York (1997)
##[2]
J. Banks, J. Brooks, G. Cairns, G. Davis, P. Stacey , On Devaney’s definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334
##[3]
S. N. Elaydi, Nonautonomous difference equations: open problems and conjectures, Differences and differential equations, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 42 (2004), 423-428
##[4]
S. Elaydi, R. J. Sacker, Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures, J. Difference Equ. Appl., 11 (2005), 337-346
##[5]
H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, M. B. Porter Lectures, Princeton University Press, Princeton, N.J. (1981)
##[6]
S. Kolyada, M. Misiurewicz, L. Snoha, Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval , Fund. Math., 160 (1999), 161-181
##[7]
S. Kolyada, L. Snoha , Topological entropy of nonautonomous dynamical systems , Random Comput. Dynam., 4 (1996), 205-233
##[8]
Y.-Y. Lan, Chaos in nonautonomous discrete fuzzy dynamical systems, J. Nonlinear Sci. Appl., 9 (2016), 404-412
##[9]
J. Li, P. Oprocha, X.-X. Wu, Furstenberg families, sensitivity and the space of probability measures, Nonlinearity, 30 (2017), 987-1005
##[10]
T. K. S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20 (2007), 2115-2126
##[11]
Y.-G. Wang, G. Wei, W. H. Campbell, Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems , Topology Appl., 156 (2009), 803-811
##[12]
X.-X. Wu, Chaos of transformations induced onto the space of probability measures, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1-12
##[13]
X.-X. Wu , A remark on topological sequence entropy, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1-7
##[14]
X.-X. Wu, G.-R. Chen, Sensitivity and transitivity of fuzzified dynamical systems, Inform. Sci., 396 (2017), 14-23
##[15]
X.-X. Wu, P. Oprocha, G.-R. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity, 29 (2016), 1942-1972
##[16]
X.-X. Wu, X. Wang, On the iteration properties of large deviations theorem, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1-6
##[17]
X.-X. Wu, X. Wang, Topological dynamics of Zadeh’s extension on the space of upper semi-continuous fuzzy sets, ArXiv, 2016 (2016), 1-14
##[18]
X.-X. Wu, J.-J. Wang, G.-R. Chen, F-sensitivity and multi-sensitivity of hyperspatial dynamical systems, J. Math. Anal. Appl., 429 (2015), 16-26
##[19]
X.-X. Wu, X. Wang, G.-R. Chen, On the large deviations theorem of weaker types, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1-12
##[20]
X.-X. Wu, L.-D. Wang, G.-R. Chen, Weighted backward shift operators with invariant distributionally scrambled subsets, Ann. Funct. Anal., 8 (2017), 199-210
##[21]
X.-X. Wu, P.-Y. Zhu, Chaos in a class of non-autonomous discrete systems, Appl. Math. Lett., 26 (2013), 431-436
##[22]
X.-D. Ye, R.-F. Zhang, On sensitive sets in topological dynamics, Nonlinearity, 21 (2008), 1601-1620
]
Two-step Maruyama schemes for nonlinear stochastic differential delay equations
Two-step Maruyama schemes for nonlinear stochastic differential delay equations
en
en
This work concerns the two-step Maruyama schemes for nonlinear stochastic differential delay equations (SDDEs). We first examine the strong convergence rates of the split two-step Maruyama scheme and linear two-step Maruyama scheme (including Adams-Bashforth and Adams-Moulton schemes) for nonlinear SDDEs with highly nonlinear delay variables, then we investigate the exponential mean square stability and exponential decay rates of the two classes of two-step Maruyama schemes. These results are important for three reasons: first, the convergence rates are established under the non-global Lipschitz condition; second, these stability results show that these two-step Maruyama schemes can not only reproduce the exponential mean square stability, but also preserve the bound of Lyapunov exponent for sufficient small stepsize; third, they are also suitable for the corresponding two-step Maruyama methods of stochastic ordinary differential equations (SODEs).
5245
5260
Dongxia
Lei
School of Mathematics and Statistics
Huazhong University of Science and Technology
China
dongxialei@hust.edu.cn
Xiaofeng
Zong
School of Automation
China University of Geosciences
China
xfzong87816@gmail.com
Junhao
Hu
School of Mathematics and Statistics
South-Central University for Nationalities
China
junhaohu74@163.com
Stochastic differential equations (SDEs)
two-step Maruyama schemes
strong convergence rate
exponential mean square stability
Article.11.pdf
[
[1]
Y. Ait-Sahalia, Testing continuous-time models of the spot interest rate, Rev. Financial Stud., 9 (1996), 385-426
##[2]
J.-H. Bao, C.-G. Yuan, Convergence rate of EM scheme for SDDEs, Proc. Amer. Math. Soc., 141 (2013), 3231-3243
##[3]
E. Beretta, V. Kolmanovskii, L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations, Delay systems, Lille, (1996), Math. Comput. Simulation, 45 (1998), 269-277
##[4]
A. Bryden, D. J. Higham , On the boundedness of asymptotic stability regions for the stochastic theta method, BIT, 43 (2003), 1-6
##[5]
E. Buckwar, R. Horvath-Bokor, R. Winkler, Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations, BIT, 46 (2006), 261-282
##[6]
E. Buckwar, R. Winkler, On two-step schemes for SDEs with small noise , PAMM, 4 (2004), 15-18
##[7]
E. Buckwar, R. Winkler , Multistep methods for SDEs and their application to problems with small noise, SIAM J. Numer. Anal., 44 (2006), 779-803
##[8]
E. Buckwar, R. Winkler, Multi-step Maruyama methods for stochastic delay differential equations, Stoch. Anal. Appl., 25 (2007), 933-959
##[9]
W.-R. Cao, Z.-Q. Zhang, On exponential mean-square stability of two-step Maruyama methods for stochastic delay differential equations, J. Comput. Appl. Math., 245 (2013), 182-193
##[10]
C. W. Eurich, J. G. Milton , Noise-induced transitions in human postural sway, Phys. Rev. E, 54 (1996), 6681-6684
##[11]
D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal., 38 (2000), 753-769
##[12]
D. J. Higham, X.-R. Mao, A. M. Stuart , Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), 1041-1063
##[13]
D. J. Higham, X.-R. Mao, A. M. Stuart , Exponential mean-square stability of numerical solutions to stochastic differential equations, LMS J. Comput. Math., 6 (2003), 297-313
##[14]
D. G. Hobson, L. C. G. Rogers, Complete models with stochastic volatility, Math. Finance, 8 (1998), 27-48
##[15]
C.-M. Huang, Exponential mean square stability of numerical methods for systems of stochastic differential equations, J. Comput. Appl. Math., 236 (2012), 4016-4026
##[16]
C.-M. Huang, Mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations, J. Comput. Appl. Math., 259 (2014), 77-86
##[17]
M. Hutzenthaler, A. Jentzen , Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Amer. Math. Soc., 236 (2015), 1-99
##[18]
P. E. Kloeden, E. Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), Springer-Verlag, Berlin (1992)
##[19]
M.-Z. Liu, W.-R. Cao, Z.-C. Fan, Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation, J. Comput. Appl. Math., 170 (2004), 255-268
##[20]
A. Longtin, J. G. Milton, J. E. Bos, M. C. Mackey, Noise and critical behavior of the pupil light reflex at oscillation onset, Phys. Rev. A, 41 (1990), 6992-7005
##[21]
X.-R. Mao, Stochastic differential equations and their applications, Horwood Publishing Series in Mathematics & Applications, Horwood Publishing Limited, Chichester (1997)
##[22]
G. N. Milstein, Numerical integration of stochastic differential equations, Translated and revised from the 1988 Russian original, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (1995)
##[23]
G. N. Milstein, M. V. Tretyakov , Stochastic numerics for mathematical physics, Scientific Computation, Springer- Verlag, Berlin (2004)
##[24]
Y. Saito, T. Mitsui , Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), 2254-2267
##[25]
T. Sickenberger, Mean-square convergence of stochastic multi-step methods with variable step-size , J. Comput. Appl. Math., 212 (2008), 300-319
##[26]
P. K. Tapaswi, A. Mukhopadhyay, Effects of environmental fluctuation on plankton allelopathy, J. Math. Biol., 39 (1999), 39-58
##[27]
A. Tocino, M. J. Senosiain, Asymptotic mean-square stability of two-step Maruyama schemes for stochastic differential equations, J. Comput. Appl. Math., 260 (2014), 337-348
##[28]
X.-J. Wang, S.-Q. Gan, The improved split-step backward Euler method for stochastic differential delay equations, Int. J. Comput. Math., 88 (2011), 2359-2378
##[29]
X.-J. Wang, S.-Q. Gan, D.-S. Wang , A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise, BIT, 52 (2012), 741-772
##[30]
F.-K. Wu, X.-R. Mao, K. Chen, The Cox-Ingersoll-Ross model with delay and strong convergence of its Euler-Maruyama approximate solutions, Appl. Numer. Math., 59 (2009), 2641-2658
##[31]
X.-F. Zong, F.-K. Wu , Choice of \(\theta\) and mean-square exponential stability in the stochastic theta method of stochastic differential equations, J. Comput. Appl. Math., 255 (2014), 837-847
##[32]
X.-F. Zong, F.-K. Wu, C.-M. Huang, Convergence and stability of the semi-tamed Euler scheme for stochastic differential equations with non-Lipschitz continuous coefficients, Appl. Math. Comput., 228 (2014), 240-250
##[33]
X.-F. Zong, F.-K. Wu, C.-M. Huang, Preserving exponential mean square stability and decay rates in two classes of theta approximations of stochastic differential equations, J. Difference Equ. Appl., 20 (2014), 1091-1111
##[34]
X.-F. Zong, F.-K. Wu, C.-M. Huang, Theta-Euler schemes for SDEs with non-global Lipschitz continuous coeffcients, , (Submitted), -
]
Common fixed points of generalized rational contractions on a closed ball in partial metric spaces
Common fixed points of generalized rational contractions on a closed ball in partial metric spaces
en
en
The notion of generalized contractions of rational type on a closed ball is introduced and used to establish some common fixed point theorems for two, three and four mappings in complete ordered partial metric spaces. These results improve several well-known, primary and conventional results. We give an example to illustrate the main idea of our results that there are mappings which have only fixed points inside or on the closed ball instead of whole space.
5261
5270
Muhammad
Nazam
Department of Mathematics
International Islamic University
Pakistan
nazim.phdma47@iiu.edu.pk
Muhammad
Arshad
Department of Mathematics and Statistics
International Islamic University
Pakistan
marshadzia@iiu.edu.pk
Choonkil
Park
Research Institute for Natural Sciences
Hanyang University
Republic of Korea
baak@hanyang.ac.kr
Sungsik
Yun
Department of Financial Mathematics
Hanshin University
Republic of Korea
ssyun@hs.ac.kr
Common fixed point
closed ball
generalized contraction
partial metric space
Article.12.pdf
[
[1]
T. Abdeljawad, Meir-Keeler \(\alpha\)-contractive fixed and common fixed point theorems , Fixed Point Theory Appl., 2013 (2013), 1-10
##[2]
T. Abdeljawad, E. Karapınar, K. Taş , Existence and uniqueness of a common fixed point on partial metric spaces, Appl. Math. Lett., 24 (2011), 1900-1904
##[3]
A . Almeida, A. F. Roldán-López-de-Hierro, K. Sadarangani , On a fixed point theorem and its application in dynamic programming, Appl. Anal. Discrete Math., 9 (2015), 221-244
##[4]
I. Altun, A. Erduran, Fixed point theorems for monotone mappings on partial metric spaces, Fixed Point Theory Appl., 2011 (2011), 1-10
##[5]
I. Altun, S. Romaguera, Characterizations of partial metric completeness in terms of weakly contractive mappings having fixed point, Appl. Anal. Discrete Math., 6 (2012), 247-256
##[6]
I. Altun, F. Sola, H. Simsek, Generalized contractions on partial metric spaces, Topology Appl., 157 (2010), 2778-2785
##[7]
M. Arshad, A. Azam, P. Vetro, Some common fixed point results in cone metric spaces, Fixed Point Theory Appl., 2009 (2009), 1-11
##[8]
M. Bukatin, R. Kopperman, S. Matthews, H. Pajoohesh, Partial metric spaces , Amer. Math. Monthly, 116 (2009), 708-718
##[9]
M. A. Bukatin, S. Y. Shorina, Partial metrics and co-continuous valuations, Foundations of software science and computation structures, Lisbon, (1998), Lecture Notes in Comput. Sci., Springer, Berlin, 1378 (1998), 125-139
##[10]
B. K. Dass, S. Gupta, An extension of Banach contraction principle through rational expression, Indian J. Pure Appl. Math., 6 (1975), 1455-1458
##[11]
´I. M. Erhan, E. Karapinar, D. Türkoğlu, Different types Meir-Keeler contractions on partial metric spaces, J. Comput. Anal. Appl., 14 (2012), 1000-1005
##[12]
T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379-1393
##[13]
R. H. Haghi, S. Rezapour, N. Shahzad , Some fixed point generalizations are not real generalizations, Nonlinear Anal., 74 (2011), 1799-1803
##[14]
R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71-76
##[15]
K. N. Leibovic, The principle of contraction mapping in nonlinear and adaptive control systems, IEEE Trans. Automatic Control, 9 (1964), 393-398
##[16]
S. G. Matthews, Partial metric topology, Papers on general topology and applications, Flushing, NY, (1992), Ann. New York Acad. Sci., New York Acad. Sci., New York, 728 (1994), 183-197
##[17]
H. K. Nashine, E. Karapinar , Fixed point results in orbitally complete partial metric spaces, Bull. Malays. Math. Sci. Soc., 36 (2013), 1185-1193
]
Second-order differential superordination for analytic functions in the upper half-plane
Second-order differential superordination for analytic functions in the upper half-plane
en
en
Let \(\Omega\) be a set in the complex plane \(\mathbb{C}\). Also let \(p(z)\) be analytic in the upper half-plane \(\Delta=\{z:z\in\mathbb{C}\ \text{and}\ \Im(z)>0\}\) and suppose that \(\psi: \mathbb{C}^3\times\Delta\rightarrow\mathbb{C}\).
In this paper, we investigate the problem of determining properties of functions \(p(z)\) that satisfy the following second-order differential superordination:
\[\Omega\subset\left\{\psi\left(p(z),p'(z),p''(z);z\right): z\in\Delta\right\}.\]
Applications of these results to second-order differential superordination for analytic functions in \(\Delta\) are also presented.
5271
5280
Huo
Tang
School of Mathematics and Statistics
Chifeng University
People's Republic of China
thth2009@163.com
H. M.
Srivastava
Department of Mathematics and Statistics
Department of Medical Research
University of Victoria
China Medical University Hospital, China Medical University
Canada
Republic of China
harimsri@math.uvic.ca
Guan-Tie
Deng
School of Mathematical Sciences
Beijing Normal University
People's Republic of China
denggt@bnu.edu.cn
Shu-Hai
Li
School of Mathematics and Statistics
Chifeng University
People's Republic of China
lishms66@sina.com
Differential subordination
differential superordination
analytic functions
admissible functions
upper half-plane
Article.13.pdf
[
[1]
I. A. Aleksander, V. V. Sobolev, Extremal problems for some classes of univalent functions in the half-plane, Ukr. Math. J., 22 (1970), 291-307
##[2]
R. M. Ali, V. Ravichandran, N. Seenivasagan, Subordination and superordination of the Liu-Srivastava linear operator on meromorphic functions, Bull. Malays. Math. Sci. Soc., 31 (2008), 193-207
##[3]
R. M. Ali, V. Ravichandran, N. Seenivasagan, Subordination and superordination on Schwarzian derivatives, J. Inequal. Appl., 2008 (2008), 1-18
##[4]
R. M. Ali, V. Ravichandran, N. Seenivasagan, Differential subordination and superordination of analytic functions defined by the multiplier transformation, Math. Inequal. Appl., 12 (2009), 123-139
##[5]
R. M. Ali, V. Ravichandran, N. Seenivasagan, Differential subordination and superordination of analytic functions defined by the Dziok-Srivastava operator, J. Franklin Inst., 347 (2010), 1762-1781
##[6]
R. M. Ali, V. Ravichandran, N. Seenivasagan , On subordination and superordination of the multiplier transformation for meromorphic functions, Bull. Malays. Math. Sci. Soc., 33 (2010), 311-324
##[7]
M. K. Aouf, T. M. Seoudy, Subordination and superordination of a certain integral operator on meromorphic functions, Comput. Math. Appl., 59 (2010), 3669-3678
##[8]
N. E. Cho, O. S. Kwon, S. Owa, H. M. Srivastava, A class of integral operators preserving subordination and superordination for meromorphic functions, Appl. Math. Comput., 193 (2007), 463-474
##[9]
N. E. Cho, H. M. Srivastava, A class of nonlinear integral operators preserving subordination and superordination, Integral Transforms Spec. Funct., 18 (2007), 95-107
##[10]
G. Dimkov, J. Stankiewicz, Z. Stankiewicz, On a class of starlike functions defined in a half-plane, Annales. Polonici. Math., 55 (1991), 81-86
##[11]
S. S. Miller, P. T. Mocanu , Differential Subordinations: Theory and Applications, CRC Press, New York (2000)
##[12]
S. S. Miller, P. T. Mocanu , Subordinations of differential superordinations, Complex Variables Theory Appl., 48 (2003), 815-826
##[13]
V. G. Moskvin, T. N. Selakova, V. V. Sobolev, Extremal properties of some classes of conformal self-mapping of the half plane with fixed coefficients, Sibirsk. Mat. Zh., 21 (1980), 139-154
##[14]
D. Raducanu, N. N. Pascu , Differential subordinations for holomorphic functions in the upper half-plane, Mathematica (Cluj), 36 (1994), 215-217
##[15]
T. N. Shanmugam, S. Sivasubramanian, H. Srivastava, Differential sandwich theorems for certain subclasses of analytic functions involving multiplier transformations, Integral Transforms Spec. Funct., 17 (2006), 889-899
##[16]
H. M. Srivastava, D.-G. Yang, N.-E. Xu, Subordinations for multivalent analytic functions associated with the Dziok- Srivastava operator, Integral Transforms Spec. Funct., 20 (2009), 581-606
##[17]
J. Stankiewicz, Geometric properties of functons regular in a half-plane, World Sci. Publ., New Jersey (1992)
##[18]
J. Stankiewicz, Z. Stankiewicz, On the classes of functions regular in a half-plane, Folia Sci. Univ. Tech. Resoviensis Math., 60 (1989), 111-123
##[19]
J. Stankiewicz, Z. Stankiewicz, On the classes of functions regular in a half-plane, Bull. Polish Acad. Sci. Math., 39 (1991), 49-56
##[20]
H. Tang, M. K. Aouf, G.-T. Deng, S.-H. Li, Differential subordination results for analytic functions in the upper half-plane, Abstr. Appl. Anal., 2014 (2014), 1-6
##[21]
H. Tang, E. Deniz, Third-order differential subordination results for analytic functions involving the generalized Bessel functions, Acta Math. Sci. Ser. B Engl. Ed., 34 (2014), 1707-1719
##[22]
H. Tang, H. M. Srivastava, G.-T. Deng, Some families of analytic functions in the upper half-plane and their associated differential subordination and differential superordination properties and problems, Appl. Math. Inf. Sci., 11 (2017), 1247-1257
##[23]
H. Tang, H. M. Srivastava, E. Deniz, S.-H. Li, Third-order differential superordination involving the generalized Bessel functions, Bull. Malays. Math. Sci. Soc., 38 (2015), 1669-1688
##[24]
H. Tang, H. M. Srivastava, S.-H. Li, L.-N. Ma , Third-order differential subordination and superordination results for meromorphically multivalent functions associated with the Liu-Srivastava operator, Abstr. Appl. Anal., 2014 (2014), 1-11
]
Positive solution for a coupled system of nonlinear fractional differential equations with fractional integral conditions
Positive solution for a coupled system of nonlinear fractional differential equations with fractional integral conditions
en
en
By studying the properties of Green's function, constructing a
special cone and applying fixed point theorem of cone expansion and
compression of norm type, this paper investigates the existence of
at least one and two positive solutions for a coupled system of
nonlinear fractional differential equations involving fractional
integral conditions and derivatives of arbitrary order. Two examples
are given to illustrate our results.
