]>
2016
9
10
ISSN 2008-1898
222
On the well-posedness of the generalized split quasi-inverse variational inequalities
On the well-posedness of the generalized split quasi-inverse variational inequalities
en
en
In this paper, a generalized split quasi-inverse variational inequality ((GSQIVI), for short) is considered
and investigated in Hilbert spaces. Since the well-posedness results, not only show us the qualitative
properties of problem (GSQIVI), but also it gives us an outlook to the convergence analysis of the solutions
for (GSQIVI). Therefore, we first introduce the concepts concerning with the approximating sequences,
well-posedness and well-posedness in the generalized sense of (GSQIVI). Then, under those definitions, we
establish several metric characterizations and equivalent conditions of well-posedness for the (GSQIVI) by
using the measure of noncompactness theory and the generalized Cantor theorem.
5497
5509
Liang
Cao
Guangxi University of Finance and Economics
P. R. China
bdhzxcaoliang@163.com
Hua
Kong
Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science
Neijiang Normal University
P. R. China
konghua2008@126.com
Sheng-Da
Zeng
Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science
Institute of Computer Science, Faculty of Mathematics and Computer Science
Neijiang Normal University
Jagiellonian University
P. R. China
Poland
shdzeng@hotmail.com;zengshengda@163.com
Generalized split quasi-inverse variational inequality
measure of noncompactness
well-posedness
Painlevé-Kuratowski limits.
Article.1.pdf
[
[1]
E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145
##[2]
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
##[3]
K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann., 266 (1984), 519-537
##[4]
Y.-P. Fang, N.-J. Huang, Variational-like inequalities with generalized monotone mappings in Banach spaces, J. Optim. Theory Appl., 118 (2003), 327-338
##[5]
R. Glowinski, J. L. Lions, R. Trémoliéres, Numerical analysis of variational inequalities, Translated from the French, Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York (1981)
##[6]
S. M. Guu, J. Li, Vector variational-like inequalities with generalized bifunctions defined on nonconvex sets, Nonlinear Anal., 71 (2009), 2847-2855
##[7]
B. S. He, X.-Z. He, H. X. Liu, Solving a class of constrained 'black-box' inverse variational inequalities, European J. Oper. Res., 204 (2010), 391-401
##[8]
B. S. He, H. X. Liu, Inverse variational inequalities in economics-applications and algorithms, Sciencepaper Online, (2006)
##[9]
B. S. He, H. X. Liu, M. Li, X.-Z. He, PPA-based methods for monotone inverse variational inequalities, Sciencepaper Online, (2006)
##[10]
R. Hu, Y.-P. Fang, Well-posedness of inverse variational inequalities, J. Convex Anal., 15 (2008), 427-437
##[11]
R. Hu, Y.-P. Fang, Levitin-Polyak well-posedness by perturbations of inverse variational inequalities, Optim. Lett., 7 (2013), 343-359
##[12]
R. Hu, Y.-P. Fang, Well-posedness of the split inverse variational inequality problem, Bull. Malays. Math. Sci. Soc., 2015 (2015), 1-12
##[13]
L. L. Huang, Matrix Lagrange multipliers, J. Comput. Complex. Appl., 2 (2016), 86-88
##[14]
N.-J. Huang, J. Li, B. H. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Modelling, 43 (2006), 1267-1274
##[15]
K. Kimura, Y.-C. Liou, S.-Y. Wu, J.-C. Yao, Well-posedness for parametric vector equilibrium problems with applications, J. Ind. Manag. Optim., 4 (2008), 313-327
##[16]
K. Kuratowski, Topology, Vol. I, New edition, revised and augmented, Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw (1966)
##[17]
B. Lemaire, Well-posedness, conditioning and regularization of minimization, inclusion and fixed-point problems, Proceedings of the 4th International Conference on Mathematical Methods in Operations Research and 6th Workshop onWell-posedness and Stability of Optimization Problems, Sozopol, (1997), Pliska Stud. Math. Bulgar., 12 (1998), 71-84
##[18]
X. Li, X.-S. Li, N.-J. Huang, A generalized f-projection algorithm for inverse mixed variational inequalities, Optim. Lett., 8 (2014), 1063-1076
##[19]
X.-J. Long, N.-J. Huang, Metric characterizations of \(\alpha\)-well-posedness for symmetric quasi-equilibrium problems, J. Global Optim., 45 (2009), 459-471
##[20]
N. K. Mahato, C. Nahak, Weakly relaxed \(\alpha\)-pseudomonotonicity and equilibrium problem in Banach spaces, J. Appl. Math. Comput., 40 (2012), 499-509
##[21]
M. A. Noor, K. I. Noor, V. Gupta, On equilibrium-like problems, Appl. Anal., 86 (2007), 807-818
##[22]
M. A. Noor, K. I. Noor, S. Zainab, On a predictor-corrector method for solving invex equilibrium problems, Nonlinear Anal., 71 (2009), 3333-3338
##[23]
L. Scrimali, An inverse variational inequality approach to the evolutionary spatial price equilibrium problem, Optim. Eng., 13 (2012), 375-387
##[24]
A. N. Tykhonov, On the stability of the functional optimization problem, USSR J. Comput. Math. Math. Phys., 6 (1966), 631-634
##[25]
G. C. Wu, New correction functional and Lagrange multipliers for two point value problems, J. Comput. Complex. Appl., 2 (2016), 46-48
##[26]
J. F. Yang, Dynamic power price problem: an inverse variational inequality approach, J. Ind. Manag. Optim., 4 (2008), 673-684
]
Schur-Convexity for Lehmer mean of n variables
Schur-Convexity for Lehmer mean of n variables
en
en
Schur-convexity, Schur-geometric convexity and Schur-harmonic convexity for Lehmer mean of n variables
are investigated, and some mean value inequalities of n variables are established.
