]>
2018
11
8
ISSN 2008-1898
67
Cylindrical Carleman's formula of subharmonic functions and its application
Cylindrical Carleman's formula of subharmonic functions and its application
en
en
Our aim in this paper is to prove the cylindrical Carleman's formula for subharmonic functions in a truncated cylinder. As an
application, we prove that if the positive part of a harmonic function in a cylinder satisfies a slowly growing condition, then
its negative part can also be dominated by a similar slowly growing condition, which
improves some classical results about harmonic functions in a cylinder.
947
952
Lei
Qiao
School of Mathematics and Information Science
Henan University of Economics and Law
China
qiaocqu@163.com
Cylindrical Carleman's formula
subharmonic function
cylinder
Article.1.pdf
[
[1]
D. H. Armitage, A Nevanlinna theorem for superharmonic functions in half-spaces, with applications, J. London Math. Soc., 23 (1981), 137-157
##[2]
D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin-New York (1977)
##[3]
Ü. Kuran, Harmonic majorizations in half-balls and half-spaces, Proc. London Math. Soc., 21 (1970), 614-636
##[4]
B. Y. Levin , Entire and subharmonic functions, Adv. Soviet Math., Amer. Math. Soc., Providence, RI (1992)
##[5]
I. Miyamoto, Harmonic functions in a cylinder which vanish on the boundary, Japan. J. Math., 22 (1996), 241-255
##[6]
I. Miyamoto, H. Yoshida , Harmonic functions in a cylinder with normal derivatives vanishing on the boundary, Ann. Polon. Math., 74 (2000), 229-235
##[7]
I. Miyamoto, H. Yoshida, On a covering property of minimally thin sets at infinity in a cylinder, Math. Montisnigri, 20/21 (2007/08), 35-54
##[8]
L. Qiao, Asymptotic behavior of Poisson integrals in a cylinder and its application to the representation of harmonic functions, Bull. Sci. Math., 144 (2018), 39-54
##[9]
A. Y. Rashkovskiı, L. I. Ronkin, Subharmonic functions of finite order in a cone. III., Functions of completely regular growth, J. Math. Sci., 77 (1995), 2929-2940
##[10]
L. I. Ronkin, Functions of completely regular growth , Kluwer Academic Publishers Group , Dordrecht (1992)
##[11]
G. V. Rozenblyum, M. Z. Solomyak, M. A. Shubin, Spectral theory of differential operators, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (1989)
##[12]
H. Yoshida, Harmonic majorization of a subharmonic function on a cone or on a cylinder , Pacific J. Math., 148 (1991), 369-395
]
Ulam-Hyers stability of fractional impulsive differential equations
Ulam-Hyers stability of fractional impulsive differential equations
en
en
In this paper, we first prove the existence and uniqueness for a fractional differential equation with time delay and finite impulses on a compact interval. Secondly, Ulam-Hyers stability of the equation is established by Picard operator and abstract Gronwall's inequality.
953
959
Yali
Ding
School of Mathematics and Statistics
Beijing Institute of Technology
P. R. China
dingyaliding@126.com
Ulam-Hyers stability
fractional order impulsive equation
delay differential equation
Article.2.pdf
[
[1]
S. Abbas, M. Benchohra , Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses, Appl. Math. Comput., 257 (2015), 190-198
##[2]
N. Ali, B. B. Fatima, K. Shah, R. A. Khan, Hyers-Ulam stability of a class of nonlocal boundary value problem having triple solutions, Int. J. Appl. Comput. Math., 4 (2018), 1-12
##[3]
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
##[4]
S. András, A. R. Mészáros , Ulam-Hyers stability of dynamic equations on time scales via Picard operators, Appl. Math. Comput., 219 (2013), 4853-4864
##[5]
J. Brzdęk, N. Eghbali, On approximate solutions of some delayed fractional differential equations, Appl. Math. Lett., 54 (2016), 31-35
##[6]
Z. Gao, X. Yu, J. Wang, Exp-type Ulam-Hyers stability of fractional differential equations with positive constant coefficient, Adv. Difference Equ., 2015 (2015), 1-10