5281
5291
Yaohong
Li
Center of Statistical Survey and Advising
School of Mathematics and Statistics
Suzhou University
Suzhou University
P. R. China
P. R. China
liz.zhanghy@163.com
Haiyan
Zhang
School of Mathematics and Statistics
Suzhou University
P. R. China
liz.zhang@yeah.net
Positive solution
fractional differential equations
fractional integral conditions
fixed point theorem
Article.14.pdf
[
[1]
B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl., 58 (2009), 1838-1843
##[2]
B. Ahmad, S. K. Ntouyas, A. Alsaedi , On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions , Chaos Solitons Fractals, 83 (2016), 234-241
##[3]
A. K. Alomari, F. Awawdeh, N. Tahat, F. Bani Ahmad, W. Shatanawi, Multiple solutions for fractional differential equations: Analytic approach, Appl. Math. Comput., 219 (2013), 8893-8903
##[4]
P. Arena, R. Caponetto, L. Fortuna, D. Porto, Chaos in a fractional order Duffing system, In: Proceedings of the 1997 European conference on circuit theory and design ,Technical University of Budapest, Budapest (1997)
##[5]
C.-Z. Bai, J.-X. Fang, The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations , Appl. Math. Comput., 150 (2004), 611-621
##[6]
Y. Chen, D. Chen, Z. Lv, The existence results for a coupled system of nonlinear fractional differential equations with multi-point boundary conditions, Bull. Iranian Math. Soc., 38 (2012), 607-624
##[7]
K. Deimling , Nonlinear Functional Analysis, Springer-Verlag, Berlin (1985)
##[8]
K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3-22
##[9]
M. Faieghi, S. Kuntanapreeda, H. Delavari, D. Baleanu , LMI-based stabi-lization of a class of fractional-order chaotic systems, Nonlinear Dynam., 72 (2013), 301-309
##[10]
Z.-M. Ge, W.-R. Jhuang, Chaos, control and synchronization of a fractional order rotational mechanical system with a centrifugal governor, Chaos Solitons. Fractals., 33 (2007), 270-289
##[11]
I. Grigorenko, E. Grigorenko, Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett., 91 (2003), 1-4
##[12]
D. J. Guo, V. Lakshmikanthan, Nonlinear Problems in Abstract Cones, Academic Press, Boston (1988)
##[13]
Z. Jiao, Y.-Q. Chen, I. Podlubny , Distributed-order dynamic systems, Springer, London (2012)
##[14]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo , Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006)
##[15]
Y. Li, Z. Wei , Positive solutions for a coupled systems of mixed higher-order nonlinear singular differential equations, Fixed Point Theory, 15 (2014), 167-178
##[16]
K. Ma, Z. Han, Y. Zhang , Stability conditions of a coupled system of fractional q-difference Lotka-Volterra model, Int. J. Dyn. Syst. Differ. Equ., 6 (2016), 305-317
##[17]
R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion:a fractional dynamics approach, Phys. Rep., 2000 (2000), 1-77
##[18]
S. K. Ntouyas, M. Obaid, A coupled system of fractional differential equations with nonlocal integral boundary conditions, Adv. Difference Equ., 2012 (2012), 1-8
##[19]
A. Pedas, E. Tamme , Numerical solution of nonlinear fractional differential equations by spline collocation methods, J. Comput. Appl. Math., 255 (2014), 216-230
##[20]
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999)
##[21]
I. M. Sokolov, J. Klafter, A. Blumen, Fractional kinetics , Phys. Today, 55 (2002), 48-54
##[22]
X.-W. Su , Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 22 (2009), 64-69
##[23]
C. Yuan , Multiple positive solutions for (n-1,1) type semipositone conjugate boundary value problems for coupled systems of nonlinear fractional differential equations , Electron. J. Qual. Theory Differ. Equ., 2011 (2011), 1-12
##[24]
Y. Zhang, Z. Bai, T. Feng , Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance, Comput. Math. Appl., 61 (2011), 1032-1047
##[25]
H. Zhang, W. Gao , Existence of Solutions to a Coupled System of Higher-order Nonlinear Fractional Differential Equations with Anti-periodic Boundary Conditions, J. Comput. Anal. Appl., 22 (2017), 262-270
##[26]
X. Zhang, C. Zhu, Z. Wu, Solvability for a coupled system of fractional differential equations with impulses at resonance, Bound. Value Probl., 2013 (2013), 1-23
##[27]
Y. Zhao, S. Sun, Z. Han, W. Feng, Positive solutions for for a coupled system of nonlinear differential equations of mixed fractional orders, Adv. Difference Equ., 2011 (2011), 1-13
##[28]
X. Zhou, C. Xu, Numerical Solution of the Coupled System of Nonlinear Fractional Ordinary Differential Equations, Adv. Appl. Math. Mech., 9 (2017), 574-595
##[29]
C. Zhu, X. Zhang, Z. Wu , Solvability for a coupled system of fractional differential equations with integral boundary conditions, Taiwanese J. Math., 17 (2013), 2039-2054
]
Uniform convexity in \(\ell_{p(\cdot)}\)
Uniform convexity in \(\ell_{p(\cdot)}\)
en
en
In this work, we investigate the variable exponent sequence space \(\ell_{p(\cdot)}\). In particular, we prove a geometric property similar to uniform convexity without the assumption \(\limsup_{n \to \infty} p(n) < \infty\). This property allows us to prove the analogue to Kirk's fixed point theorem in the modular vector space \(\ell_{p(\cdot)}\) under Nakano's formulation.
5292
5299
Mostafa
Bachar
Department of Mathematics, College of Sciences
King Saud University
Saudi Arabia
mbachar@ksu.edu.sa
Messaoud
Bounkhel
Department of Mathematics, College of Sciences
King Saud University
Saudi Arabia
bounkhel@ksu.edu.sa
Mohamed A.
Khamsi
Department of Mathematics & Statistics
Department of Mathematical Sciences
King Fahd University of Petroleum and Minerals
University of Texas at El Paso
Saudi Arabia
USA
mohamed@utep.edu
Fixed point
modular vector spaces
nonexpansive mapping
uniformly convex
variable exponent spaces
Article.15.pdf
[
[1]
B. Beauzamy , Introduction to Banach Spaces and Their Geometry, North-Holland, Amsterdam (1985)
##[2]
J. A. Clarkson , Uniformly Convex Spaces, Trans. Amer. Math. Soc., 40 (1936), 396-414
##[3]
L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Berlin (2011)
##[4]
M. A. Khamsi, W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, Wiley-Interscience, New York (2011)
##[5]
M. A. Khamsi, W. M. Kozlowski , Fixed Point Theory in Modular Function Spaces, Birkhauser, New York (2015)
##[6]
M. A. Khamsi, W. K. Kozlowski, S. Reich, Fixed point theory in modular function spaces, Nonlinear Anal., 14 (1990), 935-953
##[7]
W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004-1006
##[8]
V. Klee, Summability in \(\ell(p_{11}, p_{21}, ...)\) Spaces, Studia Math., 25 (1965), 277-280
##[9]
O. Kováčik, J. Rákosník, On spaces \(L^{p(x)}\) and \(W^{1,p(x)}\) , Czechoslovak Math. J., 41 (1991), 592-618
##[10]
W. M. Kozlowski, Modular Function Spaces, Marcel Dekker, New York (1988)
##[11]
J. Musielak , Orlicz spaces and modular spaces, Springer-Verlag, Berlin (1983)
##[12]
H. Nakano, Modulared Semi-ordered Linear Spaces, Maruzen Co., Tokyo (1950)
##[13]
H. Nakano , Modulared sequence spaces, Proc. Japan Acad., 27 (1951), 508-512
##[14]
H. Nakano , Topology of linear topological spaces, Maruzen Co. Ltd., Tokyo ( 1951)
##[15]
W. Orlicz, Über konjugierte Exponentenfolgen, Studia Math., 3 (1931), 200-211
##[16]
K. Rajagopal, M. Růžička , On the modeling of electrorheological materials, Mech. Research Comm., 23 (1996), 401-407
##[17]
M. Růžička, Electrorheological fluids: modeling and mathematical theory, Springer-Verlag, Berlin (2000)
##[18]
K. Sundaresan, Uniform convexity of Banach spaces \(\ell(\{p_i\})\), Studia Math., 39 (1971), 227-231
##[19]
D. Waterman, T. Ito, F. Barber, J. Ratti, Reflexivity and Summability: The Nakano \(\ell(p_i)\) spaces, Studia Math., 33 (1969), 141-146
]
Caustics of translation surfaces in Euclidean \(3\)-space
Caustics of translation surfaces in Euclidean \(3\)-space
en
en
The aim of this paper is to classify the singularities of caustics, which have implications for a wide range of physical applications, of translation
surfaces. In addition, we give a particular study on ridge point,
sub-parabolic ridge point, and constant curvature line on
translation surface and we find that there is no elliptic umblic
on translation surface.
5300
5310
Jingren
Chen
School of Mathematical Sciences
Harbin Normal University
P. R. China
chenjingren@ruc.edu.cn
Haiming
Liu
School of Mathematical Sciences
Mudanjiang Normal University
P. R. China
liuhm468@nenu.edu.cn
Jiajing
Miao
School of Mathematical Sciences
Mudanjiang Normal University
P. R. China
jiajing0407@126.com
Translation surface
caustics
singularity theory
Article.16.pdf
[
[1]
J. A. Adam, The mathematical physics of rainbows and glories , Physics Reports, 356 (2002), 229-365
##[2]
V. I. Arnold, Symplectic geometry and topology, J. Math. Phys., 41 (2000), 3307-3343
##[3]
V. I. Arnol’d, S. M. Gueseın-zade, A. N. Varchenko, Singularities of Differentiable Maps, Birkhäuser, Boston (1986)
##[4]
M. Avendaño-Alejo, L. Castañeda, I. Moreno , Caustics and wavefronts by multiple reflections in a circular surface, Am. J. Physics., 78 (2010), 1195-1198
##[5]
M. Avendaño-Alejo, D. González-Utrera, L. Castañeda, Caustics in a meridional plane produced by plano-convex conic lenses, J. Opt. Soc. Am. A, 28 (2011), 2619-2628
##[6]
T. Banchoff, T. Gaffney, C. McCrory, Cusps of Gauss mappings, Pitman, Boston (1982)
##[7]
M. Bekkar, B. Senoussi, Translation surfaces in the 3-dimensional space satisfying \(\Delta^{III}r_i = \mu_ir_i\), J. Geom., 103 (2012), 367-374
##[8]
M. V. Berry , Disruption of images; the caustic touching theorem, J. Opt. Soc. Am. A, 4 (1987), 561-569
##[9]
J. Ehlers, E. T. Newman, The theory of caustics and wave front singularities with physical applications, J. Math. Phys., 41 (2000), 3344-3378
##[10]
T. Fukui, M. Hasegawa, Singularities of parallel surfaces , Tohoku Math. J., 64 (2012), 387-408
##[11]
W. Goemans, I. V. Woestyne, Translation Surfaces with Vanishing Second Gaussian Curvature in Euclidean and Minkowski 3-Space, Shaker Verlag, Aachen (2007)
##[12]
W. Hasse, M. Kriele, V. Perlick, Caustics of wavefronts in general relativity , Classical Quantum Gravity, 13 (1996), 1161-1182
##[13]
S. Izumiya , Differential Geometry from the viewpoint of Lagrangian or Legendrian singularity theory , World Scientific publishing Co., Singapore (2007)
##[14]
L. Kong, D. Pei , On spacelike curves in hyperbolic space times sphere, Int. J. Geom. Methods Mod. Phys., 2014 (2014), 1-12
##[15]
H. Liu , Translation surfaces with dependent Gauss and mean curvature in 3-dimensional space, J. Neut., 14 (1993), 88-93
##[16]
H. Liu, Translation surfaces with constant mean curvature in Euclidean 3-space , J. Geom., 64 (1999), 141-149
##[17]
H. Liu, Y. Yu , Affine translation surfaces in 3-dimensional spaces, Proc. Japan Acad. Ser. A Math. Sci., 89 (2013), 111-113
##[18]
R. López, M. I. Munteanu, Minimal translation surfaces in Sol3, J. Math. Soc. Japan, 64 (2012), 985-1003
##[19]
I. R. Porteous, Geometric Differentiation for the Intelligence of Curves and Surfaces, Cambridge University Press, Cambridge (1994)
##[20]
L. Verstraelen, J. Walrave, S. Yaprak, The minimal translation surfaces in Euclidean space, Soochow J. Math., 20 (1994), 77-82
##[21]
J.Weiss, Bäcklund transformations, focal surfaces and the two-dimensional Toda lattice, Phys. Lett. A, 138 (1989), 365-368
##[22]
D. W. Yoon, On the Gauss map of translation surfaces in Minkowski 3-space, Taiwanese J. Math., 6 (2002), 389-398
##[23]
Y. Yuan, H. L. Liu , Some new translation surfaces in 3-Minkowski space, J. Math. Res. Exposition, 31 (2011), 1123-1128
]
Application of penalty methods to generalized variational inequalities in Banach spaces
Application of penalty methods to generalized variational inequalities in Banach spaces
en
en
In this paper, we consider a class of generalized variational
inequalities (GVI) in infinite dimensional Banach spaces, in which
only approximation sequences for GVI are known instead of exact
values of the cost mapping and feasible set. A sequence of inexact
solutions of auxiliary problems involving general penalty method is
introduced. We obtain some convergence properties of the perturbed
version of the regularized penalty method under mild coercive
conditions, which extend some well-known results of variational
inequalities in many respects.