5510
5520
Chun-Ru
Fu
Basic courses department
Beijing Vocational College of Electronic Technology
P. R. China
fuchunru2008@163.com
Dongsheng
Wang
Applied college of science and technology
Beijing Union University
P. R. China
wds000651225@sina.com
Huan-Nan
Shi
Department of Electronic Information, Teacher's College
Beijing Union University
P. R. China
sfthuannan@buu.com.cn;shihuannan2014@qq.com
Schur convexity
Schur geometric convexity
Schur harmonic convexity
n variables Lehmer mean
majorization
inequalities.
Article.2.pdf
[
[1]
H. Alzer, Über Lehmers Mittelwertfamilie, (German) [On Lehmer's family of mean values], Elem. Math., 43 (1988), 50-54
##[2]
H. Alzer, Bestmögliche Abschätzungen für spezielle Mittelwerte, (German) [Best possible estimates for special mean values], Zb. Rad. Prirod.-Mat. Fak. Ser. Mat., 23 (1993), 331-346
##[3]
E. F. Beckenbach , A class of mean value functions, Amer. Math. Monthly, 57 (1950), 1-6
##[4]
Y.-M. Chu, G.-D. Wang, X.-H. Zhang , The Schur multiplicative and harmonic convexities of the complete symmetric function , Math. Nachr., 284 (2011), 653-663
##[5]
Y.-M. Chu, W.-F. Xia , Necessary and sufficient conditions for the Schur harmonic convexity of the generalized Muirhead mean, Proc. A. Razmadze Math. Inst., 152 (2010), 19-27
##[6]
Y.-M. Chu, X.-M. Zhang, Necessary and sufficient conditions such that extended mean values are Schur-convex or Schur-concave, J. Math. Kyoto Univ., 48 (2008), 229-238
##[7]
Y.-M. Chu, X. M. Zhang, G.-D. Wang , The Schur geometrical convexity of the extended mean values, J. Convex Anal., 15 (2008), 707-718
##[8]
L.-L. Fu, B.-Y. Xi, H. M. Srivastava, Schur-convexity of the generalized Heronian means involving two positive numbers, Taiwanese J. Math., 15 (2011), 2721-2731
##[9]
W.-M. Gong, H. Shen, Y.-M. Chu , The Schur convexity for the generalized Muirhead mean, J. Math. Inequal., 8 (2014), 855-862
##[10]
H. W. Gould, M. E. Mays, Series expansions of means, J. Math. Anal. Appl., 101 (1984), 611-621
##[11]
C. Gu, H. N. Shi , The Schur-convexity and the Schur-geometric concavity of Lehme means, (Chinese) Math. Pract. Theory, 39 (2009), 183-188
##[12]
Z.-J. Guo, X.-H. Shen, Y.-M. Chu, The best possible Lehmer mean bounds for a convex combination of logarithmic and harmonic means , Int. Math. Forum, 8 (2013), 1539-1551
##[13]
D. H. Lehmer, On the compounding of certain means, J. Math. Anal. Appl., 36 (1971), 183-200
##[14]
Z. Liu, Remark on inequalities between Hölder and Lehmer means, J. Math. Anal. Appl., 247 (2000), 309-313
##[15]
V. Lokesha, N. Kumar, K. M. Nagaraja, S. Padmanabhan, Schur geometric convexity for ratio of difference of means, J. Sci. Res. Reports, 3 (2014), 1211-1219
##[16]
V. Lokesha, K. M. Nagaraja, N. kumar, Y.-D. Wu , Shur convexity of Gnan mean for positive arguments, Notes Number Theory Discrete Math., 17 (2011), 37-41
##[17]
A. W. Marshall, I. Olkin , Inequalities: theory of majorization and its applications, Mathematics in Science and Engineering, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London (1979)
##[18]
J.-X. Meng, Y.-M. Chu, X.-M. Tang, The Schur-harmonic-convexity of dual form of the Hamy symmetric function, Mat. Vesnik, 62 (2010), 37-46
##[19]
K. M. Nagaraja, S. K. Sahu, Schur harmonic convexity of Stolarsky extended mean values, Scientia Magna, 9 (2013), 18-29
##[20]
C. P. Niculescu , Convexity according to the geometric mean, Math. Inequal. Appl., 3 (2000), 155-167
##[21]
Z. Páles , Inequalities for sums of powers, J. Math. Anal. Appl., 131 (1988), 265-270
##[22]
F. Qi, J. Sándor, S. S. Dragomir, A. Sofo , Notes on the Schur-convexity of the extended mean values, Taiwanese J. Math., 9 (2005), 411-420
##[23]
Y.-F. Qiu, M.-K. Wang, Y.-M. Chu, G.-D. Wang, Two sharp inequalities for Lehmer mean, identric mean and logarithmic mean, J. Math. Inequal., 5 (2011), 301-306