##[7]
I. A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 10 (2009), 305-320
##[8]
K. Shah, H. Khalil, R. A. Khan, Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations, Chaos Solitons Fractals, 77 (2015), 240-246
##[9]
J.-R. Wang, M. Fečkan, Y. Zhou , Ulam’s type stability of impulsive ordinary differential equations , J. Math. Anal. Appl., 395 (2012), 258-264
##[10]
J.-R. Wang, X. Li , Ulam-Hyers stability of fractional Langevin equations, Appl. Math. Comput., 258 (2015), 72-83
##[11]
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
##[12]
J.-R. Wang, K. Shah, A. Ali, Existence and Hyers-Ulam stability of fractional nonlinear impulsive switched coupled evolution equations, Math. Methods Appl. Sci., 41 (2018), 2392-2402
##[13]
C. Wang, T.-Z. Xu, , Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives, Appl. Math., 60 (2015), 383-393
##[14]
J.-R. Wang, Y. Zhang, Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations, Optimization, 63 (2014), 1181-1190
##[15]
J.-R. Wang, Y. Zhou, Z. Lin, On a new class of impulsive fractional differential equations, Appl. Math. Comput., 242 (2014), 649-657
##[16]
X.-J. Yang, C.-D. Li, T.-W. Huang, Q.-K. Song , Mittag-Leffler stability analysis of nonlinear fractional-order systems with impulses , Appl. Math. Comput., 293 (2017), 416-422
##[17]
A. Zada, W. Ali, S. Farina , Hyers-Ulam stability of nonlinear differential equations with fractional integrable impulses, Math. Methods Appl. Sci., 40 (2017), 5502-5514
##[18]
A. Zada, S. Faisal, Y.-J. Li , On the Hyers-Ulam stability of first-order impulsive delay differential equations, J. Funct. Spaces, 2016 (2016), 1-6
]
Iterative methods for solving the split common fixed point problem of demicontractive mappings in Hilbert spaces
Iterative methods for solving the split common fixed point problem of demicontractive mappings in Hilbert spaces
en
en
The split common fixed point problem was proposed in recent years
which required to find a common fixed point of a family of mappings
in one space whose image under a linear transformation is a common
fixed point of another family of mappings in the image space. In
this paper, we study two iterative algorithms for solving this split
common fixed point problem for the class of demicontractive mappings
in Hilbert spaces. Under mild assumptions on the parameters, we
prove the convergence of both iterative algorithms. As a consequence, we obtain new convergence
theorems for solving the split
common fixed point problem for the class of directed mappings. We compare the performance of the proposed iterative
algorithms with the existing iterative algorithms and conclude from the numerical experiments that our iterative algorithms converge faster than
these existing iterative algorithms in terms of iteration numbers.
960
970
Chunxiang
Zong
Department of Mathematics
Nanchang University
P. R. China
2295358036@qq.com
Yuchao
Tang
Department of Mathematics
Nanchang University
P. R. China
hhaaoo1331@163.com
Split common fixed point problem
demicontractive mappings
cyclic iteration method
simultaneous iteration method
Article.3.pdf
[
[1]
H. H. Bauschke, J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), 367-426
##[2]
H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York (2011)
##[3]
C. Byrne , Iterative oblique projection onto convex sets and the split feasibility problem , Inverse Problems, 18 (2002), 441-453
##[4]
C. Byrne , A unified treatment of some iterative algorithms in signal procesing and image reconstruction, Inverse Problems, 20 (2004), 103-120
##[5]
L.-C. Ceng, Q. H. Ansari, J.-C. Yao, Mann type iterative methods for finding a common solution of split feasibility and fixed point problems, Positivity, 16 (2012), 471-495
##[6]
Y. Censor, T. Elfving , A multiprojection algorithm using bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239