5311
5320
G. W.
Su
Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce Intelligent Information Processing, College of Information and Statistics
Guangxi University of Finance and Economics
P. R. China
suguangwang@163.com
Z. W.
Zhao
Guangxi Key Laboratory of Universities Optimization Control and Engineering Calculation, and College of Sciences
Guangxi University for Nationalities
P. R. China
zhengweizhao100@126.com
Generalized variational inequality
penalty method
regularization
coercivity conditions
equilibrium problem
Article.17.pdf
[
[1]
H. Attouch , Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA (1984)
##[2]
J. P. Aubin , Optima and equilibria, An introduction to nonlinear analysis, Translated from the French by Stephen Wilson, Second edition, Graduate Texts in Mathematics, Springer-Verlag, Berlin (1998)
##[3]
C. Baiocchi, A. Capelo, Variational and quasivariational inequalities, Applications to free boundary problems, Translated from the Italian by Lakshmi Jayakar, New York (1984)
##[4]
S.-S. Chang, Variational inequalities and related problems, (Chinese) Chongqing Publishing Group, Chongqing (2007)
##[5]
Z. Denkowski, S. Migórski, N. S. Papageorgiou, An introduction to nonlinear analysis: applications, Kluwer Academic Publishers, Boston, MA (2003)
##[6]
Z. Denkowski, S. Migórski, N. S. Papageorgiou, An introduction to nonlinear analysis: theory, Kluwer Academic Publishers, Boston, MA, (2003), -
##[7]
I. Ekeland, R. Témam , Convex analysis and variational problems, Translated from the French, Corrected reprint of the 1976 English edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1999)
##[8]
K. Fan, A minimax inequality and applications, Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin), Academic Press, New York, (1972), 103-113
##[9]
L. Gasiński, Z.-H. Liu, S. Migórski, A. Ochal, Z.-J. Peng, Hemivariational inequality approach to evolutionary constrained problems on star-shaped sets , J. Optim. Theory Appl., 164 (2015), 514-533
##[10]
F. Giannessi, A. Maugeri (Ed.) , Variational inequalities and network equilibrium problems, Proceedings of the conference held in Erice, June 19–25, (1994), Plenum Press, New York (1995)
##[11]
W.-M. Han, M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, AMS/IP Studies in Advanced Mathematics, American Mathematical Society, Providence, RI; International Press, Somerville, MA (2002)
##[12]
I. V. Konnov, Regularization method for nonmonotone equilibrium problems, J. Nonlinear. Convex. Anal., 10 (2009), 93-101
##[13]
I. V. Konnov , Application of penalty methods to non-stationary variational inequalities, Nonlinear Anal., 92 (2013), 177-182
##[14]
I. V. Konnov, An inexact penalty method for non stationary generalized variational inequalities, Set-Valued Var. Anal., 23 (2015), 239-248
##[15]
I. V. Konnov , Regularized penalty method for general equilibrium problems in Banach spaces, J. Optim. Theory Appl., 164 (2015), 500-513
##[16]
I. V. Konnov, D. A. Dyabilkin, Nonmonotone equilibrium problems: coercivity conditions and weak regularization, J. Global Optim., 49 (2011), 575-587
##[17]
Z.-H. Liu , Browder-Tikhonov regularization of non-coercive evolution hemivariational inequalities, Inverse Problems, 21 (2005), 13-20
##[18]
Z.-H. Liu , Existence results for quasilinear parabolic hemivariational inequalities, J. Differential Equations, 244 (2008), 1395-1409
##[19]
Z.-H. Liu, Anti-periodic solutions to nonlinear evolution equations, J. Funct. Anal., 258 (2010), 2026-2033
##[20]
Z.-H. Liu, X.-W. Li, Approximate controllability for a class of hemivariational inequalities , Nonlinear Anal. Real World Appl., 22 (2015), 581-591
##[21]
Z.-H. Liu, X.-W. Li, Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives, SIAM. J. Control Optim., 53 (2015), 1920-1933
##[22]
Z.-H. Liu, X.-W. Li, D. Motreanu , Approximate controllability for nonlinear evolution hemivariational inequalities in Hilbert spaces, SIAM J. Control Optim., 53 (2015), 3228-3244
##[23]
Z.-H. Liu, S. Migórski, S.-D. Zeng, Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces, J. Differential Equations, 263 (2017), 3989-4006
##[24]
Z.-H. Liu, S.-D. Zeng, D. Motreanu, Evolutionary problems driven by variational inequalities, J. Differential Equations, 260 (2016), 6787-6799
##[25]
Z.-J. Peng, Z.-H. Liu, X.-Y. Liu, Boundary hemivariational inequality problems with doubly nonlinear operators, Math. Ann., 356 (2013), 1339-1358
##[26]
M. Sofonea, Y.-B. Xiao , Fully history-dependent quasivariational inequalities in contact mechanics, Appl. Anal., 95 (2016), 2464-2484
##[27]
K.-K. Tan, J. Yu, X.-Z. Yuan, The stability of Ky Fan’s points , Proc. Amer. Math. Soc., 123 (1995), 1511-1519
##[28]
Y.-B. Xiao, N.-J. Huang, Y. J. Cho, A class of generalized evolution variational inequalities in Banach spaces, Appl. Math. Lett., 25 (2012), 914-920
##[29]
Y.-B. Xiao, N.-J. Huang, J. Lu, A system of time-dependent hemivariational inequalities with Volterra integral terms, J. Optim. Theory Appl., 165 (2015), 837-853
##[30]
Y.-B. Xiao, X.-M. Yang, N.-J. Huang, Some equivalence results for well-posedness of hemivariational inequalities, J. Global Optim., 61 (2015), 789-802
##[31]
E. Zeidler, Nonlinear functional analysis and its applications, II/A,B, Linear monotone operators, Translated from the German by the author and Leo F. Boron, Springer-Verlag, New York (1990)
##[32]
S.-D. Zeng, S. Migórski , Noncoercive hyperbolic variational inequalities with applications to contact mechanics, J. Math. Anal. Appl., 455 (2017), 619-637
]
On monotone multivalued transformations
On monotone multivalued transformations
en
en
In this work, we discuss the recently introduced monotone \(\tau\)-Opial condition in Banach spaces which admit a sequence of monotone approximations of the identity. Then we give a fixed point theorem for monotone multivalued nonexpansive mappings in Banach spaces satisfying the monotone \(\tau\)-Opial condition. This result generalizes those of Markin, Browder and Lami Dozo to monotone mappings.
5321
5327
Buthinah A.
Bin Dehaish
Department of Mathematics, Faculty of Science
King Abdulaziz University
Saudi Arabia
bbindehaish@yahoo.com;bbendehaish@kau.edu.sa
Mohamed A.
Khamsi
Department of Mathematics \(\&\) Statistics
Department of Mathematical Sciences
King Fahd University of Petroleum and Minerals
University of Texas at El Paso
Saudi Arabia
U.S.A
mohamed@utep.edu;mkhamsi@kfupm.edu.sa
Fixed point
systems of projections
monotone Opial condition
monotone nonexpansive mappings
multivalued mappings
Article.18.pdf
[
[1]
M. R. Alfuraidan, M. A. Khamsi, A fixed point theorem for monotone asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., (to appear), -
##[2]
M. Bachar, M. A. Khamsi, Recent contributions to fixed point theory of monotone mappings, J. Fixed Point Theory Appl., 19 (2017), 1953-1976
##[3]
S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133-181
##[4]
T. D. Benavides, P. L. Ramírez , Fixed-point theorems for multivalued non-expansive mappings without uniform convexity, Abstr. Appl. Anal., 2003 (2003), 375-386
##[5]
F. E. Browder , Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968), Amer. Math. Soc., Providence, R. I., (1976), 1-308
##[6]
T. Domínguez Benavides, J. García Falsett, M. A. Japón Pineda, The \(\tau\)-fixed point property for nonexpansive mappings, Abstr. Appl. Anal., 3 (1998), 343-362
##[7]
R. M. Dudley , On sequential convergence, Trans. Amer. Math. Soc., 112 (1964), 483-507
##[8]
K. Goebel, W. A. Kirk , Iteration processes for nonexpansive mappings, Topological methods in nonlinear functional analysis, Toronto, Ont., (1982), Contemp. Math., Amer. Math. Soc.,Providence, RI, 21 (1983), 115-123
##[9]
K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, 28. Cambridge University Press, Cambridge (1990)
##[10]
S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc., 59 (1976), 65-71
##[11]
M. A. Khamsi, On uniform Opial condition and uniform Kadec-Klee property in Banach and metric spaces, Nonlinear Anal., 10 (1996), 1733-1748
##[12]
M. A. Khamsi, W. A. Kirk, An introduction to metric spaces and fixed point theory, Pure and Applied Mathematics (New York), Wiley-Interscience, New York (2001)
##[13]
M. A. Krasnosel’skiı, Two remarks on the method of successive approximations, (Russian) Uspehi Mat. Nauk (N.S.), 10 (1955), 123-127
##[14]
E. Lami Dozo, Multivalued nonexpansive mappings and Opial’s condition, Proc. Amer. Math. Soc., 38 (1973), 286-292
##[15]
T.C. Lim, Asymptotic centers and nonexpansive mappings in conjugate Banach spaces, Pacific J. Math., 90 (1980), 135-143
##[16]
J. T. Markin , A fixed point theorem for set valued mappings, Bull. Amer. Math. Soc., 74 (1968), 639-640
##[17]
Z. Opial , Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597
##[18]
A. C. M. Ran, M. C. B. Reurings , A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2004), 1435-1443
]
Existence of solutions for fractional differential equations with integral boundary conditions at resonance
Existence of solutions for fractional differential equations with integral boundary conditions at resonance
en
en
This paper investigates the existence of solutions for Riemann-Stieltjes integral boundary value problems of fractional differential equation by using Mawhin's coincidence degree theory. An example is given to show the application of our result.
5328
5341
Wei
Zhang
Department of Mathematics
China University of Mining and Technology
P. R. China
zhangwei_azyw@163.com
Wenbin
Liu
Department of Mathematics
China University of Mining and Technology
P. R. China
cumt_equations@126.com
Riemann-Stieltjes integral
fractional differential equations
resonance
coincidence degree
Article.19.pdf
[
[1]
B. Ahmad, S. K. Ntouyas, Existence results for higher-order fractional differential inclusions with Riemann-Stieltjes type integral boundary conditions, Commun. Appl. Anal., 17 (2013), 87-98
##[2]
C.-Z. Bai , Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative , J. Math. Anal. Appl., 384 (2011), 211-231
##[3]
Z.-B. Bai, H.-S. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311 (2005), 495-505
##[4]
A. Cabada, G.-T. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl., 389 (2012), 403-411
##[5]
Y.-J. Cui, Solvability of second-order boundary-value problems at resonance involving integral conditions, Electron. J. Differential Equations, 2012 (2012), 1-9
##[6]
H. A. A. El-Saka , The fractional-order SIS epidemic model with variable population size, J. Egyptian Math. Soc., 22 (2014), 50-54
##[7]
W.-H. Jiang , The existence of solutions to boundary value problems of fractional differential equations at resonance, Nonlinear Anal., 74 (2011), 1987-1994
##[8]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006)
##[9]
W.-W. Liu, J.-Q. Jiang, L.-S. Liu, Y.-H. Wu, Nontrivial solutions of singular Sturm-Liouville problem with boundary conditions involving Riemann-Stieltjes integrals, Nonlinear Funct. Anal. Appl., 17 (2012), 255-271
##[10]
R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl., 59 (2010), 1586-1593
##[11]
J. Mawhin, Topological degree methods in nonlinear boundary value problems, Expository lectures from the CBMS Regional Conference held at Harvey Mudd College, Claremont, Calif., June 9–15, (1977), CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, R.I. (1979)
##[12]
J. Mawhin , Topological degree and boundary value problems for nonlinear differential equations, Topological methods for ordinary differential equations (Montecatini Terme, 1991), Lecture Notes in Math., Springer, Berlin, 1537 (1993), 74-142
##[13]
K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1993)
##[14]
I. Podlubny, Fractional differential equationss, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)
##[15]
J. Sabatier (Ed.), O. P. Agrawal (Ed.), J. A. Tenreiro Machado (Ed.), Advances in fractional calculus , Theoretical developments and applications in physics and engineering, Including papers from the Minisymposium on Fractional Derivatives and their Applications (ENOC-2005) held in Eindhoven, August 2005, and the 2nd Symposium on Fractional Derivatives and their Applications (ASME-DETC 2005) held in Long Beach, CA, September (2005), Springer, Dordrecht (2007)
##[16]
K. Szymańska-Dębowska, k-dimensional nonlocal boundary-value problems at resonance, Electron. J. Differential Equations, 2015 (2015), 1-8
##[17]
Y. Wang, L.-S. Liu, X.-G. Zhang, Y.-H. Wu, Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection, Appl. Math. Comput., 258 (2015), 312-324
##[18]
X.-G. Zhang, Y.-F. Han, Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations, Appl. Math. Lett., 25 (2012), 555-560
##[19]
X.-G. Zhang, L.-S. Liu, Y.-H. Wu, The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium, Appl. Math. Lett., 37 (2014), 26-33
]
Nonlinear contractions and fixed point theorems with modified $\omega$-distance mappings in complete quasi metric spaces
Nonlinear contractions and fixed point theorems with modified $\omega$-distance mappings in complete quasi metric spaces
en
en
Alegre and Marin [C. Alegre, J. Marin, Topol. Appl., \({\bf 203}\) (2016), 32--41] introduced
the concept of modified \(\omega\)-distance mappings on a complete
quasi metric space in which they studied some fixed point results.
In this manuscript, we prove some fixed point results of nonlinear
contraction conditions through modified \(\omega\)-distance mapping on
a complete quasi metric space in sense of Alegre and Marin.
5342
5350
Inam
Nuseir
Department of Mathematics and Statistics
Jordan University of Science and Technology
Jordan
imnuseir@just.edu.jo
Wasfi
Shatanawi
Department of Mathematics and General Courses
Department of Mathematics
Prince Sultan University
Hashemite University
Saudi Arabia
Jordan
wshatanawi@psu.edu.sa;swasfi@hu.edu.jo
Issam
Abu-Irwaq
Department of Mathematics and Statistics
Jordan University of Science and Technology
Jordan
imabuirwaq@just.edu.jo
Anwar
Bataihah
Department of Mathematics
The University of Jordan
Jordan
anwerbataihah@gmail.com
Quasi metric
fixed point theorem
nonlinear contraction
altering distance
modified \(\omega\)-distance
Article.20.pdf
[
[1]
K. Abodayeh, A. Bataihah, W. Shatanawi , Generalized \(\Omega\)-distance mappings and some fixed point theorems, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 79 (2017), 223-232
##[2]
K. Abodayeh, W. Shatanawi, A. Bataihah, Fixed point theorem through \(\Omega\)-distance of Suzuki type conraction condition, G. U. J. Sci., 29 (2016), 129-133
##[3]
K. Abodayeh, W. Shatanawi, A. Bataihah, A. H. Ansari, Some fixed point and common fixed point results through -distance under nonlinear contractions, G. U. J. Sci., 30 (2017), 293-302
##[4]
I. Abu-Irwaq, I. Nuseir, A. Bataihah , Common Fixed Point Theorems in G-metric Spaces with \(\Omega\)-distance, J. Math. Anal., 8 (2017), 120-129
##[5]
C. Alegre, J. Marin, Modified \(\omega\)-distance on quasi metric spaces and fixed point theorem on complete quasi metric spaces, Topol. Appl., 203 (2016), 32-41
##[6]
A. Amini-Harandi, H. Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal., 72 (2010), 2238-2242
##[7]
V. Berinde, Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces, Nonlinear Anal., 74 (2011), 7347-7355
##[8]
V. Berinde , Coupled fixed point theorems for \(\Phi\)-contractive mixed monotone mappings in partially ordered metric spaces, Nonlinear Anal., 75 (2012), 3218-3228
##[9]
A. Branciari , A fixed point theorem for mapping satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci., 29 (2002), 531-536
##[10]
B. S. Choudhury, A. Kundu , A coupled coincidence point result in partially ordered metric spaces for compatible mappings, Nonlinear Anal., 73 (2010), 2524-2531
##[11]
L. B. Ćirić, A generalization of Banch’s contraction principle, Proc. Amer. Math. Soc., 45 (1974), 267-273
##[12]
M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604-608
##[13]
M. Jleli, B. Samet , Remarks on G-metric spaces and fixed point theorems, Fixed Point Theory Appl., 2012 (2012), 1-7
##[14]
R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71-76
##[15]
M. S. Khan, M. Swaleh, S. Sessa, fixed point theorems by altering distances between the points , Bull. Austral. Math. Soc., 30 (1984), 1-9
##[16]
M. Kikkawa, T. Suzuki , Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal., 69 (2008), 2942-2949
##[17]
W. Shatanawi, G. Maniu, A. Bataihah, F. Bani Ahmad, Common fixed points for mappings of cyclic form satisfying linear contractive conditions with Omega-distance, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 79 (2017), 11-20
##[18]
W. Shatanawi, M. S. Noorani, H. Alsamir, A. Bataihah, Fixed and common fixed point theorems in partially ordered quasi-metric spaces, J. Math. Computer Sci., 16 (2016), 516-528
##[19]
W. Shatanawi, A. Pitea , Some coupled fixed point theorems in quasi partial-metric spaces, Fixed Point Theory Appl., 2013 (2013), 1-15
##[20]
P. V. Subrahmanyam, Completeness and fixed points, Monatsh. Math., 80 (1975), 325-330
##[21]
T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc., 136 (2008), 1861-1869
##[22]
W. A. Wilson, On quasi-metric spaces, Amer. J. Math., 53 (1931), 675-684
]
Viscosity approximation of solutions of a split feasibility problem in Hilbert spaces
Viscosity approximation of solutions of a split feasibility problem in Hilbert spaces
en
en
In this paper, we study two viscosity approximation iterative methods for solving solutions of a split feasibility problem. Strong convergence theorems are established in the framework of infinite dimensional Hilbert spaces.
5351
5359
Yantao
Yang
College of Mathematics and Computer Science
Yanan University
China
yadxyyt@163.com
Yunpeng
Zhang
Inst. Fundamental \(\&\) Frontier Sci.