##[24]
J. Sándor, The Schur-convexity of Stolarsky and Gini means, Banach J. Math. Anal., 1 (2007), 212-215
##[25]
H.-N. Shi, Y.-M. Jiang, W.-D. Jiang, Schur-convexity and Schur-geometrically concavity of Gini means, Comput. Math. Appl., 57 (2009), 266-274
##[26]
K. B. Stolarsky, Hölder means, Lehmer means, and \(x^{-1} \log \cosh x\) , J. Math. Anal. Appl., 202 (1996), 810-818
##[27]
B.-Y. Wang , Foundations of Majorization Inequalities, (in Chinese) Beijing Normal Univ. Press, Beijing, China (1990)
##[28]
M. K. Wang, Y.-M. Chu, G.-D. Wang, A sharp double inequality between the Lehmer and arithmetic-geometric means, Pac. J. Appl. Math., 4 (2012), 1-25
##[29]
M.-K. Wang, Y.-F. Qiu, Y.-M. Chu , Sharp bounds for seiffert means in terms of Lehmer means, J. Math. Inequal., 4 (2010), 581-586
##[30]
S. R. Wassell , Rediscovering a family of means, Math. Intelligencer, 24 (2002), 58-65
##[31]
A. Witkowski , Convexity of weighted Stolarsky means, JIPAM. J. Inequal. Pure Appl. Math., 7 (2006), 1-6
##[32]
A. Witkowski , On Schur-convexity and Schur-geometric convexity of four-parameter family of means, Math. Inequal. Appl., 14 (2011), 897-903
##[33]
Y. Wu, F. Qi , Schur-harmonic convexity for differences of some means, Analysis (Munich), 32 (2012), 263-270
##[34]
Y. Wu, F. Qi, H.-N. Shi, Schur-harmonic convexity for differences of some special means in two variables, J. Math. Inequal., 8 (2014), 321-330
##[35]
W.-F. Xia, Y.-M. Chu , The Schur convexity of the weighted generalized logarithmic mean values according to harmonic mean, Int. J. Mod. Math., 4 (2009), 225-233
##[36]
W.-F. Xia, Y.-M. Chu, The Schur harmonic convexity of Lehmer means, Int. Math. Forum, 4 (2009), 2009-2015
##[37]
W.-F. Xia, Y.-M. Chu, The Schur multiplicative convexity of the generalized Muirhead mean values, Int. J. Funct. Anal. Oper. Theory Appl., 1 (2009), 1-8
##[38]
W.-F. Xia, Y.-M. Chu, The Schur convexity of Gini mean values in the sense of harmonic mean, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 1103-1112
##[39]
W.-F. Xia, Y.-M. Chu, G.-D. Wang , Necessary and sufficient conditions for the Schur harmonic convexity or concavity of the extended mean values, Rev. Un. Mat. Argentina, 51 (2010), 121-132
##[40]
Z.-H. Yang, Necessary and sufficient conditions for Schur geometrical convexity of the four-parameter homogeneous means, Abstr. Appl. Anal., 2010 (2010), 1-16
##[41]
Z.-H. Yang , Schur harmonic convexity of Gini means, Int. Math. Forum, 6 (2011), 747-762
##[42]
Z.-H. Yang, Schur power convexity of Stolarsky means, Publ. Math. Debrecen, 80 (2012), 43-66
##[43]
H.-P. Yin, H.-N. Shi, F. Qi , On Schur m-power convexity for ratios of some means, J. Math. Inequal., 9 (2015), 145-153
##[44]
X.-M. Zhang , Geometrically convex functions, (Chinese) Anhui University Press, Hefei (2004)
##[45]
T.-Y. Zhang, A.-P. Ji , Schur-Convexity of Generalized Heronian Mean, International Conference on Information Computing and Applications, Springer, Berlin, Heidelberg, 244 (2011), 25-33
]
Viscosity approximation methods for hierarchical optimization problems of multivalued nonexpansive mappings in CAT(0) spaces
Viscosity approximation methods for hierarchical optimization problems of multivalued nonexpansive mappings in CAT(0) spaces
en
en
The purpose of this paper is to prove some strong convergence theorems for hierarchical optimization
problems of multivalued nonexpansive mappings in CAT(0) spaces by using the viscosity approximation
method. Our results generalize the results of [X.-D. Liu, S.-S. Chang, J. Inequal. Appl., 2013 (2013), 14
pages], [R. Wangkeeree, P. Preechasilp, J. Inequal. Appl., 2013 (2013), 15 pages], and many others. Some
related results in R-trees are also given.