##[7]
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
##[8]
Y. Censor, A. Segal , The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587-600
##[9]
S. S. Chang, H. W. Joseph Lee, C. K. Chan, L. Wang, L. J. Qin, Split feasibility problem for quasi-nonexpansive multi-valued mappings and total asymptotically strict pseudo-contracive mapping, Appl. Math. Comput., 219 (2013), 10416-10424
##[10]
S. S. Chang, L. Wang, Y. K. Tang, L. Yang , The split common fixed point problems for total asymptotically strictly pseudocontracive mappings, J. Appl. Math., 2012 (2012), 1-13
##[11]
H.-H. Cui, M.-L. Su, F.-H. Wang, Damped projection method for split common fixed point problems, J. Inequal. Appl., 2013 (2013), 1-10
##[12]
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
##[13]
Q.-L. Dong, Y.-H. Yao, S.-N. He, Weak convergence theorems of the modified relaxed projection algorithms for the split feasibility problem in Hilbert spaces, Optim. Lett., 8 (2014), 1031-1046
##[14]
Y.-Y. Huang, C.-C. Hong , Approximating common fixed points of averaged self-mappings with application to the split feasibility problem and maximal monotone operators in Hilbert spaces , Fixed Point Theory Appl., 2013 (2013), 1-20
##[15]
R. Kraikaew, S. Saejung, On split common fixed point problems, J. Math. Anal. Appl., 415 (2014), 513-524
##[16]
A. Moudafi, The split common fixed point problem for demicontractive mappings, Inverse Problems, 26 (2010), 1-6
##[17]
A. Moudafi , A note on the split common fixed-point problem for quasi-nonexpansive operators, Nonlinear Anal., 74 (2011), 4083-4087
##[18]
A. Moudafi, Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal., 15 (2014), 809-818
##[19]
Y.-C. Tang, L.-W. Liu, Several iterative algorithms for solving the split common fixed point problem of directed operators with applications, Optimization, 65 (2016), 53-65
##[20]
Y.-C. Tang, J.-G. Peng, L.-W. Liu , A cyclic algorithm for the split common fixed point problem of demicontractive mappings in Hilbert spaces, Math. Model. Anal., 17 (2012), 457-466
##[21]
Y.-C. Tang, J.-G. Peng, L.-W. Liu, A cyclic and simultaneous iterative algorithm for the multiple split common fixed point problem of demicontractive mappings, Bull. Korean. Math. Soc., 51 (2014), 1527-1538
##[22]
D. V. Thong, D. V. Hieu , An inertial method for solving split common fixed point problems, J. Fixed Point Theory Appl., 19 (2017), 3029-3051
##[23]
F.-H. Wang, H.-H. Cui, Convergence of a cyclic algorithm for the split common fixed point problem without continuity assumption, Math. Model. Anal., 18 (2013), 537-542
##[24]
F.-H. Wang, H.-K. Xu, Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Anal., 74 (2011), 4105-4111
##[25]
H.-K. Xu, A variable krasnoselskii-mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021-2034
##[26]
H.-K. Xu, Iterative methods for the split feasibility problem in infinite dimensional Hilbert spaces, Inverse Problems, 26 (2010), 1-17
##[27]
L.-J. Zhu, Y.-C. Liou, S. M. Kang, Y.-H. Yao, Algorithmic and analytical approach to the split common fixed points problem, Fixed Point Theory Appl., 2014 (2014), 1-10
##[28]
L.-J. Zhu, Y.-C. Liu, J.-C. Yao, Y.-H. Yao , New algorithms for designed for the split common fixed point problem of quasi-pseudocontractions, J. Inequal. Appl., 2014 (2014), 1-13
]
The \(q\)-Stirling numbers of the second kind and its applications
The \(q\)-Stirling numbers of the second kind and its applications
en
en
The study of \(q\)-Stirling numbers of the second kind began with Carlitz [L. Carlitz, Duke Math. J., \(\textbf{15}\) (1948), 987--1000] in 1948.
Following Carlitz, we derive some identities and relations related to \(q\)-Stirling numbers of the second kind
which appear to be either new or else new ways of expressing older ideas more comprehensively.