Univ. Elect. Sci. \(\&\) Technol. China
China
zhangypliyl@yeah.net
Convergence analysis
Hilbert space
monotone mapping
split feasibility problem
Article.21.pdf
[
[1]
B. A. Bin Dehaish, A. Latif, H. O. Bakodah, X.-L. Qin, A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces, J. Inequal. Appl., 2015 (2015), 1-14
##[2]
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120
##[3]
Y. Censor, T. Bortfeld, B. Martin, A. Trofimov , A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365
##[4]
Y. Censor, T. Elfving , A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239
##[5]
Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084
##[6]
S. S. Chang, L. Wang, Y. Zhao, On a class of split equality fixed point problems in Hilbert spaces, J. Nonlinear Var. Anal., 1 (2017), 201-212
##[7]
S. Y. Cho, B. A. Bin Dehaish, X.-L. Qin, Weak convergence of a splitting algorithm in Hilbert spaces, J. Appl. Anal. Comput., 7 (2017), 427-438
##[8]
S. Y. Cho, X.-L. Qin , On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems, Appl. Math. Comput., 235 (2014), 430-438
##[9]
S. Y. Cho, X.-L. Qin, L. Wang, Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl., 2014 (2014), 1-15
##[10]
N. Fang , Some results on split variational inclusion and fixed point problems in Hilbert spaces, Commun. Optim. Theory, 2017 (2017), 1-13
##[11]
P. E. Maingé, A viscosity method with no spectral radius requirements for the split common fixed point problem, European J. Oper. Res., 235 (2014), 17-27
##[12]
A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl., 28 (1969), 326-329
##[13]
A. Moudafi , Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55
##[14]
X.-L. Qin, S. Y. Cho, Convergence analysis of a monotone projection algorithm in reflexive Banach spaces, Acta Math. Sci. Ser. B Engl. Ed., 37 (2017), 488-502
##[15]
X.-L. Qin, J.-C. Yao , Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators, J. Inequal. Appl., 2016 (2016), 1-9
##[16]
X.-L. Qin, J.-C. Yao, Projection splitting algorithms for nonself operators, J. Nonlinear Convex Anal., 18 (2017), 925-935
##[17]
T. Suzuki , Moudafi’s viscosity approximations with Meir-Keeler contractions, J. Math. Anal. Appl., 321 (2007), 342-352
##[18]
J.-F. Tang, S.-S. Chang, J. Dong, Split equality fixed point problem for two quasi-asymptotically pseudocontractive mappings, J. Nonlinear Funct. Anal., 2017 (2017), 1-15
##[19]
H. Zhang, Iterative processes for fixed points of nonexpansive mappings, Commun. Optim. Theory, 2013 (2013), 1-7
##[20]
Y.-F. Zhang, S.-H. Wang, H.-Q. Zhao, Explicit and implicit iterative algorithms for strict pseudo-contractions in Banach spaces, J. Nonlinear Funct. Anal., 2017 (2017), 1-14
]
Positive solutions to nonlinear fractional differential equations involving Stieltjes integrals conditions
Positive solutions to nonlinear fractional differential equations involving Stieltjes integrals conditions
en
en
In this paper, we consider the existence of positive solutions for a
class of nonlinear fractional semipositone differential equations
involving integral boundary conditions. Some existence results of
positive solutions are obtained by means of Leray-Schauder's
alternative and Krasnoselskii's fixed point theorem. An example is
given to demonstrate the application of our main results.
5360
5372
Jiqiang
Jiang
School of Mathematical Sciences
Qufu Normal University
People's Republic of China
qfjjq@163.com
Lishan
Liu
School of Mathematical Sciences
Department of Mathematics and Statistics
Qufu Normal University
Curtin University
People's Republic of China
Australia
lls@mail.qfnu.edu.cn
Yonghong
Wu
Department of Mathematics and Statistics
Curtin University
Australia
yhwu@maths.curtin.edu.au
Integral boundary conditions
semipositone
fractional differential equation
positive solutions
fixed point theory
Article.22.pdf
[
[1]
B. Ahmad, Sharp estimates for the unique solution of two-point fractional-order boundary value problems, Appl. Math. Lett., 65 (2017), 77-82
##[2]
B. Ahmad, J. J. Nieto, Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations, Abstr. Appl. Anal., 2009 (2009), 1-9
##[3]
Z.-B. Bai, Eigenvalue intervals for a class of fractional boundary value problem, Comput. Math. Appl., 64 (2012), 3253-3257
##[4]
D. Baleanu, R. P. Agarwal, H. Mohammadi, S. Rezapour, Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces, Bound. Value Probl., 2013 (2013), 1-8
##[5]
D. Baleanu, H. Mohammadi, S. Rezapour , On a nonlinear fractional differential equation on partially ordered metric spaces, Adv. Difference Equ., 2013 (2013), 1-10
##[6]
D. Baleanu, O. G. Mustafa, R. P. Agarwal , On \(L^p\)-solutions for a class of sequential fractional differential equations, Appl. Math. Comput., 218 (2011), 2074-2081
##[7]
M. Benchohra, S. Hamani, S. K. Ntouyas , Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal., 71 (2009), 2391-2396
##[8]
A. Cabada, T. Kisela , Existence of positive periodic solutions of some nonlinear fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 50 (2017), 51-67
##[9]
J. R. Cannon , The solution of the heat equation subject to the specification of energy , Quart. Appl. Math., 21 (1963), 155-160
##[10]
R. Y. Chegis , Numerical solution of a heat conduction problem with an integral boundary condition, Litovsk. Mat. Sb., 24 (1984), 209-215
##[11]
Y.-J. Cui , Existence results for singular boundary value problem of nonlinear fractional differential equation, Abstr. Appl. Anal., 2011 (2011), 1-9
##[12]
Y.-J. Cui, Uniqueness of solution for boundary value problems for fractional differential equations, Appl. Math. Lett., 51 (2016), 48-54
##[13]
K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin (1985)
##[14]
M. El-Shahed, J. J. Nieto, Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order, Comput. Math. Appl., 59 (2010), 3438-3443
##[15]
C. S. Goodrich, Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions, Comput. Math. Appl., 61 (2011), 191-202
##[16]
D. J. Guo, V. Lakshmikantham, Nonlinear problems in abstract cones, Notes and Reports in Mathematics in Science and Engineering, Academic Press, Inc., Boston, MA (1988)
##[17]
N. I. Ionkin , The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition, (Russian) Differencial’nye Uravnenija, 13 (1977), 294-304
##[18]
T. Jankowski , Positive solutions to fractional differential equations involving Stieltjes integral conditions, Appl. Math. Comput., 241 (2014), 200-213
##[19]
J.-Q. Jiang, L.-S. Liu, Existence of solutions for a sequential fractional differential system with coupled boundary conditions, Bound. Value Probl., 2016 (2016), 1-15
##[20]
J.-Q. Jiang, L.-S. Liu, Y.-H. Wu , Multiple positive solutions of singular fractional differential system involving Stieltjes integral conditions, Electron. J. Qual. Theory Differ. Equ., 2012 (2012), 1-18
##[21]
J.-Q. Jiang, L.-S. Liu, Y.-H. Wu, Positive solutions for nonlinear fractional differential equations with boundary conditions involving Riemann-Stieltjes integrals , Abstr. Appl. Anal., 2012 (2012), 1-21
##[22]
J.-Q. Jiang, L.-S. Liu, Y.-H. Wu, Positive solutions to singular fractional differential system with coupled boundary conditions, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3061-3074
##[23]
D.-Q. Jiang, C.-J. Yuan, The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application, Nonlinear Anal., 72 (2010), 710-719
##[24]
H. Khalil, R. A. Khan, D. Baleanu, S. H. Saker , Approximate solution of linear and nonlinear fractional differential equations under m-point local and nonlocal boundary conditions , Adv. Difference Equ., 2016 (2016), 1-28
##[25]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo , Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006)
##[26]
H.-D. Li, L.-S. Liu, Y.-H. Wu, Positive solutions for singular nonlinear fractional differential equation with integral boundary conditions, Bound. Value Probl., 2015 (2015), 1-15
##[27]
I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)
##[28]
S. Y. Reutskiy , A new numerical method for solving high-order fractional eigenvalue problems, J. Comput. Appl. Math., 317 (2017), 603-623
##[29]
G.-T. Wang, S.-Y. Liu, D. Baleanu, L.-H. Zhang, Existence results for nonlinear fractional differential equations involving different Riemann-Liouville fractional derivatives, Adv. Difference Equ., 2013 (2013), 1-7
##[30]
Y.-Q. Wang, L.-S. Liu, Y.-H. Wu, Positive solutions for a nonlocal fractional differential equation, Nonlinear Anal., 74 (2011), 3599-3605
##[31]
J.-W. Wu, X.-G. Zhang, L.-S. Liu, Y.-H. Wu, Positive solutions of higher-order nonlinear fractional differential equations with changing-sign measure, Adv. Difference Equ., 2012 (2012), 1-14
##[32]
S.-Q. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. Math. Appl., 59 (2010), 1300-1309
##[33]
X.-G. Zhang, Y.-F. Han, Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations, Appl. Math. Lett., 25 (2012), 555-560
##[34]
X.-G. Zhang, L.-S. Liu, Y.-H. Wu, Multiple positive solutions of a singular fractional differential equation with negatively perturbed term, Math. Comput. Modelling, 55 (2012), 1263-1274
##[35]
X.-G. Zhang, L.-S. Liu, Y.-H. Wu , The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives, Appl. Math. Comput., 218 (2012), 8526-8536
##[36]
X.-G. Zhang, L.-S. Liu, Y.-H. Wu, B. Wiwatanapataphee , The spectral analysis for a singular fractional differential equation with a signed measure, Appl. Math. Comput., 257 (2015), 252-263
]
Well-posedness for systems of generalized mixed quasivariational inclusion problems and optimization problems with constraints
Well-posedness for systems of generalized mixed quasivariational inclusion problems and optimization problems with constraints
en
en
In this paper, several metric characterizations of well-posedness for
systems of generalized mixed quasivariational inclusion problems and for optimization problems with systems of generalized mixed quasivariational
inclusion problems as constraints are given. The equivalence between the well-posedness of systems of generalized mixed quasivariational inclusion
problems and the existence of solutions of systems of generalized mixed quasivariational inclusion problems is given under suitable conditions.
5373
5392
Lu-Chuan
Ceng
Department of Mathematics
Shanghai Normal University
China
zenglc@hotmail.com
Yeong-Cheng
Liou
Department of Healthcare Administration and Medical Informatics, Center for Big Data Analytics \(\&\) Intelligent Healthcare, and Research Center of Nonlinear Analysis and Optimization
Department of Medical Research
Kaohsiung Medical University
Kaohsiung Medical University Hospital
Taiwan
Taiwan
simplex liou@hotmail.com
Jen-Chih
Yao
Center for General Education
China Medical University
Taiwan
yaojc@kmu.edu.tw
Yonghong
Yao
Department of Mathematics
Tianjin Polytechnic University
China
yaoyonghong@aliyun.com
Ching-Hua
Lo
Department of Mathematics
Tianjin Polytechnic University
China
bde_lo@sina.com
Well-posedness
metric characterization
system of generalized mixed quasivariational inclusion problems
optimization problem with constraint
Article.23.pdf
[
[1]
J.-P. Aubin, I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, New York (1984)
##[2]
M. Bianchi, G. Kassay, R. Pini, Well-posed equilibrium problems , Nonlinear Anal., 72 (2010), 460-468
##[3]
L.-C. Ceng, Y.-C. Lin, Metric characterizations of \(\alpha\)-well-posedness for a system of mixed quasivariational-like inequalities in Banach spaces, J. Appl. Math., 2012 (2012), 1-22
##[4]
L.-C. Ceng, Y.-C. Liou, J.-C. Yao, Y.-H. Yao , Well-posedness for systems of time-dependent hemivariational inequalities in Banach spaces, J. Nonlinear Sci. Appl., 10 (2017), 4318-4336
##[5]
L.-C. Ceng, J.-C. Yao, Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems, Nonlinear Anal., 69 (2008), 4585-4603
##[6]
G. P. Crespi, A. Guerraggio, M. Rocca, Well-posedness in vector optimization problems and vector variational inequalities, J. Optim. Theory Appl., 132 (2007), 213-226
##[7]
M. Durea, Scalarization for pointwise well-posed vectorial problems, Math. Methods Oper. Res., 66 (2007), 409-418
##[8]
Y.-P. Fang, R. Hu, N.-J. Huang, Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints, Comput. Math. Appl., 55 (2008), 89-100
##[9]
Y.-P. Fang, N.-J. Huang, J.-C. Yao, Well-posedness of mixed variational inequalities, inclusion problems and fixed-point problems, J. Global Optim., 41 (2008), 117-133
##[10]
Y.-P. Fang, N.-J. Huang, J.-C. Yao, Well-posedness by perturbations of mixed variational inequalities in Banach spaces, European J. Oper. Res., 201 (2010), 682-692
##[11]
F. Giannessi, P. M. Pardalos, T. Rapcsak , New Trends in Equilibrium Systems, Kluwer Academic Publishers, Dordrecht (2001)
##[12]
N. X. Hai, P. Q. Khanh, The solution existence of general variational inclusion problems, J. Math. Anal. Appl., 328 (2007), 1268-1277
##[13]
X. X. Huang, X. Q. Yang, D. L. Zhu , Levitin-Polyak well-posedness of variational inequality problems with functional constraints, J. Global Optim., 44 (2009), 159-174
##[14]
K. Kuratowski, Topology, Academic Press, New York (1968)
##[15]
B. Lemaire, C. O. A. Salem, J. P. Revalski, Well-posedness by perturbations of variational problems, J. Optim. Theory Appl., 115 (2002), 345-368
##[16]
M. B. Lignola, Well-posedness and L-well-posedness for quasivariational inequalities, J. Optim. Theory Appl., 128 (2006), 119-138
##[17]
L.-J. Lin, C.-S. Chuang, Well-posedness in the generalized sense for variational inclusion and disclusion problems and well-posedness for optimization problems with constraint, Nonlinear Anal., 70 (2009), 3609-3617
##[18]
X.-J. Long, N.-J. Huang, Metric characterizations of \(\alpha\)-well-posedness for symmetric quasiequilibrium problems , J. Global Optim., 45 (2009), 459-471
##[19]
R. Lucchetti, F. Patrone, A characterization of Tykhonov well-posedness for minimum problems with applications to variational inequalities, Numer. Funct. Anal. Optim., 3 (1981), 461-476
##[20]
M. Margiocco, F. Patrone, L. Pusillo Chicco, A new approach to Tikhonov well-posedness for Nash equilibria, Optim., 40 (1997), 385-400
##[21]
J. Morgan , Approximations and well-posedness in multicriteria games, Ann. Oper. Res., 137 (2005), 257-268
##[22]
A. Petruşel, I. A. Rus, J.-C. Yao, Well-posedness in the generalized sense of the fixed point problems for multivalued operators, Taiwanese J. Math., 11 (2007), 903-914
##[23]
J. P. Revalski , Hadamard and strong well-posedness for convex programs, SIAM J. Optim., 7 (1997), 519-526
##[24]
P. H. Sach, L. A. Tuan, Generalizations of vector quasivariational inclusion problems with set-valued maps, J. Global Optim., 43 (2009), 23-45
##[25]
A. N. Tikhonov, On the stability of the functional optimization problems, USSR J. Comput. Math. Math. Phys., 6 (1966), 28-33
##[26]
S.-H. Wang, N.-J. Huang, D. O’Regan, Well-posedness for generalized quasi-variational inclusion problems and for optimization problems with constraints, J. Global Optim., 55 (2013), 189-208
##[27]
Y.-H. Yao, M. A. Noor, Y.-C. Liou, S. M. Kang, Iterative algorithms for general multi-valued variational inequalities, Abstr. Appl. Anal., 2012 (2012), 1-10
##[28]
Y.-H. Yao, M. Postolache, Y.-C. Liou, Z.-S. Yao, Construction algorithms for a class of monotone variational inequalities , Optim. Lett., 10 (2016), 1519-1528
##[29]
Y.-H. Yao, N. Shahzad, Strong convergence of a proximal point algorithm with general errors , Optim. Lett., 6 (2012), 621-628
##[30]
Y.-H. Yao, N. Shahzad, An algorithmic approach to the split variational inequality and fixed point problem, J. Nonlinear Convex Anal., 18 (2017), 977-991
##[31]
H. Zegeye, N. Shahzad, Y.-H. Yao, Minimum-norm solution of variational inequality and fixed point problem in Banach spaces, Optimization, 64 (2015), 453-471
##[32]
T. Zolezzi, Well-posedness criteria in optimization with application to the calculus of variations, Nonlinear Anal., 25 (1995), 437-453
##[33]
T. Zolezzi, Extended well-posedness of optimization problems, J. Optim. Theory Appl., 91 (1996), 257-266
]
An inverse nodal problem for \({p}\)-Laplacian Sturm-Liouville equation with Coulomb potential
An inverse nodal problem for \({p}\)-Laplacian Sturm-Liouville equation with Coulomb potential
en
en
We deal with an inverse nodal problem for
\(p\)-Laplacian Sturm-Liouville equation which
includes Coulomb type potential function under boundary condition depends on
polynomial spectral parameter. Here, we get some asymptotic formulas of
eigenvalues and nodal parameters by using a suitable Prüfer substitution. Eventually, we construct Coulomb potential by using nodal
lengths.