5521
5535
Jinhua
Zhu
College of Mathematics
Yibin University
China
Shih-Sen
Chang
Center for General Education
China Medical University
Taiwan
changss2013@163.com
Min
Liu
College of Mathematics
Yibin University
China
Viscosity approximation method
fixed point
variational inequality
hierarchical optimization problems
multivalued nonexpansive mapping
CAT(0) space.
Article.3.pdf
[
[1]
B. Ahmadi Kakavandi, Weak topologies in complete CAT(0) metric spaces, Proc. Amer. Math. Soc., 141 (2013), 1029-1039
##[2]
A. G. Aksoy, M. A. Khamsi, A selection theorem in metric trees, Proc. Amer. Math. Soc., 134 (2006), 2957-2966
##[3]
I. D. Berg, I. G. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, 133 (2008), 195-218
##[4]
M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin (1999)
##[5]
K. S. Brown, Buildings, Springer-Verlag, New York (1989)
##[6]
D. Burago, Y. Burago, S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, American Mathematical Society, Providence (2001)
##[7]
P. Chaoha, A. Phon-on, A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl., 320 (2006), 983-987
##[8]
S. Dhompongsa, A. Kaewkhao, B. Panyanak, Browder's convergence theorem for multivalued mappings without endpoint condition, Topology Appl., 159 (2012), 2757-2763
##[9]
S. Dhompongsa, A. Kaewkhao, B. Panyanak, On Kirk's strong convergence theorem for multivalued nonexpansive mappings on CAT(0) spaces, Nonlinear Anal., 75 (2012), 459-468
##[10]
S. Dhompongsa, W. A. Kirk, B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal., 8 (2007), 35-45
##[11]
S. Dhompongsa, W. A. Kirk, B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., 65 (2006), 762-772
##[12]
S. Dhompongsa, B. Panyanak, On \(\Delta\)-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (2008), 2572-2579
##[13]
B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957-961
##[14]
M. A. Khamsi, W. A. Kirk, C. Martinez Yáñez, Fixed point and selection theorems in hyperconvex spaces, Proc. Amer. Math. Soc., 128 (2000), 3275-3283
##[15]
W. A. Kirk, Geodesic geometry and fixed point theory, II, International Conference on Fixed Point Theory and Applications, Yokohama Publ., Yokohama, 2004 (2004), 113-142
##[16]
W. A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689-3696
##[17]
P. Kumam, G. S. Saluja, H. K. Nashine, Convergence of modified S-iteration process for two asymptotically nonexpansive mappings in the intermediate sense in CAT(0) spaces, J. Inequal. Appl., 2014 (2014), 1-15
##[18]
X.-D. Liu, S.-S. Chang, Viscosity approximation methods for hierarchical optimization problems in CAT(0) spaces, J. Inequal. Appl., 2013 (2013), 1-14
##[19]
J. Markin, Fixed points for generalized nonexpansive mappings in R-trees, Comput. Math. Appl., 62 (2011), 4614-4618
##[20]
A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55
##[21]
Jr. S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math., 30 (1968), 475-488
##[22]
P. Saipara, P. Chaipunya, Y. J. Cho, P. Kumam, On strong and \(\Delta\)-convergence of modified S-iteration for uniformly continuous total asymptotically nonexpansive mappings in \(CAT(\kappa)\) spaces, J. Nonlinear Sci. Appl., 8 (2015), 965-975
##[23]
K. Samanmit, B. Panyanak, On multivalued nonexpansive mappings in R-trees, J. Appl. Math., 2012 (2012), 1-13
##[24]
N. Shioji, W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc., 125 (1997), 3641-3645
##[25]
R. Wangkeeree, P. Preechasilp, Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces, J. Inequal. Appl., 2013 (2013), 1-15
##[26]
H.-K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116 (2003), 659-678
]
On the split equality common fixed point problem for quasi-nonexpansive multi-valued mappings in Banach spaces
On the split equality common fixed point problem for quasi-nonexpansive multi-valued mappings in Banach spaces
en
en
The purpose of this paper is to study the split equality common fixed point problems of quasi-nonexpansive multi-valued mappings in the setting of Banach spaces. For solving this kind of problems, some new
iterative algorithms are proposed. Under suitable conditions, some weak and strong convergence theorems
for the sequences generated by the proposed algorithm are proved. The results presented in this paper are
new which also improve and extend some recent results announced by some authors.
5536
5543
Xuejin
Tian
College of Statistics and Mathematics
Yunnan University of Finance and Economics
P. R. China
1099439905@qq.com
Lin
Wang
College of Statistics and Mathematics
Yunnan University of Finance and Economics
P. R. China
WL64mail@aliyun.com
Zhaoli
Ma
Department of General Education
The College of Arts and Sciences Yunnan Normal University
P. R. China
kmszmzl@126.com
Split equality problem
quasi-nonexpansive multi-valued mapping
weak convergence
strong convergence.