971
983
Min-Soo
Kim
Division of Mathematics, Science, and Computers
Kyungnam University
Republic of Korea
mskim@kyungnam.ac.kr
Daeyeoul
Kim
Department of Mathematics and Institute of Pure and Applied Mathematics
Chonbuk National University
Republic of Korea
kdaeyeoul@jbnu.ac.kr
\(q\)-Stirling numbers of the second kind
\(q\)-factorial
Article.4.pdf
[
[1]
K. N. Boyadzhiev, Close encounters with the Stirling numbers of the second kind, Math. Mag., 85 (2012), 252-266
##[2]
Y. Cai, M. A. Readdy, q-Stirling numbers: a new view, Adv. Appl. Math., 86 (2017), 50-80
##[3]
L. Carlitz, q-Bernoulli numbers and polynomials , Duke Math. J., 15 (1948), 987-1000
##[4]
L. Carlitz, On abelian fields, Trans. Amer. Math. Soc., 35 (1933), 122-136
##[5]
C. A. Charalambides, On the q-differences of the generalized q-factorials, J. Statist. Plann. Inference, 54 (1996), 31-43
##[6]
J. Cigler , Operatormethoden fur q-Identitaten (German) , Monatsh. Math., 88 (1979), 87-105
##[7]
L. Comtet, Advanced Combinatorics , Springer Netherlands, Dordrecht (1974)
##[8]
K. Conrad , A q-analogue of Mahler expansions I, Adv. Math., 153 (2000), 185-230
##[9]
R. B. Corcino, On p, q-binomial coefficients, Integers, 8 (2008), 1-16
##[10]
C. B. Corcino, R. B. Corcino, J. M. Ontolan, C. M. Perez-Fernandez, E. R. Cantallopez, The Hankel transform of q-noncentral Bell numbers, Int. J. Math. Math. Sci., 2015 (2015), 1-10
##[11]
B. R. Corcino, C. B. Montero, On p, q-difference operator, J. Korean Math. Soc., 49 (2012), 537-547
##[12]
H. W. Gould, Euler’s formula for nth differences of powers, Amer. Math. Monthly, 85 (1978), 450-467
##[13]
H. W. Gould, The q-Stirling numbers of first and second kinds, Duke Math. J., 28 (1961), 281-289
##[14]
J. A. Grunert , Uber die Summerung der Reihen von der Form \(A\phi(0), A_1\phi(1)x, A_2\phi(2)x^2, . . . A_n\phi(n)x^n, . . . ,\) wo A eine beliebige constante Gre, \(A_n\) eine beliebige und \(\phi(n)\) eine ganze rationale algebraische Function der positiven ganzen Zahl n bezeichnet, J. Reine Angew. Math., 25 (1843), 240-279
##[15]
F. H. Jackson, On q-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh, 46 (1909), 253-281
##[16]
C. Jordan, Calculus of finite differences, Chelsea, New York (1950)
##[17]
M.-S. Kim, J.-W. Son, A note on q-difference operators, Commun. Korean Math. Soc., 17 (2002), 423-430
##[18]
P. M. Knopf , The Operator \((x \frac{d}{ dx})^n\) and Its Applications to Series, Math. Mag., 76 (2003), 364-371
##[19]
I. Kucukoglu, A. Bayad, Y. Simsek, k-ary Lyndon words and necklaces arising as rational arguments of Hurwitz-Lerch zeta function and Apostol-Bernoulli polynomials, Mediterr. J. Math., 14 (2017), 1-16
##[20]
T. Mansour, M. Schork, M. Shattuck, The Generalized Stirling and Bell Numbers Revisited, J. Integer Seq., 15 (2012), 1-47
##[21]
S. Milne, A q-analogue of restricted growth functions, Dobinski’s equality, and Charlier polynomials, Trans. Amer. Math. Soc., 245 (1978), 89-118
##[22]