5393
5401
Tuba
Gulsen
Department of Mathematics
Firat University
Turkey
tubagulsen87@hotmail.com
Inverse nodal problem
Prüfer substitution
Coulomb potential
Article.24.pdf
[
[1]
V. Ambarzumian, Über eine frage der eigenwerttheorie, Z. Phys., 53 (1929), 690-695
##[2]
R. K. Amirov, Y. Cakmak, S. Gulyaz, Boundary value problem for second-order differential equations with Coulomb singularity on a finite interval, Indian J. Pure Appl. Math., 37 (2006), 125-140
##[3]
R. K. Amirov, N. Topsakal, A representation for solutions of Sturm-Liouville equations with Coulomb potential inside finite interval, J. Cumhuriyet Univ. Nat. Sci., 28 (2007), 11-38
##[4]
R. K. Amirov, N. Topsakal , Sturm-Liouville operators with Coulomb potential which have discontinuity conditions inside an interval, Integral Transforms Spec. Funct., 19 (2008), 923-937
##[5]
P. Binding, P. Drábek, Sturm-Liouville theory for the p-Laplacian , Studia Sci. Math. Hungar., 40 (2003), 373-396
##[6]
G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Bestimmung der Differentialgleichung durch die Eigenwerte, (German) Acta Math., 78 (1945), 1-96
##[7]
P. J. Browne, B. D. Sleeman, Inverse nodal problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions, Inverse Problems, 12 (1996), 377-381
##[8]
S. A. Buterin, On inverse spectral problem for non-selfadjoint Sturm-Liouville operator on a finite interval, J. Math. Anal. Appl., 335 (2007), 739-749
##[9]
R. Carlson, Inverse spectral theory for some singular Sturm-Liouville problems, J. Differential Equations, 106 (1993), 121-140
##[10]
R. Carlson, An inverse spectral problem for Sturm-Liouville operators with discontinuous coefficients, Proc. Amer. Math. Soc., 120 (1994), 475-484
##[11]
H. Y. Chen, On generalized trigonometric functions, Master of Science thesis, National Sun Yat-sen University, Kaohsiung, Taiwan (2009)
##[12]
Á . Elbert, On the half-linear second order differential equations , Acta Math. Hungar., 49 (1987), 487-508
##[13]
G. Freiling, V. A. Yurko, Inverse problems for Sturm-Liouville equations with boundary conditions polynomially dependent on the spectral parameter, Inverse Problems, 26 (2010), 1-17
##[14]
N. J. Guliyev, Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions, Inverse Problems, 21 (2005), 1315-1330
##[15]
T. Gulsen, E. Yilmaz , Inverse nodal problem for p-Laplacian diffusion equation with polynomially dependent spectral parameter, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 65 (2016), 23-36
##[16]
O. H. Hald, Discontinuous inverse eigenvalue problems, Comm. Pure Appl. Math., 37 (1984), 539-577
##[17]
O. H. Hald, J. R. McLaughlin, Solutions of inverse nodal problems, Inverse Problems, 5 (1989), 307-347
##[18]
H. Koyunbakan, Erratum: Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation, Bound. Value Probl., 2014 (2014), 1-2
##[19]
H. Koyunbakan, E. Yilmaz, Reconstruction of the potential function and its derivatives for the diffusion operator, Z. Naturforsch. A, 63 (2008), 127-130
##[20]
C. K. Law, W.-C. Lian, W.-C. Wang, The inverse nodal problem and the Ambarzumyan problem for the p-Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 1261-1273
##[21]
C. K. Law, C.-F. Yang, Reconstructing the potential function and its derivatives using nodal data, Inverse Problems, 14 (1998), 299-312
##[22]
N. Levinson, The inverse Sturm-Liouville problem, Mat. Tidsskr. B., 1949 (1949), 25-30
##[23]
B. M. Levitan , On the determination of the Sturm-Liouville operator from one and two spectra , Math. USSR Izv., 12 (1978), 179-193
##[24]
V. A. Marchenko, Sturm-Liouville operators and applications, Translated from the Russian by A. Iacob, Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel (1986)
##[25]
J. R. McLaughlin, Inverse spectral theory using nodal points as data–a uniqueness result, J. Differential Equations, 73 (1988), 354-362
##[26]
A. S. Ozkan, B. Keskin, Inverse nodal problems for Sturm-Liouville equation with eigenparameter-dependent boundary and jump conditions, Inverse Probl. Sci. Eng., 23 (2015), 1306-1312
##[27]
W. Y. Ping, C. T. Shieh, Inverse problems for Sturm-Liouville equations with boundary conditions linearly dependent on the spectral parameter from partial information, Results Math., 65 (2014), 105-119
##[28]
J. Pöschel, E. Trubowitz, Inverse spectral theory , Pure and Applied Mathematics, Academic Press, Inc., Boston, MA (1987)
##[29]
J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary condition, Math. Z., 133 (1973), 301-312
##[30]
W.-C. Wang, Direct and inverse problems for one dimensional p-Laplacian operators, PhD thesis, National Sun Yat-sen University, Taiwan (2010)
##[31]
W. C.Wang, Y. H. Cheng,W. C. Lian, Inverse nodal problems for the p-Laplacian with eigenparameter dependent boundary conditions, Math. Comput. Modelling, 54 (2011), 2718-2724
##[32]
C.-F. Yang, X.-P. Yang, Ambarzumyan’s theorem with eigenparameter in the boundary conditions, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 1561-1568
##[33]
C.-F. Yang, X.-P. Yang, Inverse nodal problems for the Sturm-Liouville equation with polynomially dependent on the eigenparameter, Inverse Probl. Sci. Eng., 19 (2011), 951-961
##[34]
A. Yantır, Oscillation theory for second order differential equations and dynamic equations on time scales, Master of Science thesis, Izmir institue of Technology, Izmir, Turkey (2004)
##[35]
V. A. Yurko , Inverse spectral problems for differential operators and their applications, Analytical Methods and Special Functions, Gordon and Breach Science Publishers, Amsterdam (2000)
##[36]
V. A. Yurko, Inverse nodal problems for Sturm-Liouville operators on star-type graphs, J. Inverse Ill-Posed Probl., 16 (2008), 715-722
]
The rapid convergence for nonlinear singular differential systems with "maxima''
The rapid convergence for nonlinear singular differential systems with "maxima''
en
en
This paper investigates the initial value problem for a class of nonlinear singular differential systems with "maxima''. By using the
comparison principle and the approximate quasilinearization method, we obtain two monotone iterative sequences of approximate solutions
which converge uniformly and rapidly to the solution of such systems.
5402
5421
Peiguang
Wang
College of Mathematics and Information Science
Hebei University
China
pgwang@hbu.edu.cn
Xiang
Liu
College of Mathematics and Information Science
Hebei University
China
1065265114@qq.com
Tongxing
Li
School of Information Science and Engineering
Linyi University
China
litongx2007@163.com
Singular system
maxima
approximate quasilinearization
rapid convergence
Article.25.pdf
[
[1]
A. R. Abd-Ellateef Kamar, G. M. Attia, K. Vajravelu, M. Mosaad , Generlized quasilinearization for singular system of differential equations, Appl. Math. Comput., 114 (2000), 69-74
##[2]
R. P. Agarwal, S. Hristova, Quasilinearization for initial value problems involving differential equations with “maxim” , Math. Comput. modelling, 55 (2012), 2096-2105
##[3]
B. Ahmad, A. Alsaedi, An extended method of quasilinearization for nonlinear impulsive differential equations with a nonlinear three-point boundary condition, Electron. J. Qual. Theory Differ. Equ., 2007 (2007), 1-19
##[4]
B. Ahmad, R. A. Khan, S. Sivasundaram, Generalized quasilinearization method for nonlinear functional differential equations, J. Appl. Math. Stochastic Anal., 16 (2003), 33-43
##[5]
P. Amster, P. De Nápoli , A quasilinearization method for elliptic problems with a nonlinear boundary condition , Nonlinear Anal., 66 (2007), 2255-2263
##[6]
D. D. Bainov, S. G. Hristova, The method of quasilinearization for the periodic boundary value problem for systems of impulsive differential equations, Appl. Math. Comput., 117 (2001), 73-85
##[7]
D. D. Bainov, S. G. Hristova, Differential Equations with Maxima, CRC Press Taylor & Francis, New York (2011)
##[8]
R. E. Bellman, R. E. Kalaba, Quasilinearization and Nonlinear Boundary Value Problems, Elsevier, New York (1965)
##[9]
A. Buică , Quasilinearization method for nonlinear elliptic boundary value problems, J. Optim. Theory Appl., 124 (2005), 323-338
##[10]
S. L. Campbell, Singular systems of differential equations, Pitman Advanced Publishing Program (I), London (1982)
##[11]
S. L. Campbell , Singular systems of differential equations, Pitman Advanced Publishing Program (II), London (1982)
##[12]
Z. Drici, F. A. McRae, J. V. Devi, Quasilinearization for functional differential equations with retardation and anticipation, Nonlinear Anal., 70 (2009), 1763-1775
##[13]
M. A. El-Gebeily, D. O’Regan, Upper and lower solutions and quasilinearization for a class of second order singular nonlinear differential equations with nonlinear boundary conditions, Nonlinear Anal. Real World Appl., 8 (2007), 636-645
##[14]
M. A. El-Gebeily, D. O’Regan, Existence and quasilinearization for a class of nonlinear elliptic second order partial differential equations, Dynam. Systems Appl., 17 (2008), 445-458
##[15]
S. Hristova, A. Golev, K. Stefanova, Quasilinearization of the initial value problem for difference equations with “maxim”, J. Appl. Math., 2012 (2012), 1-17
##[16]
V. Lakshmikantham, S. Köksal , Monotone flows and rapid convergence for nonlinear partial differential equations, Taylor & Francis, London (2003)
##[17]
V. Laksmikantham, S. Leela, Z. Drici, F. A. McRae, Theory of causal differential equations, World Scientific, Hackensack (2009)
##[18]
V. Lakshmikantham, A. S. Vatsala, Generalized quasilinearization for nonlinear problems, Kluwer Academic Publishers, Dordrecht (1998)
##[19]
D. O’Regan, M. El-Gebeily, Existence, upper and lower solutions and quasilinearization for singular differential equations, IMA J. Appl. Math., 73 (2008), 323-344
##[20]
H. H. Rosenbrock, Structural properties of linear dynamical systems, Internat. J. Control, 20 (1974), 191-202
##[21]
J. Vasundhara Devi, F. A. McRae, Z. Drici, Generalized quasilinearization for fractional differential equations, Comput. Math. Appl., 59 (2010), 1057-1062
##[22]
P.-G. Wang, W. Gao, Quasilinearization of an initial value problem for a set valued integro-differential equation , Comput. Math. Appl., 61 (2011), 2111-2115
##[23]
P.-G. Wang, T.-T. Kong , Quasilinearization for the boundary value problem of second-order singular differential system, Abstr. Appl. Anal., 2013 (2013), 1-7
##[24]
P.-G. Wang, P. Li , Kth Order convergence for a semilinear elliptic boundary value problem in the divergence form, Appl. Math. Comput., 217 (2011), 8547-8551
##[25]
P.-G. Wang, X. Liu, A periodic boundary value problem for nonlinear singular differential systems with “maxim”, Bound. Value Probl., 2015 (2015), 1-18
##[26]
P.-G. Wang, X. Liu, The quadratic convergence of approximate solutions for singular difference systems with ”maxima”, J. Math. Computer Sci., 16 (2016), 227-238
##[27]
P.-G. Wang, H.-X. Wu, Y.-H. Wu, Higher even-order convergence and coupled solutions for second-order boundary value problems on time scales, Comput. Math. Appl., 55 (2008), 1693-1705
]
Modelling the movement of groundwater pollution with variable order derivative
Modelling the movement of groundwater pollution with variable order derivative
en
en
In this paper, a new concept of variable differentiation is used to revisit the model of groundwater pollution. The new variable order derivation has a non-singular kernel and can be used for analytical and numerical purposes. The novel model is solved via Fourier transform method. We solve numerically the new equation using the implicit finite difference scheme and study the stability and convergence of that scheme.
5422
5432
S. N.
Kameni
African Institute for Mathematical Sciences (AIMS)
Cameroon
sedrick.ngwamou@aims-cameroon.org
J. D.
Djida
African Institute for Mathematical Sciences (AIMS)
Cameroon
jeandaniel.djida@aims-cameroon.org
A.
Atangana
Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences
University of the Free State
South Africa
abdonatangana@yahoo.fr
New variable order derivative
advection-dispersion equation with variable order
stability
convergence
finite difference scheme
Article.26.pdf
[
[1]
L. M. Abriola, , Modeling contaminant transport in the subsurface: An interdisciplinary challenge, Rev. Geophys., 25 (1987), 125-134
##[2]
A. N. Angelakis, T. N. Kadir, D. E. Rolston, Solutions for transport of two sorbed solutes with differing dispersion coefficients in soil , Soil Sci. Soc. Am. J., 51 (1987), 1428-1434
##[3]
G. B. Arfken, H. J. Weber , Mathematical Methods for Physicists, Am. J. Phys., 67 (1999), 165-169
##[4]
A. Atangana, A Derivative with variable order with no singular kernel, J. Comput. Phys. , preprint (2016)
##[5]
A. Atangana, A. Kilicman , Analytical solutions of the space-time fractional derivative of advection dispersion equation, Math. Probl. Engin., 2013 (2013), 1-9
##[6]
D. A. Benson, S. W. Wheatcraft, M. M. Meerschaert , Application of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000), 1403-1412
##[7]
B. R. Bicknell, J. C. Imhoff, J. Kittle, L. John, J. Donigian, S. Anthony, R. C. Johanson, Hydrological simulation program–Fortran, User’s manual for version 11, US EPA (1996)
##[8]
D. K. Jaiswal, A. Kumar, N. Kumar, M. K. Singh, Solute transport along temporally and spatially dependent flows through horizontal semi-infinite media: dispersion proportional to square of velocity, J. Hydrol. Eng., 16 (2011), 228-238
##[9]
D. K. Jaiswal, A. Kumar, N. Kumar, R. Yadav, Analytical solutions for temporally and spatially dependent solute dispersion of pulse type input concentration in one-dimensional semi-infinite media, J. Hydro. Environ. Res., 2 (2009), 254-263
##[10]
K. R. Lassey , Unidimensional solute transport incorporating equilibrium and ratelimited isotherms with firstorder loss: 1. Model conceptualizations and analytic solutions, Water Resour. Res., 24 (1988), 343-350
##[11]
R. Leonard, W. Knisel, F. Davis, A. Johnson , Validating GLEAMS with field data for fenamiphos and its metabolites, J. Irrig. Drain. Eng., 116 (1990), 24-35
##[12]
M. M. Meerschaert, C. Tadjeran , Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65-77
##[13]
T. G. Naymik, R. A. Freeze , Mathematical modeling of solute transport in the subsurface, Crit. Rev. Environ. Sci. Technol., 17 (1987), 229-251
##[14]
I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic press (1998)
##[15]
B. K. Rao, D. L. Hathaway , A Three-Dimensional Mixing Cell Solute Transport Model and Its Application, Ground Water, 27 (1989), 509-516
##[16]
L. L. Shoemaker, W. L. Magette, A. Shirmohammadi, Modeling management practice effects on pesticide movement to ground water, Groundwater Monit. Remediat., 10 (1990), 109-115
##[17]
C. S. Slichter, Field measurements of the rate of movement of underground waters, Govt. Print. Off., Washington (1905)
##[18]
W. J. Weber, C. T. Miller , Modeling the sorption of hydrophobic contaminants by aquifer materialsI. Rates and equilibria, Water Res., 22 (1988), 457-464
##[19]
L.Wen, X. L. Tang, Numerical Solving Two-dimensional Variable-order Fractional Advection-dispersion Equation, WSEAS Trans. Math., 2013 (2013), 1-7
##[20]
R. Yadav, D. K. Jaiswal , Two-dimensional solute transport for periodic flow in isotropic porous media: an analytical solution, Hydrol. Process., 26 (2012), 3425-3433
]
Strong convergence theorems for the general split common fixed point problem in Hilbert spaces
Strong convergence theorems for the general split common fixed point problem in Hilbert spaces
en
en
In this paper, we propose and investigate a new iterative algorithm for solving the general split common fixed point problem in the setting of infinite-dimensional Hilbert spaces. We also prove the sequence generated by the proposed algorithm converge strongly to a common solution of the general split common fixed point problem. As application, some particular cases of directed operator and quasi-nonexpansive operator are also considered. Finally, we present several numerical results for general split common fixed point problem to demonstrate the efficiency of the proposed algorithm.
5433
5444
Rudong
Chen
Department of Mathematics, Tian jin Polytechnic University, Tian jin 300387, China
chenrd@tjpu.edu.cn
Tao
Sun
Department of Mathematics
Tian jin Polytechnic University
China
1639690598@qq.com
Huimin
He
School of Mathematics and Statistics
Xidian University
China
huiminhe@126.com
Jen-Chih
Yao
Center for General Education
China Medical University
ROC
yaojc@mail.cmu.edu.tw
General split common fixed point problem
demicontractive operator
quasi-nonexpansive operator
directed operator
Article.27.pdf
[
[1]
C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441-453
##[2]
C. Byrne , A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120
##[3]
Y. Censor, T. Bortfeld, B. Martin, A. Trofimov , A unified approach for inversion problems in intensity-modulated radiation therapy , Phys. Med. Biol., 51 (2005), 2353-2365
##[4]
Y. Censor, Y. Elfving, A multiprojection algorithm using Bregman projections in a product space , Numer. Algorithms, 8 (1994), 221-239
##[5]
Y. Censor, Y. Elfving, N. Kopf, T. Bortfeld , The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084
##[6]
Y. Censor, A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587-600
##[7]
R. D. Chen, Fixed point Theory and Applications, National Defence Industry Press, (2012)
##[8]
P. L. Combettes, V. R.Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168-2000
##[9]
H.-H. Cui, F.-H. Wang, Iterative methods for the split common fixed point problem in Hilbert spaces, Fixed Point Theory Appl., 2014 (2014), 1-8
##[10]
P. E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912
##[11]
A. Moudafi , The split common fixed point problem for demicontractive mappings, Inverse Problems, 26 (2010), 1-6
##[12]
A. Moudafi, A note on the split common fixed-point problem for quasi-nonexpansive operators, Nonlinear Anal., 74 (2011), 4083-4087
##[13]
W. Takahashi, Nonlinear functional analysis, Fixed point theory and its applications, Yokohama Publishers, Yokohama (2000)
##[14]
F.-H. Wang, H.-K. Xu , Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Anal., 74 (2011), 4105-4111
##[15]
H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256
##[16]
Y.-H. Yao, R. P. Agarwal, M. Postolache, Y.-C. Liou , Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem, Fixed Point Theory Appl., 2014 (2014), 1-14
##[17]
Y.-H. Yao, M. Postolache, Y.-C. Liou , Strong convergence of a self-adaptive method for the split feasibility problem, Fixed Point Theory Appl., 2013 (2013), 1-12
]
Existence, non-existence and multiplicity results for a third order eigenvalue three-point boundary value problem
Existence, non-existence and multiplicity results for a third order eigenvalue three-point boundary value problem
en
en
This paper provides sufficient conditions to guarantee the existence, non-existence and multiplicity of solutions for a third order eigenvalue fully differential equation, coupled with three point boundary value conditions.