Article.4.pdf
[
[1]
H. Attouch, J. Bolte, P. Redont, A. Soubeyran, Alternating proximal algorithms for weakly coupled convex minimization problems, Applications to dynamical games and PDE's, J. Convex Anal., 15 (2008), 485-506
##[2]
C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441-453
##[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]
Y. Censor, A. Motova, A. Segal, Perturbed projections and subgradient projections for the multiple-sets split feasibility problem, J. Math. Anal. Appl., 327 (2007), 1244-1256
##[7]
S.-S. Chang, J. K. Kim, Y. J. Cho, J. Y. Sim, Weak- and strong-convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces, Fixed Point Theory Appl., 2014 (2014), 1-12
##[8]
S.-S. Chang, L. Wang, Y. K. Tang, L. Yang, The split common fixed point problem for total asymptotically strictly pseudocontractive mappings, J. Appl. Math., 2012 (2012), 1-13
##[9]
I. Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (1990)
##[10]
A. Moudafi, The split common fixed-point problem for demicontractive mappings, Inverse Problems, 26 (2010), 1-6
##[11]
A. Moudafi, A note on the split common fixed-point problem for quasi-nonexpansive operators, Nonlinear Anal., 74 (2011), 4083-4087
##[12]
A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Anal., 79 (2013), 117-121
##[13]
B. Qu, N. H. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, 21 (2005), 1655-1665
##[14]
J. Quan, S.-S. Chang, X. Zhang, Multiple-set split feasibility problems for \(\kappa\)-strictly pseudononspreading mapping in Hilbert spaces, Abstr. Appl. Anal., 2013 (2013), 1-5
##[15]
J. F. Tang, S.-S. Chang, L. Wang, X. R. Wang, On the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces, J. Inequal. Appl., 2015 (2015), 1-11
##[16]
Y. J. Wu, R. D. Chen, L. Y. Shi, Split equality problem and multiple-sets split equality problem for quasi- nonexpansive multi-valued mappings, J. Inequal. Appl., 2014 (2014), 1-8
##[17]
H.-K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127-1138
##[18]
H.-K. Xu, A variable Krasnoselskiĭ-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021-2034
##[19]
Q. Z. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), 1261-1266
##[20]
X.-F. Zhang, L. Wang, Z. L. Ma, L. J. Qin, The strong convergence theorems for split common fixed point problem of asymptotically nonexpansive mappings in Hilbert spaces, J. Inequal. Appl., 2015 (2015), 1-11
]
Fixed points for \(\alpha\)-admissible contractive mappings via simulation functions
Fixed points for \(\alpha\)-admissible contractive mappings via simulation functions
en
en
Based on concepts of \(\alpha\)-admissible mappings and simulation functions, we establish some fixed point
results in the setting of metric-like spaces. We show that many known results in the literature are simple
consequences of our obtained results. We also provide some concrete examples to illustrate the obtained
results.
5544
5560
Abdelbasset
Felhi
Department of Mathematics and Statistics, College of Sciences
King Faisal University
Saudi Arabia
afelhi@kfu.edu.sa
Hassen
Aydi
Department of Mathematics, College of Education of Jubail
Department of Medical Research
University of Dammam
China Medical University Hospital, China Medical University
Saudi Arabia
Taiwan
hmaydi@uod.edu.sa
Dong
Zhang
School of Mathematical Sciences
Peking University
China
dongzhang@pku.edu.cn;zd20082100333@163.com
Metric-like
fixed point
simulation functions
\(\alpha\)-admissible mappings.
Article.5.pdf
[
[1]
T. Abdeljawad, E. Karapınar, K. Taş, A generalized contraction principle with control functions on partial metric spaces, Comput. Math. Appl., 63 (2012), 716-719
##[2]
A. Amini-Harandi, Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory Appl., 2012 (2012), 1-10
##[3]
H. Argoubi, B. Samet, C. Vetro, Nonlinear contractions involving simulation functions in a metric space with a partial order, J. Nonlinear Sci. Appl., 8 (2015), 1082-1094
##[4]
H. Aydi, A. Felhi, E. Karapınar, S. Sahmim, A Nadler-type fixed point theorem in metric-like spaces and applications, Miskolc Math. Notes, accepted (2015), -
##[5]
H. Aydi, A. Felhi, S. Sahmim, Fixed points of multivalued nonself almost contractions in metric-like spaces, Math. Sci. (Springer), 9 (2015), 103-108
##[6]
H. Aydi, M. Jellali, E. Karapınar, On fixed point results for \(\alpha\)-implicit contractions in quasi-metric spaces and consequences, Nonlinear Anal. Model. Control, 21 (2016), 40-56
##[7]
H. Aydi, E. Karapınar, Fixed point results for generalized \(\alpha-\psi\)-contractions in metric-like spaces and applications, Electron. J. Differential Equations, 2015 (2015), 1-15
##[8]
H. Aydi, E. Karapınar, C. Vetro, On Ekeland's variational principle in partial metric spaces, Appl. Math. Inf. Sci., 9 (2015), 257-262
##[9]
R. George, R. Rajagopalan, S. Vinayagam, Cyclic contractions and fixed points in dislocated metric spaces, Int. J. Math. Anal., 7 (2013), 403-411