H. Ozden, I. N. Cangul, Y. Simsek, Generalized q-Stirling numbers and their interpolation functions, Axioms, 2 (2013), 10-19
##[23]
B. E. Sagan, Congruence properties of q-analogs , Adv. Math., 95 (1992), 127-143
##[24]
Y. Simsek, Combinatorial inequalities and sums involving Bernstein polynomials and basis functions, J. Inequal. Spec. Funct., 8 (2017), 15-24
##[25]
Y. Simsek, Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications, Fixed Point Theory Appl., 2013 (2013), 1-28
##[26]
Y. Simsek, On q-deformed Stirling numbers, Int. J. Math. Comput., 15 (2012), 70-80
##[27]
Y. Simsek, A. Bayad, V. Lokesha , q-Bernstein polynomials related to q-Frobenius-Euler polynomials, l-functions, and q-Stirling numbers, Math. Methods Appl. Sci., 35 (2012), 877-884
##[28]
S.-Z. Song, G.-S. Cheon, Y.-B. Jun, L. B. Beasley, A q-analogue of the generalized factorial numbers, J. Korean Math. Soc., 47 (2010), 645-657
##[29]
M. Wachs, D. White, p, q-Stirling numbers and set partition statistics, J. Combin. Theory Ser. A, 56 (1991), 27-46
##[30]
M. Ward , A calculus of sequences, Amer. J. Math., 58 (1936), 255-266
##[31]
J. Zeng, The q-Stirling numbers, continued fractions and the q-Charlier and q-Laguerre polynomials, J. Comput. Appl. Math., 57 (1995), 413-424
]
Fractional integral associated to Schrödinger operator on the Heisenberg groups in central generalized Morrey spaces
Fractional integral associated to Schrödinger operator on the Heisenberg groups in central generalized Morrey spaces
en
en
Let \(L=-\Delta_{\mathbb{H}_n}+V\) be a Schrödinger operator on the Heisenberg groups \(\mathbb{H}_n\), where the non-negative potential \(V\) belongs to the reverse Hölder class \(RH_{Q/2}\)
and \(Q\) is the homogeneous dimension of \(\mathbb{H}_n\). Let \(b\) belong to a new \(BMO_{\theta}(\mathbb{H}_n,\rho)\) space, and let \({\cal I}_{\beta}^{L}\) be the fractional integral operator associated with \(L\).
In this paper, we study the boundedness of the operator \({\cal I}_{\beta}^{L}\) and its commutators \([b,{\cal I}_{\beta}^{L}]\) with \(b \in BMO_{\theta}(\mathbb{H}_n,\rho)\)
on central generalized Morrey spaces \(LM_{p,\varphi}^{\alpha,V}(\mathbb{H}_n)\) and generalized Morrey spaces \(M_{p,\varphi}^{\alpha,V}(\mathbb{H}_n)\) associated with Schrödinger operator.
We find the sufficient conditions on the pair \((\varphi_1,\varphi_2)\) which ensures the boundedness of the operator \({\cal I}_{\beta}^{L}\)
from \(LM_{p,\varphi_1}^{\alpha,V}(\mathbb{H}_n)\) to \(LM_{q,\varphi_2}^{\alpha,V}(\mathbb{H}_n)\) and from \(M_{p,\varphi_1}^{\alpha,V}(\mathbb{H}_n)\) to \(M_{q,\varphi_2}^{\alpha,V}(\mathbb{H}_n)\), \(1/p-1/q=\beta/Q\).
When \(b\) belongs to \(BMO_{\theta}(\mathbb{H}_n,\rho)\) and \((\varphi_1,\varphi_2)\) satisfies some conditions, we also show that the commutator operator \([b,{\cal I}_{\beta}^{L}]\) is bounded
from \(LM_{p,\varphi_1}^{\alpha,V}(\mathbb{H}_n)\) to \(LM_{q,\varphi_2}^{\alpha,V}(\mathbb{H}_n)\) and from \(M_{p,\varphi_1}^{\alpha,V}\) to \(M_{q,\varphi_2}^{\alpha,V}\), \(1/p-1/q=\beta/Q\).