Although the change of sign, some bounds for the second derivative of the Green's function are obtained, which allow to define a different kind of cone that, as far as we know, has not been previously used in the literature.
The main arguments are based on the fixed point index theory for bounded and unbounded sets. Some examples are presented in order to show that the different existence theorems proved are not comparable.
5445
5463
Alberto
Cabada
Instituto de Matemáticas, Facultade de Matemáticas
Universidade de Santiago de Compostela
Spain
alberto.cabada@usc.es
Lucía
López-Somoza
Instituto de Matemáticas, Facultade de Matemáticas
Universidade de Santiago de Compostela
Spain
lucia.lopez.somoza@usc.es
Feliz
Minhós
Departamento de Matematica, Escola de Ciencias e Tecnologia
Centro de Investigação em Matemática e Aplicações (CIMA), Instituto de Investigação e Formação Avançada
Universidade de Évora
Universidade de Évora
Portugal
Portugal
fminhos@sapo.pt
Nonlinear boundary value problems
parameter dependence
multipoint boundary value problems
Green functions
degree theory
fixed points in cones
Article.28.pdf
[
[1]
H. Amann , Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM. Rev., 18 (1976), 620-709
##[2]
A. Cabada, G. Infante, F. A. F. Tojo, Nontrivial solutions of Hammerstein integral equations with reflections, Bound. Value Probl., 2013 (2013), 1-22
##[3]
A. Cabada, F. Minhós, A. I. Santos, Solvability for a third order discontinuous fully equation with nonlinear functional boundary conditions, J. Math. Anal. Appl., 322 (2006), 735-748
##[4]
J. Graef, L. Kong, F. Minhós, Generalized Hammerstein equations and applications, Results Math., 72 (2017), 369-383
##[5]
A. Granas, J. Dugundji, Fixed Point Theory, Springer, New York (2003)
##[6]
M. Greguš, Third Order Linear Differential Equations, Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht (1987)
##[7]
D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York (1988)
##[8]
L. Guo, J. Sun, Y. Zhao, Existence of positive solutions for nonlinear third-order three-point boundary value problems, Nonlinear Anal., 68 (2008), 3151-3158
##[9]
G. Infante, P. Pietramala, A third order boundary value problem subject to nonlinear boundary conditions, Math. Bohem., 135 (2010), 113-121
##[10]
G. Infante, P. Pietramala, F. A. F. Tojo, Nontrivial solutions of local and nonlocal Neumann boundary-value problems, Proc. Royal Soc. Edinburgh, 146 (2016), 337-369
##[11]
G. Infante, J. R. L. Webb , Three point boundary value problems with solutions that change sign, J. Integral Equations Appl., 15 (2003), 37-57
##[12]
Y. Liu, Z. Weiguo, L. Xiping, S. Chunfang, C. Hua , Positive solutions for a nonlinear third order multipoint boundary value problem, Pac. J. Math., 249 (2011), 177-188
##[13]
F. Minhós, R. Sousa , On the solvability of third-order three point systems of differential equations with dependence on the first derivative, Bull. Braz. Math. Soc., 48 (2017), 485-503
]
New criteria on exponential synchronization and existence of periodic solutions of complex BAM networks with delays
New criteria on exponential synchronization and existence of periodic solutions of complex BAM networks with delays
en
en
In this paper, we study a class of time-delayed BAM neural networks with discontinuous activations. Base on the framework of differential inclusion theory and set-valued analysis, by designing discontinuous feedback controller and using some analytic methods, easily verifiable delay-independent criteria are established to guarantee the existence of periodic solution and global exponential synchronization of the drive-response system. Finally, we give a numerical example to illustrate our theoretical analysis. The obtained results are essentially new and they extend previously known results.
5464
5482
Chao
Yang
Department of Mathematics and Computer Science
Changsha University
P. R. China
yang0915@hnu.edu.cn
Lihong
Huang
School of Mathematical and Statistics
Changsha University of Science and Technology
P. R. China
lhhuang@csust.edu.cn
BAM neural networks
time-delayed
discontinuous activations
periodic solution
exponential synchronization
Article.29.pdf
[
[1]
J. P. Aubin, A. Cellina, Differential inclusions, Set-valued maps and viability theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin (1984)
##[2]
A. Berman, R. J. Plemmons, Nonnegative matrices in the mathematical sciences, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1979)
##[3]
V. I. Blagodatskikh, A. F. Filippov , Differential inclusions and optimal control , (Russian) Topology, ordinary differential equations, dynamical systems, Trudy Mat. Inst. Steklov., 169 (1985), 194-252
##[4]
Z.-W. Cai, L.-H. Huang , Functional differential inclusions and dynamic behaviors for memristor-based BAM neural networks with time-varying delays , Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1279-1300
##[5]
Z.-W. Cai, L.-H. Huang, Z.-Y. Guo, X.-Y. Chen, On the periodic dynamics of a class of time-varying delayed neural networks via differential inclusions, Neural Netw., 33 (2012), 97-113
##[6]
Z.-W. Cai, L.-H. Huang, Z.-Y. Guo, L.-L. Zhang, X.-T. Wan, Periodic synchronization control of discontinuous delayed networks by using extended Filippov-framework, Neural Netw., 68 (2015), 96-110
##[7]
J.-D. Cao, J. Wang , Global asymptotic stability of a general class of recurrent neural networks with time-varying delays, IEEE Trans. Circuits Systems I Fund. Theory Appl., 50 (2003), 34-44
##[8]
F. H. Clarke , Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1983)
##[9]
L. Duan, L.-H. Huang, Z.-W. Cai, Existence and stability of periodic solution for mixed time-varying delayed neural networks with discontinuous activations, Neurocomputing, 123 (2014), 255-265
##[10]
A. F. Filippov, Differential equations with discontinuous righthand sides, Translated from the Russian, Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht (1988)
##[11]
M. Forti, M. Grazzini, P. Nistri, L. Pancioni, Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations, Phys. D, 214 (2006), 88-99
##[12]
M. Forti, P. Nistri, D. Papini, Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain, IEEE Trans. Neural Netw., 16 (2005), 1449-1463
##[13]
C.-H. Hou, J.-X. Qian, Stability analysis for neural dynamics with time-varying delays, IEEE Trans. Neural Netw., 9 (1998), 221-223
##[14]
L.-H. Huang, J.-F. Wang, X.-N. Zhou , Existence and global asymptotic stability of periodic solutions for Hopfield neural networks with discontinuous activations, Nonlinear Anal. Real World Appl., 10 (2009), 1651-1661
##[15]
B. Kosko, Adaptive bidirectional associative memories, Appl. Opt., 26 (1987), 4947-4960
##[16]
B. Kosko, Bidirectional associative memories, IEEE Trans. Systems Man Cybernet., 18 (1988), 46-60
##[17]
S. Lakshmanan, J. H. Park, T. H. Lee, H. Y. Jung, R. Rakkiyappan , Stability criteria for BAM neural networks with leakage delays and probabilistic time-varying delays, Appl. Math. Comput., 219 (2013), 9408-9423
##[18]
J. P. LaSalle, The stability of dynamical systems, With an appendix: ”Limiting equations and stability of nonautonomous ordinary differential equations” by Z. Artstein, Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa. (1976)
##[19]
L.-P. Li, L.-H. Huang, Dynamical behaviors of a class of recurrent neural networks with discontinuous neuron activations, Appl. Math. Model., 33 (2009), 4326-4336
##[20]
H.-F. Li, H.-J. Jiang, C. Hu , Existence and global exponential stability of periodic solution of memristor-based BAM neural networks with time-varying delays, Neural Netw., 75 (2016), 97-109
##[21]
Y. Li, Z. H. Lin, Periodic solutions of differential inclusions, Nonlinear Anal., 24 (1995), 631-641
##[22]
B.-W. Liu , Global exponential stability for BAM neural networks with time-varying delays in the leakage terms, Nonlinear Anal. Real World Appl., 14 (2013), 559-566
##[23]
X.-Y. Liu, N. Jiang, J.-D. Cao, S.-M. Wang, Z.-X. Wang, Finite-time stochastic stabilization for BAM neural networks with uncertainties, J. Franklin Inst., 350 (2013), 2109-2123
##[24]
N. G. Lloyd, Degree theory, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge-New York- Melbourne (1978)
##[25]
R. Rakkiyappan, R. Sasirekha, Asymptotic synchronization of continuous/discrete complex dynamical networks by optimal partitioning method, Complexity, 21 (2015), 193-210
##[26]
D.-S. Wang, L.-H. Huang, Periodicity and global exponential stability of generalized Cohen-Grossberg neural networks with discontinuous activations and mixed delays , Neural Netw., 51 (2014), 80-95
##[27]
D.-S. Wang, L.-H. Huang, Z.-W. Cai, On the periodic dynamics of a general Cohen-Grossberg BAM neural networks via differential inclusions, Neurocomputing, 118 (2013), 203-214
##[28]
L.-M. Wang, Y. Shen, G.-D. Zhang , General decay synchronization stability for a class of delayed chaotic neural networks with discontinuous activations, Neurocomputing, 179 (2016), 169-175
##[29]
H.-Q. Wu, Y.-W. Li, Existence and stability of periodic solution for BAM neural networks with discontinuous neuron activations , Comput. Math. Appl., 56 (2008), 1981-1993
##[30]
H.-Q. Wu, C.-H. Shan , Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses, Appl. Math. Model., 33 (2009), 2564-2574
##[31]
L.-L. Zhang, L.-H. Huang, Z.-W. Cai, Finite-time stabilization control for discontinuous time-delayed networks: new switching design, Neural Netw., 75 (2016), 84-96
]
Weak convergence of a modified subgradient extragradient algorithm for monotone variational inequalities in Banach spaces
Weak convergence of a modified subgradient extragradient algorithm for monotone variational inequalities in Banach spaces
en
en
Applying the generalized projection operator, we introduce a modified subgradient extragradient algorithm in Banach spaces for a variational inequality involving a monotone Lipschitz continuous
mapping which is more general than an inverse-strongly-monotone mapping. Weak convergence of the iterative algorithm is also proved. An advantage of the algorithm is the computation of only one value of the inequality
mapping and one projection onto the admissible set per one iteration.
5483
5494
Ying
Liu
College of Mathematics and Information Science
Hebei University
China
ly_cyh2013@163.com
Hang
Kong
College of Mathematics and Information Science
Hebei University
China
1969849957@qq.com
Subgradient extragradient method
generalized projection operator
monotone mapping
variational inequality
Lipschitz continuous
weakly sequentially continuous
Article.30.pdf
[
[1]
Y. I. Al’ber, S. Reich , An iterative method for solving a class of nonlinear operator equations in Banach spaces , Panamer. Math. J., 4 (1994), 39-54
##[2]
K. Ball, E. A. Carlen, E. H. Lieb, Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math., 115 (1994), 463-482
##[3]
N. Buong , Strong convergence theorem of an iterative method for variational inequalities and fixed point problems in Hilbert spaces , Appl. Math. Comput., 217 (2010), 322-329
##[4]
L.-C. Ceng, N. Hadjisavvas, N.-C. Wong , Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems , J. Global Optim., 46 (2010), 635-646
##[5]
Y. Censor, A. Gibali, S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Methods Softw., 26 (2011), 827-845
##[6]
Y. Censor, A. Gibali, S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space , J. Optim. Theory Appl., 148 (2011), 318-335
##[7]
J.-M. Chen, L.-J. Zhang, T.-G. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, J. Math. Anal. Appl., 334 (2007), 1450-1461
##[8]
C.-J. Fang, Y.Wang, S.-K. Yang , Two algorithms for solving single-valued variational inequalities and fixed point problems, J. Fixed Point Theory Appl., 18 (2016), 27-43
##[9]
H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive nonself-mappings and inverse-strongly-monotone mappings, J. Convex Anal., 11 (2004), 69-79
##[10]
H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal., 61 (2005), 341-350
##[11]
H. Iiduka, W. Takahashi , Weak convergence of a projection algorithm for variational inequalities in a Banach space, J. Math. Anal. Appl., 339 (2008), 668-679
##[12]
G. M. Korpelevič, An extragradient method for finding saddle points and for other problems, (Russian) Èkonom. i Mat. Metody, 12 (1976), 747-756
##[13]
R. Kraikaew, S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163 (2014), 399-412
##[14]
J.-L. Lions, Optimal control of systems governed by partial differential equations, Translated from the French by S. K. Mitter, Die Grundlehren der mathematischenWissenschaften, Band 170 Springer-Verlag, New York-Berlin (1971)
##[15]
Y. Liu, Strong convergence theorem for relatively nonexpansive mapping and inverse-strongly-monotone mapping in a Banach space, Appl. Math. Mech. (English Ed.), 30 (2009), 925-932
##[16]
Y. Liu, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Banach spaces, J. Nonlinear Sci. Appl., 10 (2017), 395-409
##[17]
Y. V. Malitsky, V. V. Semenov , An extragradient algorithm for monotone variational inequalities, Translation of Kibernet. Sistem. Anal., 2014 (2014), 125–131, Cybernet. Systems Anal., 50 (2014), 271-277
##[18]
S.-Y. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory, 134 (2005), 257-266
##[19]
N. Nadezhkina, W. Takahashi , Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitzcontinuous monotone mappings, SIAM J. Optim., 16 (2006), 1230-1241
##[20]
K. Nakajo, Strong convergence for gradient projection method and relatively nonexpansive mappings in Banach spaces , Appl. Math. Comput., 271 (2015), 251-258
##[21]
L. D. Popov, A modification of the Arrow-Hurwitz method of search for saddle points, (Russian) Mat. Zametki, 28 (1980), 777-784
##[22]
Y. Takahashi, K. Hashimoto, M. Kato, On sharp uniform convexity, smoothness, and strong type, cotype inequalities, J. Nonlinear Convex Anal., 3 (2002), 267-281
##[23]
W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428
##[24]
K.-K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301-308
##[25]
A. R. Tufa, H. Zegeye, An algorithm for finding a common point of the solutions of fixed point and variational inequality problems in Banach spaces, Arab. J. Math. (Springer), 4 (2015), 199-213
]
Existence of fixed points for \(\gamma\)-\(FG\)-contractive condition via cyclic \((\alpha,\beta)\)-admissible mappings in \(b\)-metric spaces
Existence of fixed points for \(\gamma\)-\(FG\)-contractive condition via cyclic \((\alpha,\beta)\)-admissible mappings in \(b\)-metric spaces
en
en
In this paper, we introduce a new concept of cyclic \((\alpha,\beta)\)-type \(\gamma\)-\(FG\)-contractive mapping
and we prove some fixed point theorems for such mappings in complete \(b\)-metric spaces.
Suitable examples are introduced to verify the main results.
As an application, we obtain sufficient conditions for the existence of solutions for
nonlinear integral equation which are illustrated by an example.
5495
5508
Saroj Kumar
Padhan
Department of Mathematics
Veer Surendra Sai University of Technology
INDIA
skpadhan_math@vssut.ac.in
G V V Jagannadha
Rao
Department of Mathematics
Veer Surendra Sai University of Technology
INDIA
gvvjagan1@gmail.com
Ahmed
Al-Rawashdeh
Department of Mathematical sciences
UAE University
U.A.E
aalrawashdeh@uaeu.ac.ae
Hemant Kumar
Nashine
Department of Mathematics
Texas A \(\&\) M University
U.S.A
drhknashine@gmail.com
Ravi P.
Agarwal
Department of Mathematics
Texas A \(\&\) M University
U.S.A
agarwal@tamuk.edu
\(b\)-metric space
cyclic \((\alpha
\beta)\)-admissible
\(\gamma\)-\(FG\)-contractive
Article.31.pdf
[
[1]
A. Aghajani, M. Abbas, J. R. Roshan , Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca, 64 (2014), 941-960
##[2]
J. Ahmad, A. Al-Rawashdeh, A. Azam, New fixed point theorems for generalized F-contractions in complete metric spaces, Fixed Point Theory Appl., 2015 (2015), 1-18
##[3]
S. Alizadeh, F. Moradlou, P. Salimi , Some fixed point results for \((\alpha,\beta)-(\psi,\phi )\)-contractive mappings, Filomat, 28 (2014), 635-647
##[4]
A. Al-Rawashdeh, J. Ahmad , Common fixed point theorems for JS-contractions, Bull. Math. Anal. Appl., 8 (2016), 12-22
##[5]
M. Boriceanu, M. Bota, A. Petruşel , Multivalued fractals in b-metric spaces, Cent. Eur. J. Math., 8 (2010), 367-377
##[6]
S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5-11
##[7]
W. A. Kirk, P. S. Srinivasan, P. Veeramani , Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4 (2003), 79-89
##[8]
V. Parvaneh, N. Hussain, Z. Kadelburg, Generalized Wardowski type fixed point theorems via \(\alpha\)-admissible FG- contractions in b-metric spaces, Acta Math. Sci. Ser. B Engl. Ed., 36 (2016), 1445-1456
##[9]
H. Piri, P. Kumam , Some fixed point theorems concerning F-contraction in complete metric spaces, Fixed Point Theory Appl., 2014 (2014), 1-11
##[10]
B. Samet, C. Vetro, P. Vetro, Fixed point theorems for \(\alpha-\psi\)-contractive type mappings, Nonlinear Anal., 75 (2012), 2154-2165
##[11]
D. Wardowski , Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), 1-6
##[12]
D. Wardowski, N. Van Dung, Fixed points of F-weak contractions on complete metric spaces, Demonstr. Math., 47 (2014), 146-155
]
Some fixed point results for \(\alpha\)-nonexpansive maps on partial metric spaces
Some fixed point results for \(\alpha\)-nonexpansive maps on partial metric spaces
en
en
In this paper, we prove some fixed point results for a class of \(\alpha\)-nonexpansive single and multi-valued mappings in the setting of partial metric spaces.