##[10]
P. Hitzler, A. K. Seda, Dislocated topologies, J. Electr. Eng., 51 (2000), 3-7
##[11]
E. Karapınar, P. Kumam, P. Salimi, On \(\alpha-\psi\)-Meir-Keeler contractive mappings, Fixed Point Theory Appl., 2013 (2013), 1-12
##[12]
F. Khojasteh, S. Shukla, S. Radenović, A new approach to the study of fixed point theory for simulation functions, Filomat, 29 (2015), 1189-1194
##[13]
S. G. Matthews, Partial metric topology, Papers on general topology and applications, Flushing, NY, (1992), Ann. New York Acad. Sci., New York Acad. Sci., 1994 (1994), 183-197
##[14]
B. Mohammadi, Sh. Rezapour, N. Shahzad, Some results on fixed points of \(\alpha-\psi\)-Ciric generalized multifunctions, Fixed Point Theory Appl., 2013 (2013), 1-10
##[15]
A. Nastasi, P. Vetro, Fixed point results on metric and partial metric spaces via simulation functions, J. Nonlinear Sci. Appl., 8 (2015), 1059-1069
##[16]
S. J. O'Neill, Partial metrics, valuations, and domain theory, Papers on general topology and applications, Gorham, ME, (1995), 304-315, Ann. New York Acad. Sci., New York Acad. Sci., New York (1996)
##[17]
B. Samet, C. Vetro, P. Vetro, Fixed point theorems for \(\alpha-\psi\)-contractive type mappings, Nonlinear Anal., 75 (2012), 2154-2165
]
Proximal point algorithms involving fixed point of nonspreading-type multivalued mappings in Hilbert spaces
Proximal point algorithms involving fixed point of nonspreading-type multivalued mappings in Hilbert spaces
en
en
In this paper, a new modified proximal point algorithm involving fixed point of nonspreading-type
multivalued mappings in Hilbert spaces is proposed. Under suitable conditions, some weak convergence and
strong convergence to a common element of the set of minimizers of a convex function and the set of fixed
points of the nonspreading-type multivalued mappings in Hilbert space are proved. The presented results
in the paper are new.
5561
5569
Shih-Sen
Chang
Center for General Educatin
China Medical University
Taiwan
changss2013@163.com
Ding Ping
Wu
School of Applied Mathematics
Chengdu University of Information Technology Chengsu
China
wdp68@163.com
Lin
Wang
College of Statistics and Mathematics
Yunnan University of Finance and Economics
China
wl64mail@aliyun.com
Gang
Wang
College of Statistics and Mathematics
Yunnan University of Finance and Economics
China
wg631208@sina.com
Convex minimization problem
resolvent identity
proximal point algorithm
weak and strong convergence theorem
nonspreading-type multivalued mapping.
Article.6.pdf
[
[1]
R. P. Agarwal, D. O'Regan, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8 (2007), 61-79
##[2]
L. Ambrosio, N. Gigli, G. Savaré, Gradient ows in metric spaces and in the space of probability measures, Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (2008)
##[3]
D. Ariza-Ruiz, L. Leuştean, G. Lóez, Firmly nonexpansive mappings in classes of geodesic spaces, Trans. Amer. Math. Soc., 366 (2014), 4299-4322
##[4]
M. Bačák, The proximal point algorithm in metric spaces, Israel J. Math., 194 (2013), 689-701
##[5]
O. A. Boikanyo, G. Moroşanu, A proximal point algorithm converging strongly for general errors, Optim. Lett., 4 (2010), 635-641
##[6]
W. Cholamjiak, Shrinking projection methods for a split equilibrium problem and a nonspreading-type multivalued mapping, J. Nonlinear Sci. Appl., ((in press)), -
##[7]
P. Cholamjiak, A. A. N. Abdou, Y. J. Cho, Proximal point algorithms involving fixed points of nonexpansive mappings in CAT(0) spaces, Fixed Point Theory and Appl., 2015 (2015), 1-13
##[8]
O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), 403-419
##[9]
B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957-961
##[10]
J. Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature, Comment. Math. Helv., 70 (1995), 659-673
##[11]
S. Kamimura, W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory, 106 (2000), 226-240
##[12]
F. Kohsaka, W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. Optim., 19 (2008), 824-835
##[13]
G. Marino, H.-K. Xu, Convergence of generalized proximal point algorithms, Commun. Pure Appl. Anal., 3 (2004), 791-808
##[14]
B. Martinet, Régularisation d'inéquations variationnelles par approximations successives, (French) Rev. Franaise Informat. Recherche Opérationnelle, 4 (1970), 154-158
##[15]
R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optimization, 14 (1976), 877-898
]
Common fixed point theorems in Menger PMT--spaces with applications
Common fixed point theorems in Menger PMT--spaces with applications
en
en
In this paper, we introduce the concept of Menger PMT-spaces. Further, we prove common fixed point
theorems in a complete Menger probabilistic metric type space and, by using the main result, we give
applications on the existence and uniqueness of a solution for a class of integral equations.