984
993
Ahmet
Eroglu
Department of Mathematics
Nigde Omer Halisdemir University
Turkey
aeroglu@ohu.edu.tr
Tahir
Gadjiev
Institute of Mathematics and Mechanics
NAS of Azerbaijan
Azerbaijan
tgadjiev@mail.az
Faig
Namazov
Baku State University
Azerbaijan
f-namazov@mail.ru
Schrödinger operator
Heisenberg group
central generalized Morrey space
fractional integral
commutator
BMO
Article.5.pdf
[
[1]
A. Akbulut, R. V. Guliyev, S. Celik, M. N. Omarova, Fractional integral associated with Schrödinger operator on vanishing generalized Morrey spaces, accepted in J. Math. Ineq., (2018)
##[2]
A. Akbulut, V. S. Guliyev, M. N. Omarova, Marcinkiewicz integrals associated with Schrödinger operators and their commutators on vanishing generalized Morrey spaces, Bound. Value Probl., 2017 (2017), 1-16
##[3]
J. Alvarez, J. Lakey, M. Guzman-Partida, Spaces of bounded \(\lambda\)-central mean oscillation, Morrey spaces, and \(\lambda\)-central Carleson measures, Collect. Math., 51 (2000), 1-47
##[4]
B. Bongioanni, E. Harboure, O. Salinas, Commutators of Riesz transforms related to Schödinger operators, J. Fourier Anal. Appl., 17 (2011), 115-134
##[5]
T. Bui, Weighted estimates for commutators of some singular integrals related to Schrödinger operators, Bull. Sci. Math., 138 (2014), 270-292
##[6]
L. Capogna, D. Danielli, S. D. Pauls, J. T. Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, Birkhuser Verlag, Basel (2007)
##[7]
G. Di Fazio, M. A. Ragusa, Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal., 112 (1993), 241-256
##[8]
A. Eroglu, J. V. Azizov, A note of the fractional integral operators in generalized Morrey spaces on Heisenberg groups, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 37 (2017), 86-91
##[9]
A. Eroglu, V. S. Guliyev, C. V. Azizov, Characterizations for the fractional integral operators in generalized Morrey spaces on Carnot groups, Math. Notes, 102 (2017), 127-139
##[10]
D. S. Fan, S. Z. Lu, D. C. Yang, Boundedness of operators in Morrey spaces on homogeneous spaces and its applications, Acta Math. Sinica (Chin. Ser.), 14 (1998), 625-634
##[11]
G. B. Folland, E. M. Stein, Estimates for the \(\partial_b\)-complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27 (1974), 429-522
##[12]
G. B. Folland, E. M. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, Princeton (1982)
##[13]
V. S. Guliyev, Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl., 2009 (2009), 1-20
##[14]
V. S. Guliyev , Function spaces and integral operators associated with Schrödinger operators: an overview, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 40 (2014), 178-202
##[15]
V. S. Guliyev, Function spaces, integral operators and two weighted inequalities on homogeneous groups, Some applications, Elm, Baku (1999)
##[16]
V. S. Guliyev, Generalized local Morrey spaces and fractional integral operators with rough kernel, J. Math. Sci. (N.Y.), 193 (2013), 211-227
##[17]
V. S. Guliyev, Integral operators on function spaces on the homogeneous groups and on domains in \(R^n\), Doctors degree dissertation, Mat. Inst. Steklov, Moscow (1994)
##[18]
V. S. Guliyev, A. Eroglu, Y. Y. Mammadov , Riesz potential in generalized Morrey spaces on the Heisenberg group, J. Math. Sci. (N. Y.), 189 (2013), 365-382
##[19]
V. S. Guliyev, T. S. Gadjiev, S. Galandarova, Dirichlet boundary value problems for uniformly elliptic equations in modified local generalized Sobolev-Morrey spaces, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), 1-17
##[20]
V. S. Guliyev, R. V. Guliyev, M. N. Omarova, M. A. Ragusa, Schrödinger type operators on local generalized Morrey spaces related to certain nonnegative potentials, , (submitted), -
##[21]
V. S. Guliyev, Y. Y. Mammadov, Boundedness of the fractional maximal operator in generalized Morrey space on the Heisenberg group, Indian J. Pure Appl. Math., 44 (2013), 185-202
##[22]
V. S. Guliyev, M. N. Omarova, M. A. Ragusa, A. Scapellato, Commutators and generalized local Morrey spaces, J. Math. Anal. Appl., 457 (2018), 1388-1402
##[23]
H. Q. Li, Estimations \(L_p\) des oprateurs de Schrödinger sur les groupes nilpotents, J. Funct. Anal., 161 (1999), 152-218
##[24]
C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166
##[25]
S. Polidoro, M. A. Ragusa , Sobolev-Morrey spaces related to an ultraparabolic equation, Manuscripta Math., 96 (1998), 371-392
##[26]
Z. Shen, \(L_p\) estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble), 45 (1995), 513-546
##[27]
E. M. Stein, Harmonic Analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton (1993)
##[28]
L. Tang, J. Dong, Boundedness for some Schrödinger type operator on Morrey spaces related to certain nonnegative potentials, J. Math. Anal. Appl., 355 (2009), 101-109
##[29]
R. Wheeden, A. Zygmund , Measure and integral: An introduction to real analysis , Marcel Dekker, New York-Basel (1977)
]
Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel
Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel
en
en
In this paper, we analyze the fractional modeling of the giving up the smoking using the definitions of Liouville-Caputo and Atangana-Baleanu-Caputo fractional derivatives. Applying the homotopy analysis method and the Laplace transform with polynomial homotopy, the analytical solution of the smoking dynamics has obtained. Furthermore, using an iterative scheme by the Laplace transform, and the Atangana-Baleanu fractional integral, special solutions of the model are obtained. Uniqueness and existence of the solutions by the fixed-point theorem and Picard-Lindelof approach are studied. Finally, some numerical simulations are carried out for illustrating the results obtained.