Our results generalize the analogous ones of Vetro [F. Vetro, Filomat, \(\bf 29\) (2015), 2011--2020]. Some examples are presented making our results effective.
5509
5527
Hassen
Aydi
Department of Mathematics, College of Education of Jubail
Department of Medical Research
Imam Abdulrahman Bin Faisal University
China Medical University Hospital, China Medical University
Saudi Arabia
Taiwan
hmaydi@uod.edu.sa
Abdelbasset
Felhi
Department of Mathematics and Physics, Preparatory Engineering Institute, Bizerte
Carthage University
Tunisia
abdelbassetfelhi@gmail.com
Fixed point
nonexpansive mapping
partial metric space
Article.32.pdf
[
[1]
T. Abdeljawad, H. Aydi, E. Karapınar, Coupled fixed points for Meir-Keeler contractions in ordered partial metric spaces , Math. Probl. Eng., 2012 (2012), 1-20
##[2]
I. Altun, H. Simsek, Some fixed point theorems on dualistic partial metric spaces, J. Adv. Math. Stud., 1 (2008), 1-8
##[3]
H. Aydi, M. Abbas, C. Vetro, Partial Hausdorff metric and Nadler’s fixed point theorem on partial metric spaces, Topology Appl., 159 (2012), 3234-3242
##[4]
H. Aydi, M. Abbas, C. Vetro , Common fixed points for multivalued generalized contractions on partial metric spaces , Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM, 108 (2014), 483-501
##[5]
H. Aydi, M. Barakat, A. Felhi, H. Işık , On \(\phi\)-contraction type couplings in partial metric spaces , J. Math. Anal., 8 (2017), 78-89
##[6]
H. Aydi, M. Jellali, E. Karapınar , Common fixed points for generalized \(\alpha\)-implicit contractions in partial metric spaces: consequences and application, Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM, 109 (2015), 367-384
##[7]
H. Aydi, E. Karapınar , New Meir-Keeler type tripled fixed-point theorems on ordered partial metric spaces, Math. Probl. Eng., 2012 (2012), 1-17
##[8]
H. Aydi, E. Karapınar, W. Shatanawi, Coupled fixed point results for (\(\psi,\phi\))-weakly contractive condition in ordered partial metric spaces, Comput. Math. Appl., 62 (2011), 4449-4460
##[9]
H. Aydi, E. Karapınar, C. Vetro , On Ekeland’s variational principle in partial metric spaces, Appl. Math. Inf. Sci., 9 (2015), 257-262
##[10]
H. Aydi, C. Vetro, W. Sintunavarat, P. Kumam, Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces , Fixed Point Theory Appl., 2012 (2012), 1-18
##[11]
L. Ćirić, B. Samet, H. Aydi, C. Vetro, Common fixed points of generalized contractions on partial metric spaces and an application , Appl. Math. Comput., 218 (2011), 2398-2406
##[12]
M. Edelstein , On nonexpansive mappings, Proc. Amer. Math. Soc., 15 (1964), 689-695
##[13]
K. Goebel, W. A. Kirk , Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1990)
##[14]
M. A. Khamsi, S. Reich, Nonexpansive mappings and semigroups in hyperconvex spaces , Math. Japon., 35 (1990), 467-471
##[15]
S. G. Matthews, Partial metric topology, Papers on general topology and applications, Flushing, NY, (1992), Ann. New York Acad. Sci., New York Acad. Sci., New York, 728 (1994), 183-197
##[16]
O. Popescu, Some new fixed point theorems for \(\alpha\)-Geraghty contraction type maps in metric spaces, Fixed Point Theory Appl., 2014 (2014), 1-12
##[17]
S. Reich, I. Shafrir, The asymptotic behavior of firmly nonexpansive mappings, Proc. Amer. Math. Soc., 101 (1987), 246-250
##[18]
B. Samet, E. Karapınar, H. Aydi, V. Ćojbašić Rajić, Discussion on some coupled fixed point theorems, Fixed Point Theory Appl., 2013 (2013), 1-12
##[19]
T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 340 (2008), 1088-1095
##[20]
F. Vetro , Fixed point results for nonexpansive mappings on metric spaces, Filomat, 29 (2015), 2011-2020
]
On nonlinear implicit fractional differential equations without compactness
On nonlinear implicit fractional differential equations without compactness
en
en
The main purpose of this research paper is to develop some sufficient
conditions for the existence of solution of a nonlinear problem of
implicit fractional differential equations (IFDEs) with boundary conditions, using
prior estimate method. The distinction of the method applied here
is, it does not require compactness of the operator. This idea is
the result of motivation from the book of O'Regan and other [R. P. Agarwal, D. O'Regan, Y. J. Cho, Y.-Q. Chen, Taylor and Francis Group, New York, (2006)].
Devising the respective conditions, we also developed some conditions
for Hyers-Ulam type stability to the solution of the said problem.
To justify the relevant results a suitable example is provided.
5528
5539
Samia
Bushnaq
Department of Science
Princess Sumaya University for Technology
Jordan
S.Bushnaq@psut.edu.jo
Wajid
Hussain
Department of Mathematics
University of Qurtuba
Pakistan
whussain598@gmail.com
Kamal
Shah
Department of Mathematics
University of Malakand
Pakistan
kamalshah408@gmail.com
Implicit fractional differential equations
Brouwer degree
topological degree theory
boundary conditions
Banach contraction theorem
Lebesgue dominated convergence theorem
Article.33.pdf
[
[1]
S. Abbas, M. Benchohra, G. M. N'Guérékata, Topics in fractional differential equations, Springer, New York (2012)
##[2]
R. P. Agarwal, D. Baleanu, S. Rezapour, S. Salehi , The existence of solutions for some fractional finite difference equations via sum boundary conditions, Adv. Difference Equ., 2014 (2014), 1-16
##[3]
R. P. Agarwal, M. Benchohra, S. Hamani , Boundary value problems for fractional differential equations, Georgian Math. J., 16 (2009), 401-411
##[4]
R. P. Agarwal, D. O’Regan, Y. J. Cho, Y.-Q. Chen, Topological Degree Theory and its Applications, Taylor and Francis Group, New York (2006)
##[5]
Z. Bai , On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal., 72 (2010), 916-924
##[6]
D. Baleanu, R. P. Agarwal, H. Khan, R. A. Khan, H. Jafari , On the existence of solution for fractional differential equations of order \(3 < \delta\leq 4\) , Adv. Difference Equ., 2015 (2015), 1-9
##[7]
D. Baleanu, R. P. Agarwal, H. Mohammad, S. Rezapour, Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces, Bound. Value Probl., 2013 (2013), 1-8
##[8]
M. Benchohra, J. R. Graef, S. Hamani, Existence results for boundary value problems with non-linear fractional differential equations, Appl. Anal., 87 (2008), 851-863
##[9]
M. Benchohra, S. Hamani, S. K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl., 3 (2008), 1-12
##[10]
M. Benchohra, J. E. Lazreg, Existence results for nonlinear implicit fractional differential equations, Surv. Math. Appl., 9 (2014), 79-92
##[11]
K. Diethelm, N. J. Ford, Analysis of Fractional Differential Equations, J. Math. Anal. Appl., 265 (2002), 229-248
##[12]
C. S. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett., 23 (2010), 1050-1055
##[13]
D. H. Hyers , On the stability of the linear functional equations, Proc. Nat. Acad. Sci., 27 (1941), 222-224
##[14]
F. Isaia, On a nonlinear integral equation without compactness, Acta Math. Univ. Comenian., 75 (2006), 233-240
##[15]
S.-M. Jung , Hyers-Ulam Stability of linear differential equations of first order , Appl. Math. Lett., 17 (2004), 1135-1140
##[16]
R. A. Khan, M. Rehman, J. Henderson , Existence and uniqueness of solutions for nonlinear fractional differential equations with integral boundary conditions, Fract. Differ. Calc., 1 (2011), 29-43
##[17]
R. A. Khan, K. Shah, Existence and uniqueness of solutions to fractional order multi-point boundary value problems, Commun. Appl. Anal., 19 (2015), 515-526
##[18]
A. A. Kilbas, O. I. Marichev, S. G. Samko, Fractional Integrals and Derivatives , (Theory and Applications), Gordon and Breach, Switzerland (1993)
##[19]
V. Lakshmikantham, S. Leela, J. V. Devi , Theory of Fractional Dynamic Systems, CSP, UK (2009)
##[20]
L. Lv, J.-R. Wang, W. Wei, Existence and uniqueness results for fractional differential equations with boundary value conditions, Opuscula Math., 31 (2011), 629-643
##[21]
R. L. Magin, Fractional calculus in bioengineering-part 2, Crit. Rev. Biomed. Eng., 32 (2004), 105-193
##[22]
J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, Am. Math. Soc., United States of America (1979)
##[23]
K. S. Miller, B. Ross, An Introduction to the fractional Calculus and fractional Differential Equations, Wiley, New York (1993)
##[24]
N. Nyamoradi, D. Baleanu, R. P. Agarwal , Existence and uniqueness of positive solutions to fractional boundary value problems with nonlinear boundary conditions, Adv. Difference Equ., 2013 (2013), 1-11
##[25]
N. Nyamoradi, D. Baleanu, R. P. Agarwal, On a Multipoint Boundary Value Problem for a Fractional Order Differential Inclusion on an Infinite Interval , Adv. Math. Phys., 2013 (2013), 1-9
##[26]
I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, , San Diego (1999)
##[27]
T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300
##[28]
T. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62 (2000), 23-130
##[29]
W. Soedel , Vibrations of Shells and Plates, Dekker, New York (2004)
##[30]
I. M. Stamova, Mittag-Leffler stability of impulsive differential equations of fractional order, Quart. Appl. Math., 73 (2015), 525-535
##[31]
S. P. Timoshenko, Theory of Elastic Stability, McGraw-Hill Book Co., New York (1961)
##[32]
J. C. Trigeassou, N. Maamri, J. Sabatier, A. Oustaloup, A Lyapunov approach to the stability of fractional differential equations, Signal Processing, 91 (2011), 437-445
##[33]
S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publishers, New York (1960)
##[34]
J.-R. Wang, X.-Z. Li , Ulam-Hyers stability of fractional Langevin equations, Appl. Math. Comput., 258 (2015), 72-83
##[35]
J.-R. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 2011 (2011), 1-10
##[36]
J.-R. Wang, Y. Zhou, W. Wei , Study in fractional differential equations by means of topological degree methods , Numer. Func. Anal. Optim., 33 (2012), 216-238
##[37]
Y. Zhao, S. Sun, Z. Han, Q.-P. Li , The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 2086-2097
]
Minimizing the object space error for pose estimation: towards the most efficient algorithm
Minimizing the object space error for pose estimation: towards the most efficient algorithm
en
en
In this paper, we present an efficient branch-and-bound algorithm to
globally minimize the object space error for the camera pose
estimation. The key idea is to reformulate the pose estimation
model using the optimal Lagrangian multipliers.
Numerical simulation results show that our algorithm usually
terminates in the first iteration and finds an \(\epsilon\)-suboptimal
solution.
Furthermore, the efficiency of our algorithm is demonstrated by a comprehensive numerical
comparison with two well-known heuristics. We also demonstrate the computational
power of our algorithm by comparing it with the state-of-the-art
global optimization package BARON.
5540
5551
Yingwei
Han
School of Economics and Management
Beihang University
China
hanyingwei@buaa.edu.cn
Yong
Xia
School of Mathematics and System Sciences
Beihang University
China
yxia@buaa.edu.cn
Ping
Li
School of Economics and Management
Beihang University
China
liping124@buaa.edu.cn
Pose estimation
PnP
robotics
branch-and-bound
Lagrangian dual
Article.34.pdf
[
[1]
S. Agarwal, M. K. Chandraker, F. Kahl, D. Kriegman, S. Belongie, Practical global optimization for multiview geometry, Comput. Vis. ECCV, Graz, Austria, May 7–13, Springer, Berlin, Heidelberg, (2006), 592-605
##[2]
A. Ansar, K. Daniilidis, Linear pose estimation from points or lines, Comput. Vis. ECCV, Denmark, May 28–31, Springer, Berlin, Heidelberg, (2002), 282-296
##[3]
K. Anstreicher, X. Chen, H. Wolkowicz, Y.-X. Yuan , Strong duality for a trust-region type relaxation of the quadratic assignment problem, Linear Algebra Appl., 301 (1999), 121-136
##[4]
V. Balakrishnan, S. Boyd, S. Balemi, Branch and bound algorithm for computing the minimum stability degree of parameter-dependent linear systems, Internat. J. Robust Nonlinear Control, 1 (1991), 295-317
##[5]
H. P. Benson, Using concave envelopes to globally solve the nonlinear sum of ratios problem, Dedicated to Professor Reiner Horst on his 60th birthday, J. Global Optim., 22 (1936), 343-364
##[6]
C. Eckart, G. Yong, The approximation of one matrix by another of lower rank, Psychometrika, 1 (1936), 211-218
##[7]
P. D. Fiore, Efficient linear solution of exterior orientation, IEEE Trans. Pattern Anal. Mach. Intell., 23 (2001), 140-148
##[8]
R. Hartley, F. Kahl, Global Optimization through Searching Rotation Space and Optimal Estimation of the Essential Matrix, International Conference on Computer Vision, (2007), 1-8
##[9]
R. Hartley, F. Kahl , Global optimization through rotation space search, Int. J. Comput. Vis., 82 (2009), 64-79
##[10]
H. Hmam, J. Kim, Optimal non-iterative pose estimation via convex relaxation, Image Vis. Comput., 28 (2010), 1515-1523
##[11]
R. Hartley, A. Zisserman , Multiple view geometry in computer vision, Second edition, With a foreword by Olivier Faugeras, Cambridge University Press, Cambridge, (2003), -
##[12]
S.-Q. Li, C. Xu, M. Xie, A robust O(n) solution to the perspective-n-point problem, IEEE Trans. Pattern Anal. Mach. Intell., 34 (2012), 1444-1450
##[13]
C.-P. Lu, G. D. Hager, E. Mjolsness, Fast and globally convergent pose estimation from video images, IEEE Trans. Pattern Anal. Mach. Intell., 22 (2000), 610-622
##[14]
F. Moreno-Noguer, V. Lepetit, P. Fua, Accurate non-iterative O(n) solution to the PnP problem, IEEE 11th International Conference on Computer Vision, Rio de Janeiro, Brazil, (2007), 1-8
##[15]
C. Olsson, F. Kahl, M. Oskarsson, Optimal estimation of perspective camera pose , 18th International Conference on Pattern Recognition, Hong Kong, China, 2 (2006), 5-8
##[16]
L. Quan, Z.-D. Lan , Linear n-point camera pose determination, IEEE Trans. Pattern Anal. Mach. Intell, 21 (1999), 774-780
##[17]
M. Sarkis, K. Diepold, Camera-pose estimation via projective Newton optimization on the manifold, IEEE Trans. Image Process., 21 (2012), 1729-1741
##[18]
G. Schweighofer, A. Pinz, Robust pose estimation from a planar target, IEEE Trans. Pattern Anal. Mach. Intell., 28 (2006), 2024-2030
##[19]
G. Schweighofer, A. Pinz, Fast and Globally Convergent Structure and Motion Estimation for General Camera Models, Proceedings of the 17th British Machine Vision Conference, (2006), 147-156
##[20]
G. Schweighofer, A. Pinz , Globally optimal O(n) solution to the PnP problem for general camera models, Proceedings of the 19th British Machine Vision Conference, (2008), 1-10
##[21]
O. Tahri, H. Araujo, Y. Mezouar, F. Chaumette, Efficient iterative pose estimation using an invariant to rotations, IEEE Trans. Cybern., 44 (2014), 199-207
##[22]
J. F. Sturm , Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw., 11 (1990), 625-653
##[23]
M. Tawarmalani, N. V. Sahinidis , Semidefinite relaxations of fractional programs via novel convexification techniques, J. Global Optim., 20 (2001), 133-154
##[24]
Y. Xia, Global optimization of a class of nonconvex quadratically constrained quadratic programming problems, Acta Math. Sin. (Engl. Ser.), 27 (2011), 1803-1812
##[25]
X. S. Zhou, S. I. Roumeliotis, Determining 3-D relative transformations for any combination of range and bearing measurements, IEEE Trans. Robot., 29 (2013), 458-474
]
Stochastic shadowing analysis of a class of stochastic differential equations
Stochastic shadowing analysis of a class of stochastic differential equations
en
en
This paper is devoted to the feasibility of stochastic shadowing of a class of stochastic differential equations via numerical analysis tools. A general shadowing theorem of stochastic differential equations is proven, and an explicit relationship of shadowing and the coefficients of SDE, a bound for shadowing distance are both investigated. The focus is explicit regularity conditions of stochastic differential equations which can ensure the shadowing.
A numerical experiment is provided to illustrate the effectiveness of the proposed theorem by the numerical simulations of chaotic orbits of the stochastic Lorenz equations.