5570
5578
Yeol Je
Cho
Department of Mathematics Education and the RINS
Center for General Education
Gyeongsang National University
China Medical University
Korea
Taiwan
yjcho@@gnu.ac.kr
Young-Oh
Yang
Department of Mathematics
Jeju National University
Korea
yangyo@jejunu.ac.kr
Nonlinear probabilistic contractive mapping
complete probabilistic metric type space
Menger space
fixed point theorem
integral equation.
Article.7.pdf
[
[1]
C. Alsina, B. Schweizer, A. Sklar, On the definition of a probabilistic normed space, Aequationes Math., 46 (1993), 91-98
##[2]
C. Alsina, B. Schweizer, A. Sklar, Continuity properties of probabilistic norms, J. Math. Anal. Appl., 208 (1997), 446-452
##[3]
A. T. Bharucha-Reid, Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc., 82 (1976), 641-657
##[4]
S.-S. Chang, Y. J. Cho, S. M. Kang, Nonlinear operator theory in probabilistic metric spaces, Nova Science Publishers, Inc., Huntington (2001)
##[5]
J. X. Fang, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems, 46 (1992), 107-113
##[6]
O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon., 44 (1996), 381-391
##[7]
M. A. Khamsi, V. Y. Kreinovich, Fixed point theorems for dissipative mappings in complete probabilistic metric spaces, Math. Japon., 44 (1996), 513-520
##[8]
D. O'Regan, R. Saadati, Nonlinear contraction theorems in probabilistic spaces, Appl. Math. Comput., 195 (2008), 86-93
##[9]
R. Saadati, D. O'Regan, S. M. Vaezpour, J. K. Kim, Generalized distance and common fixed point theorems in Menger probabilistic metric spaces, Bull. Iranian Math. Soc., 35 (2009), 97-117
##[10]
R. Saadati, S. M. Vaezpour, Linear operators in finite dimensional probabilistic normed spaces, J. Math. Anal. Appl., 346 (2008), 446-450
##[11]
B. Schweizer, A. Sklar, Probabilistic metric spaces, North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing Co., New York (1983)
##[12]
B. Schweizer, A. Sklar, E. Throp, The metrization of statistical metric spaces, Pacific J. Math., 10 (1960), 673-675
##[13]
A. N. Šerstnev, The notion of random normed space, (Russian) Dokl. Akad. Nauk SSSR, 149 (1963), 280-283, English translation in Soviet Math. Dokl., 4 (1963), 388-390
]
Bifurcations of twisted double homoclinic loops with resonant condition
Bifurcations of twisted double homoclinic loops with resonant condition
en
en
In this paper, the bifurcation problems of twisted double homoclinic loops with resonant condition
are studied for (m + n)-dimensional nonlinear dynamic systems. In the small tubular neighborhoods of
the homoclinic orbits, the foundational solutions of the linear variational systems are selected as the local
coordinate systems. The Poincaré maps are constructed by using the composition of two maps, one is
in the small tubular neighborhood of the homoclinic orbit, and another is in the small neighborhood of
the equilibrium point of system. By the analysis of bifurcation equations, the existence, uniqueness and
existence regions of the large homoclinic loops, large periodic orbits are obtained, respectively. Moreover,
the corresponding bifurcation diagrams are given.
5579
5620
Yinlai
Jin
School of Science
Linyi University
China
jinyinlai@sina.com
Man
Zhu
School of Science
School of Mathematical Sciences
Linyi University
Shandong Normal University
China
China
zhuman01@sina.cpm
Feng
Li
School of Science
Linyi University
China
lf0539@126.com
Dandan
Xie
School of Science
School of Mathematical Sciences
Linyi University
Shandong Normal University
China
China
xiedandan01@sina.com
Nana
Zhang
School of Science
School of Mathematical Sciences
Linyi University
Shandong Normal University
China
China
zhangnana01@sina.com
Double homoclinic loops
twisted
resonance
bifurcation
higher dimensional system.