994
1014
V. F.
Morales-Delgado
Facultad de Matematicas
Universidad Autonoma de Guerrero
Mexico
J. F.
Gómez-Aguilar
CONACyT-Tecnológico Nacional de México/CENIDET
Mexico
jgomez@cenidet.edu.mx
M. A.
Taneco-Hernández
Facultad de Matematicas
Universidad Autonoma de Guerrero
Mexico
R. F.
Escobar-Jiménez
Tecnológico Nacional de México/CENIDET
Mexico
V. H.
Olivares-Peregrino
Tecnológico Nacional de México/CENIDET
Mexico
Smoking model
Liouville-Caputo fractional derivative
Atangana-Baleanu fractional derivative
Laplace transform
homotopy method
Article.6.pdf
[
[1]
O. J. Algahtani, A. Zeb, G. Zaman, S. Momani, I. H. Jung, Mathematical study of smoking model by incorporating campaign class, Wulfenia, 22 (2015), 205-216
##[2]
B. S. T. Alkahtani , Chua’s circuit model with Atangana-Baleanu derivative with fractional order, Chaos Solitons Fractals, 89 (2016), 547-551
##[3]
A. Atangana, D. Baleanu , New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model , Therm Sci., 20 (2016), 763-769
##[4]
A. Atangana, I. Koca , Model of Thin Viscous Fluid Sheet Flow within the Scope of Fractional Calculus: Fractional Derivative with and No Singular Kernel , Fund. Inform., 151 (2017), 145-159
##[5]
C. Castillo-Garsow, G. Jordan-Salivia, A. R. Herrera, Mathematical models for the dynamics of tobacco use, recovery, and relapse , Technical Report Series BU-1505-M, Cornell University, Ithaca (1997)
##[6]
J. S. Choi, D. Kumar, J. Singh, R. Swroop, Analytical techniques for system of time fractional nonlinear differential equations, J. Korean Math. Soc., 54 (2017), 1209-1229
##[7]
K. Diethelm, N. J. Ford, A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31-52
##[8]
Q. Din, M. Ozair, T. Hussain, U. Saeed, Qualitative behavior of a smoking model, Adv. Difference Equ., 2016 (2016), 1-12
##[9]
V. S. Ertürk, G. Zaman, S. Momani , A numeric-analytic method for approximating a giving up smoking model containing fractional derivatives, Comput. Math. Appl., 64 (2012), 3065-3074
##[10]
F. Haq, K. Shah, G. ur Rahman, M. Shahzad , Numerical solution of fractional order smoking model via laplace Adomian decomposition method, Alexandria Engineering Journal, 2017 (2017), 1-9
##[11]
J. Hristov , Integral balance solutions to applied models involving time-fractional derivatives-The scope of the method and results thereof, Commun. Frac. Calc., 4 (2013), 64-104
##[12]
J. Hristov, Integral-Balance Solution to Nonlinear Subdiffusion Equation, Frontiers in Fractional Calculus, 2017 (2017), 71-106
##[13]
S. Kumar, A new analytical modelling for telegraph equation via Laplace transform, Appl. Math. Model., 38 (2014), 3154-3163
##[14]
S. Kumar, A. Kumar, I. K. Argyros, A new analysis for the Keller-Segel model of fractional order, Numer. Algorithms, 75 (2017), 213-228
##[15]
S. Kumar, M. M. Rashidi , New analytical method for gas dynamic equation arising in shock fronts, Comput. Phys. Commun., 185 (2014), 1947-1954
##[16]
D. Kumar, J. Singh, M. M. Al Qurashi, D. Baleanu , Analysis of logistic equation pertaining to a new fractional derivative with non-singular kernel , Adv. Mech. Eng., 9 (2017), 1-8
##[17]
D. Kumar, J. Singh, D. Baleanu , A new numerical algorithm for fractional Fitzhugh-Nagumo equation arising in transmission of nerve impulses, Nonlinear Dynam., 91 (2018), 307-317