5552
5565
Qingyi
Zhan
College of Computer and Information Science
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science
Fujian Agriculture and Forestry University
Chinese Academy of Sciences
P. R. China
P. R. China
zhanqy@lsec.cc.ac.cn
Yuhong
Li
College of Hydropower and Information Engineering
Huazhong University of Science and Technology
P. R. China
liyuhong@hust.edu.cn
Stochastic differential equations
random dynamical systems
numerical method
shadowing
multiplicative ergodic theorem
stochastic Lorenz equations
Article.35.pdf
[
[1]
L. Arnold , Random dynamical systems, Dynamical systems, Montecatini Terme, (1994), Lecture Notes in Math., Springer, Berlin, 1609 (1995), 1-43
##[2]
L. Arnold, B. Schmalfuss, Lyapunov’s second method for random dynamical systems, J. Differential Equations, 177 (2001), 235-265
##[3]
N. D. Cong , Topological dynamics of random dynamical systems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York (1997)
##[4]
J.-Q. Duan , An introduction to stochastic dynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, New York (2015)
##[5]
A. Fakhari, A. Golmakani , Shadowing properties of random hyperbolic sets, Internat. J. Math., 23 (2012), 1-10
##[6]
C.-R. Feng, Y. Wu, H.-Z. Zhao, Anticipating random periodic solutions, I, SDEs with multiplicative linear noise, J. Funct. Anal., 271 (2016), 365-417
##[7]
G. H. Golub, C. F. Van Loan , Matrix computations, Fourth edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD (2013)
##[8]
L. V. Kantorovich, G. P. Akilov, Functional analysis, Translated from the Russian by Howard L. Silcock, Second edition, Pergamon Press, Oxford-Elmsford, N.Y. (1982)
##[9]
Y.-H. Li, Z. Brzeźniak, J.-Z. Zhou, Conceptual analysis and random attractor for dissipative random dynamical systems, Acta Math. Sci. Ser. B Engl. Ed., 28 (2008), 253-268
##[10]
B. F. Liu, Y. L. Han, X. D. Sun , Square-mean almost periodic solutions for a class of stochastic integro-differential equations, (Chinese) J. Jilin Univ. Sci., 51 (2013), 393-397
##[11]
G. N. Milstein, Numerical integration of stochastic differential equations, Translated and revised from the 1988 Russian original, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (1995)
##[12]
K. Palmer , Shadowing in dynamical systems, Theory and applications, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht (2000)
##[13]
S. Y. Pilyugin, Shadowing in dynamical systems , Lecture Notes in Mathematics, Springer-Verlag, Berlin (1999)
##[14]
T. Tél, M. Gruiz, Chaotic dynamics, An introduction based on classical mechanics, Translated from the 2002 Hungarian original by Katalin Kulacsy, Cambridge University Press, Cambridge (2006)
##[15]
D. Todorov, Stochastic shadowing and stochastic stability, ArXiv, 2014 (2014), 1-16
##[16]
T.-C. Wang, Maximum norm error bound of a linearized difference scheme for a coupled nonlinear Schrödinger equations, J. Comp. Appl. Math., 235 (2011), 4237-4250
##[17]
P. Wang, A-stable Runge-Kutta methods for stiff stochastic differential equations with multiplicative noise, Comput. Appl. Math., 34 (2015), 773-792
##[18]
X.-D. Xie, Q.-Y. Zhan, Uniqueness of limit cycles for a class of cubic system with an invariant straight line, Nonlinear Anal., 70 (2009), 4217-4225
##[19]
Q.-Y. Zhan, Mean-square numerical approximations to random periodic solutions of stochastic differential equations, Adv. Difference Equ., 2015 (2015), 1-17
##[20]
Q.-Y. Zhan, Shadowing orbits of stochastic differential equations, J. Nonlinear Sci. Appl., 9 (2016), 2006-2018
##[21]
Q.-Y. Zhan, Y.-H. Li, Numerical random periodic shadowing orbits of a class of stochastic dierential equations, Chapter 10, Dynamical Systems-Analytical and Computational Techniques, edited by Mahmut Reyhanoglu, InTech, Rijeka (2017)
]
Perturbation resilience of proximal gradient algorithm for composite objectives
Perturbation resilience of proximal gradient algorithm for composite objectives
en
en
In this paper, we study the perturbation resilience of a proximal gradient algorithm
under the general Hilbert space setting. With the assumption that the error sequence
is summable, we prove that the iterative sequence converges weakly to a solution of the composite
optimization problem. We also show the bounded perturbation resilience of this iterative
method and apply it to the lasso problem.
5566
5575
Yanni
Guo
College of Science, Civil Aviation University of China
P. R. China
ynguo@amss.ac.cn
Wei
Cui
College of Science
Civil Aviation University of China
P. R. China
851241788@qq.com
Yansha
Guo
School of Information Technology Engineering
Tianjin University of Technology and Education
P. R. China
gysh337@126.com
Perturbation resilience
proximal gradient algorithm
composite optimization
lasso problem
Article.36.pdf
[
[1]
F. Alvarez, H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Wellposedness in optimization and related topics, Gargnano, (1999), Set-Valued Anal., 9 (2001), 3-11
##[2]
J. B. Baillon, G. Haddad , Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones, (French) Israel J. Math., 26 (1977), 137-150
##[3]
H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, With a foreword by Hédy Attouch, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York (2011)
##[4]
Y. Censor, R. Davidi, G. T. Herman, Perturbation resilience and superiorization of iterative algorithms, Inverse Problems, 26 (2010), 1-12
##[5]
Y. Censor, R. Davidi, G. T. Herman, R. W. Schulte, L. Tetruashvili , Projected subgradient minimization versus superiorization, J. Optim. Theory Appl., 160 (2014), 730-747
##[6]
Y. Censor, A. J. Zaslavski, Convergence and perturbation resilience of dynamic string-averaging projection methods, Comput. Optim. Appl., 54 (2013), 65-76
##[7]
Y. C. Cheng , On the gradient-projection method for solving the nonsymmetric linear complementarity problem, J. Optim. Theory Appl., 43 (1984), 527-541
##[8]
R. Davidi, Y. Censor, R. W. Schulte, S. Geneser, L. Xing, Feasibility-seeking and superiorization algorithms applied to inverse treatment planning in radiation therapy, Infinite products of operators and their applications, Contemp. Math., Israel Math. Conf. Proc., Amer. Math. Soc., Providence, RI, 636 (2015), 83-92
##[9]
E. S. Helou, M. V. W. Zibetti, E. X. Miqueles, Superiorization of incremental optimization algorithms for statistical tomographic image reconstruction, Inverse Problems, 33 (2017), 1-26
##[10]
W.-M. Jin, Y. Censor, M. Jiang, Bounded perturbation resilience of projected scaled gradient methods, Comput. Optim. Appl., 63 (2016), 365-392
##[11]
A. Kaplan, R. Tichatschke, On inexact generalized proximal methods with a weakened error tolerance criterion, Optimization, 63 (2004), 3-17
##[12]
P.-L. Lions, B. Mercier , Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979
##[13]
Z.-Q. Luo, P. Tseng, Error bounds and convergence analysis of feasible descent methods: a general approach, Degeneracy in optimization problems, Ann. Oper. Res., 46/47 (1993), 157-178
##[14]
J. J. Moreau , Proximité et dualité dans un espace hilbertien, (French) Bull. Soc. Math. France, 93 (1965), 273-299
##[15]
Y. Nesterov, Gradient methods for minimizing composite objective function , CORE Discussion Papers, 140 (2007), 125-161
##[16]
T. Nikazad, R. Davidi, G. T. Herman, Accelerated perturbation-resilient block-iterative projection methods with application to image reconstruction , Inverse Problems, 28 (2012), 1-19
##[17]
G. B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl., 72 (1979), 383-390
##[18]
B. T. Polyak , Introduction to optimization, Translated from the Russian. With a foreword by Dimitri P. Bertsekas, Translations Series in Mathematics and Engineering, Optimization Software, Inc., Publications Division, New York (1987)
##[19]
S. A. Santos, R. C. M. Silva, An inexact and nonmonotone proximal method for smooth unconstrained minimization , J. Comput. Appl. Math., 269 (2014), 86-100
##[20]
M. J. Schrapp, G. T. Herman, Data fusion in X-ray computed tomography using a superiorization approach, Rev. Sci. Instrum., 85 (2014), 1-9
##[21]
H.-K. Xu , Properties and iterative methods for the lasso and its variants, Chin. Ann. Math. Ser. B, 35 (2014), 501-518
##[22]
H.-K. Xu, Bounded perturbation resilience and superiorization techniques for the projected scaled gradient method, Inverse Problems, 33 (2017), 1-19
##[23]
Y.-H. Yao, N. Shahzad, Strong convergence of a proximal point algorithm with general errors, Optim. Lett., 6 (2012), 621-628
##[24]
A. J. Zaslavski, Convergence of a proximal point method in the presence of computational errors in Hilbert spaces, SIAM J. Optim., 20 (2010), 2413-2421
]
Existence of solutions to boundary value problems for a higher-dimensional difference system
Existence of solutions to boundary value problems for a higher-dimensional difference system
en
en
By using critical point theory, some new criteria are obtained for the existence of a
nontrivial homoclinic orbit to a higher order difference system
containing both many advances and retardations. The proof is based
on the Mountain Pass Lemma in combination with periodic
approximations. Related results in the literature are generalized
and improved.
5576
5584
Tao
Zhou
School of Business Administration
South China University of Technology
China
zhoutaoscut@hotmail.com
Xia
Liu
Oriental Science and Technology College
Science College
Hunan Agricultural University
Hunan Agricultural University
China
China
xia991002@163.com
Haiping
Shi
Modern Business and Management Department
Guangdong Construction Polytechnic
China
shp7971@163.com
Boundary value problems
higher-dimensional
mountain pass lemma
critical point theory
Article.37.pdf
[
[1]
R. P. Agarwal, Difference equations and inequalities, Theory, methods, and applications, Second edition, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York (2000)
##[2]
X.-C. Cai, J.-S. Yu, Existence of periodic solutions for a 2nth-order nonlinear difference equation, J. Math. Anal. Appl., 329 (2007), 870-878
##[3]
P. Chen, X.-H. Tang, Existence of homoclinic orbits for 2nth-order nonlinear difference equations containing both many advances and retardations, J. Math. Anal. Appl., 381 (2011), 485-505
##[4]
P. Cull, M. Flahive, R. Robson, Difference equations , From rabbits to chaos, Undergraduate Texts in Mathematics, Springer, New York (2005)
##[5]
X.-Q. Deng, Nonexistence and existence results for a class of fourth-order difference mixed boundary value problems, J. Appl. Math. Comput., 45 (2014), 1-14
##[6]
X.-Q. Deng, H.-P. Shi, On boundary value problems for second order nonlinear functional difference equations, Acta Appl. Math., 110 (2010), 1277-1287
##[7]
X.-Q. Deng, H.-P. Shi, X.-L. Xie, Periodic solutions of second order discrete Hamiltonian systems with potential indefinite in sign, Appl. Math. Comput., 218 (2011), 148-156
##[8]
C.-J. Guo, D. O’Regan, Y.-T. Xu, R. P. Agarwal, Existence of subharmonic solutions and homoclinic orbits for a class of even higher order differential equations, Appl. Anal., 90 (2011), 1169-1183
##[9]
C.-J. Guo, D. O’Regan, Y.-T. Xu, R. P. Agarwal, Existence and multiplicity of homoclinic orbits of a second-order differential difference equation via variational methods, Appl. Math. Inform. Mech., 4 (2012), 1-15
##[10]
C.-J. Guo, D. O’Regan, Y.-T. Xu, R. P. Agarwal, Existence of homoclinic orbits for a class of first-order differential difference equations, Acta Math. Sci. Ser. B Engl. Ed., 35 (2015), 1077-1094
##[11]
C.-J. Guo, Y.-T. Xu, Existence of periodic solutions for a class of second order differential equation with deviating argument, J. Appl. Math. Comput., 28 (2008), 425-433
##[12]
R.-H. Hu, L.-H. Huang, Existence of periodic solutions of a higher order difference system, J. Korean Math. Soc., 45 (2008), 405-423
##[13]
M. Jia, Standing waves for discrete nonlinear Schrödinger equations, Electron. J. Differential Equations, 2016 (2016), 1-9
##[14]
J.-H. Leng, Existence of periodic solutions for higher-order nonlinear difference equations, Electron. J. Differential Equations, 2016 (2016), 1-10
##[15]
J.-H. Leng, Periodic and subharmonic solutions for 2nth-order \(\phi_c\)-Laplacian difference equations containing both advance and retardation, Indag. Math. (N.S.), 27 (2016), 902-913
##[16]
X. Liu, H.-P. Shi, Y.-B. Zhang, Nonexistence and existence of solutions for a fourth-order discrete mixed boundary value problem, Proc. Indian Acad. Sci. Math. Sci., 124 (2014), 179-191
##[17]
X. Liu, H.-P. Shi, Y.-B. Zhang, On the nonexistence and existence of solutions for a fourth-order discrete Dirichlet boundary value problem, Quaest. Math., 38 (2015), 203-216
##[18]
X. Liu, Y.-B. Zhang, H.-P. Shi, Existence and nonexistence results for a 2n-th order p-Laplacian discrete Dirichlet boundary value problem, translated from Izv. Nats. Akad. Nauk Armenii Mat., 49 (2014), 133–143, J. Contemp. Math. Anal., 49 (2014), 287-295
##[19]
X. Liu, Y.-B. Zhang, H.-P. Shi, Existence and nonexistence results for a fourth-order discrete Neumann boundary value problem, Studia Sci. Math. Hungar., 51 (2014), 186-200
##[20]
X. Liu, Y.-B. Zhang, H.-P. Shi, Nonexistence and existence results for a 2nth-order discrete mixed boundary value problem, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 109 (2015), 303-314
##[21]
J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, Springer- Verlag, New York (1989)
##[22]
G. Molica Bisci, D. Repovš, Existence of solutions for p-Laplacian discrete equations, Appl. Math. Comput., 242 (2014), 454-461
##[23]
P. H. Rabinowitz , Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1986)
##[24]
H. Sedaghat, Nonlinear difference equations, Theory with applications to social science models, Mathematical Modelling: Theory and Applications, Kluwer Academic Publishers, Dordrecht (2003)
##[25]
H.-P. Shi , Boundary value problems of second order nonlinear functional difference equations, J. Difference Equ. Appl., 16 (2010), 1121-1130
##[26]
H.-P. Shi, Z.-Z. Liu, Z.-G. Wang, Dirichlet boundary value problems for second order p-Laplacian difference equations, Rend. Istit. Mat. Univ. Trieste, 42 (2010), 19-29
##[27]
H.-P. Shi, X. Liu, Y.-B. Zhang, Nonexistence and existence results for a 2nth-order discrete Dirichlet boundary value problem, Kodai Math. J., 37 (2014), 492-505
##[28]
D. Smets, M. Willem, Solitary waves with prescribed speed on infinite lattices , J. Funct. Anal., 149 (1997), 266-275
##[29]
R. R. Stoll, Linear algebra and matrix theory, McGraw-Hill Book Company, Inc., New York–Toronto–London (1952)
]
Some integrability estimates for solutions of the fractional \(p\)-Laplace equation
Some integrability estimates for solutions of the fractional \(p\)-Laplace equation
en
en
For \((\alpha,p)\in(0,1)\times (1,\infty)\), this note focuses on some integrability estimates for solutions of the following Dirichlet problem
\[
\begin{cases}
L_{\alpha,p}u(x)=g(x) \,\, \hbox{as} \,\,x\in \Omega,\\
u(x)=0 \,\, \hbox{as} \,\,x\in \mathbb{R}^{n}\backslash \Omega,
\end{cases}
\]
where \(L_{\alpha,p}\) is the fractional \(p\)-Laplace operator.
5585
5592
Shaoguang
Shi
Department of Mathematics
Linyi University
China
shishaoguang@mail.bnu.edu.cn
Fractional \(p\)-Laplace equation
Dirichlet problem
solution
Article.38.pdf
[
[1]
B. Barrios, I. Peral, S. Vita, Some remarks about the summability of nonlocal nonlinear problems , Adv. Nonlinear Anal., 4 (2015), 91-107
##[2]
C. Bjorland, L. Caffarelli, A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math., 65 (2012), 337-380
##[3]
K. Bogdan, K. Burdzy, Z.-Q. Chen, Censored stable processes, Probab. Theory Related Fields, 127 (2003), 89-152
##[4]
L. A. Caffarelli, S. Salsa, L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461
##[5]
A. Di Castro, T. Kuusi, G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836
##[6]
E. Di Nezza, G. Palatucci, E. Valdinoci , Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573
##[7]
A. Garroni, R. V. Kohn, Some three-dimensional problems related to dielectric breakdown and polycrystal plasticity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 2613-2625
##[8]
Q.-Y. Guan, Z.-M. Ma, Reflected symmetric \(\alpha\)-stable processes and regional fractional Laplacian, Probab. Theory Related Fields, 134 (2006), 649-694
##[9]
R. Hurri-Syrjänen, A. V. Vähäkangas, Fractional Sobolev-Poincar and fractional Hardy inequalities in unbounded John domains, Mathematika, 61 (2015), 385-401
##[10]
A. Iannizzotto, S.-B. Liu, K. Perera, M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101-125
##[11]
N. S. Landkof, Foundations of modern potential theory , Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg (1972)
##[12]
T. Leonori, I. Peral, A. Primo, F. Soria, Basic estimates for solution of elliptic and parabolic equations for a class of nonlocal operators, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 6031-6068
##[13]
E. Lindgren, P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826
##[14]
X. Ros-Oton, J. Serra , The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302
##[15]
S.-G. Shi, J. Xiao, On fractional capacities relative to bounded open Lipschitz sets, Potential Anal., 45 (2016), 261-298
##[16]
S.-G. Shi, J. Xiao, Fractional capacities relative to bounded open Lipschitz sets complemented, Calc. Var. Partial Differential Equations, 56 (2017), 1-22
##[17]
P. Shvartsman, On Sobolev extension domains in \(R^n\), J. Funct. Anal., 258 (2010), 2205-2245
##[18]
E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. (1970)
##[19]
H. Triebel, Function spaces and wavelets on domains, EMS Tracts in Mathematics, European Mathematical Society (EMS), Zürich (2008)
##[20]
J. Xiao, Z. Zhai, Fractional Sobolev, Moser-Trudinger Morrey-Sobolev inequalities under Lorentz norms, Problems in mathematical analysis, J. Math. Sci. (N.Y.), 166 (2010), 357-376
##[21]
Y. Zhou , Fractional Sobolev extension and imbedding , Trans. Amer. Math. Soc., 367 (2015), 959-979
]