##[18]
D. Kumar, J. Singh, D. Baleanu, A new analysis of the Fornberg-Whitham equation pertaining to a fractional derivative with Mittag-Leffler-type kernel , D. Eur. Phys. J. Plus, 133 (2018), 1-10
##[19]
D. Kumar, J. Singh, D. Baleanu, Sushila, Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel , Phys. A, 492 (2018), 155-167
##[20]
C. P. Li, C. X. Tao , On the fractional Adams method, Comput. Math. Appl., 58 (2009), 1573-1588
##[21]
Z. Odibat, A. S. Bataineh, An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials, Math. Meth. Appl. Sci., 38 (2015), 991-1000
##[22]
K. M. Owolabi , Mathematical analysis and numerical simulation of chaotic noninteger order differential systems with Riemann-Liouville derivative, Numer. Methods Partial Differential Equations, 34 (2018), 274-295
##[23]
K. M. Owolabi , Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Chaos Solitons Fractals, 103 (2017), 544-554
##[24]
K. M. Owolabi , Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 304-317
##[25]
K. M. Owolabi, A. Atangana , Numerical approximation of nonlinear fractional parabolic differential equations with Caputo-Fabrizio derivative in Riemann-Liouville sense , Chaos Solitons Fractals, 99 (2017), 171-179
##[26]
K. M. Owolabi, A. Atangana, Numerical Simulation of Noninteger Order System in Subdiffusive, Diffusive, and Superdiffusive Scenarios, J. Comput. Nonlinear Dynam, 12 (2017), 1-7
##[27]
I. Podlubny, Fractional Differential Equations , Academic Press, San Diego (1999)
##[28]
Z. Rahimi, W. Sumelka, X. J. Yang, Linear and non-linear free vibration of nano beams based on a new fractional non-local theory, Engineering Computations, 34 (2017), 1754-1770
##[29]
K. M. Saad, E. H. AL-Shareef, M. S. Mohamed, X. J. Yang, Optimal q-homotopy analysis method for time-space fractional gas dynamics equation, D. Eur. Phys. J. Plus, 132 (2017), 1-23
##[30]
O. Sharomi, A. B. Gumel , Curtailing smoking dynamics: a mathematical modeling approach, Appl. Math. Comput., 195 (2008), 475-499
##[31]
J. Singh, D. Kumar, M. Al Qurashi, D. Baleanu , A new fractional model for giving up smoking dynamics, Adv. Difference Equ., 2017 (2017), 1-16
##[32]
J. Singh, D. Kumar, Z. Hammouch, A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504-515
##[33]
Y. Sun, B. Indraratna, J. P. Carter, T. Marchant, S. Nimbalkar, Application of fractional calculus in modelling ballast deformation under cyclic loading, Comput. Geotech., 82 (2017), 16-30
##[34]
A. Yadav, P. K. Srivastava, A. Kumar, Mathematical model for smoking: Effect of determination and education, Int. J. Biomath., 8 (2015), 1-14
##[35]
G. Zaman, Qualitative behavior of giving up smoking models, Bull. Malays. Math. Sci. Soc., 34 (2011), 1-12
##[36]
A. Zeb, F. Bibi, G. Zaman, Optimal control strategies in square root dynamics of smoking model, International Journal of Scientific World, 3 (2015), 91-97
##[37]
A. Zeb, G. Zaman, V. S. Erturk, B. Alzalg, F. Yousafzai, M. Khan, Approximating a Giving Up Smoking Dynamic on Adolescent Nicotine Dependence in Fractional Order, PloS one, 11 (2016), 1-10
##[38]
Y. D. Zhang, S. H. Wang, J.-F. Yang, Z. Zhang, P. Phillips, P. Sun, J. Yan, A Comprehensive Survey on Fractional Fourier Transform, Fund. Inform., 151 (2017), 1-48
]