Article.8.pdf
[
[1]
S.-N. Chow, B. Deng, B. Fiedler, Homoclinic bifurcation at resonant eigenvalues, J. Dynam. Differential Equations, 2 (1990), 177-244
##[2]
J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, Springer-Verlag, New York (1983)
##[3]
M. A. Han, D. J. Luo, D. M. Zhu, The uniqueness of limit cycles bifurcating from a singular closed orbit (I), Acta Math. Sinica (Chin. Ser.), 35 (1992), 407-417
##[4]
X. Huang, L. Y. Wang, Y. L. Jin, Stability of homoclinic loops to a saddle-focus in arbitrarily finite dimensional spaces, (Chinese), Chinese Ann. Math. Ser. A, 30 (2009), 563-574
##[5]
Y. L. Jin, Bifurcations of twisted homoclinic loops for degenerated cases, Appl. Math. J. Chinese Univ. Ser. B, 18 (2003), 186-192
##[6]
Y. L. Jin, F. Li, H. Xu, J. Li, L. Q. Zhang, B. Y. Ding, Bifurcations and stability of nondegenerated homoclinic loops for higher dimensional systems, Comput. Math. Methods Med., 2013 (2013), 1-9
##[7]
Y. L. Jin, H. Xu, Y. R. Gao, X. Zhao, D. D. Xie, Bifurcations of resonant double homoclinic loops for higher dimensional systems, J. Math. Computer Sci., 16 (2016), 165-177
##[8]
Y. L. Jin, D. M. Zhu, Degenerated homoclinic bifurcations with higher dimensions, Chinese Ann. Math. Ser. B, 21 (2000), 201-210
##[9]
Y. L. Jin, D. M. Zhu, Bifurcations of rough heteroclinic loops with three saddle points, Acta Math. Sin. (Engl. Ser.), 18 (2002), 199-208
##[10]
Y. L. Jin, D. M. Zhu, Bifurcations of rough heteroclinic loops with two saddle points, Sci. China Ser. A, 46 (2003), 459-468
##[11]
Y. L. Jin, D. M. Zhu, Twisted bifurcations and stability of homoclinic loop with higher dimensions, (Chinese) translated from Appl. Math. Mech., 25 (2004), 1076-1082, Appl. Math. Mech. (English Ed.), 25 (2004), 1176-1183
##[12]
Y. L. Jin, D. M. Zhu, Bifurcations of fine 3-point-loop in higher dimensional space, Acta Math. Sin. (Engl. Ser.), 21 (2005), 39-52
##[13]
Y. L. Jin, X. W. Zhu, Z. Guo, H. Xu, L. Q. Zhang, B. Y. Ding, Bifurcations of nontwisted heteroclinic loop with resonant eigenvalues, Sci. World J., 2014 (2014), 1-8
##[14]
Y. L. Jin, D. M. Zhu, Q. G. Zheng, Bifurcations of rough 3-point-loop with higher dimensions, Chinese Ann. Math. Ser. B, 24 (2003), 85-96
##[15]
G. Kovačić, S. Wiggins, Orbits homoclinic to resonance with an application to chaos in a model of the forced and damped sine-Gordon equation, Phys. D, 57 (1992), 185-225
##[16]
X. B. Liu, D. M. Zhu, On the stability of homoclinic loops with higher dimension, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 915-932
##[17]
Q. Y. Lu, Codimension 2 bifurcation of twisted double homoclinic loops, Comput. Math. Appl., 57 (2009), 1127-1141
##[18]
D. J. Luo, X. Wang, D. M. Zhu, M. A. Han, Bifurcation theory and methods of dynamical systems, Advanced Series in Dynamical Systems, World Scientific, Singapore (1997)
##[19]
K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256
##[20]
S. Wiggins, Global bifurcations and chaos, Analytical methods, Applied Mathematical Sciences, Springer-Verlag, New York (1988)
##[21]
S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Second edition, Texts in Applied Mathematics, Springer-Verlag, New York (2003)
##[22]
W. P. Zhang, D. M. Zhu, Codimension 2 bifurcations of double homoclinic loops, Chaos Solitons Fractals, 39 (2009), 295-303
##[23]
D. M. Zhu, Problems in homoclinic bifurcation with higher dimensions, Acta Math. Sinica (N.S.), 14 (1998), 341-352
##[24]
D. M. Zhu, Z. H. Xia, Bifurcations of heteroclinic loops, Sci. China Ser. A, 41 (1998), 837-484
]
A new numerical technique for local fractional diffusion equation in fractal heat transfer
A new numerical technique for local fractional diffusion equation in fractal heat transfer
en
en
In this paper, a new numerical approach, embedding the differential transform (DT) and Laplace trans-
form (LT), is firstly proposed. It is considered in the local fractional derivative operator for obtaining the
non-differential solution for diffusion equation in fractal heat transfer.
5621
5628
Xiao-Jun
Yang
School of Mechanics and Civil Engineering
State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering
China University of Mining and Technology
China University of Mining and Technology
China
China
J. A. Tenreiro
Machado
Department of Electrical Engineering
Institute of Engineering, Polytechnic of Porto
Portugal
Dumitru
Baleanu
Department of Mathematics
Cankya University
Institute of Space Sciences
Turkey
Romania
Feng
Gao
School of Mechanics and Civil Engineering
State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering
China University of Mining and Technology
China University of Mining and Technology
China
China
jsppw@sohu.com
Numerical solution
di usion equation
di erential transform
Laplace transform
fractal heat transfer
local fractional derivative.
Article.9.pdf
[
[1]
R. Almeida, A. B. Malinowska, D. F. M. Torres, A fractional calculus of variations for multiple integrals with application to vibrating string, J. Math. Phys., 51 (2010), 1-12
##[2]
M. Alquran, K. Al-Khaled, M. Ali, A. Ta'any, The combined Laplace transform-differential transform method for solving linear non-homogeneous PDEs, J. Math. Comput. Sci., 2 (2012), 690-701
##[3]
D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Models and numerical methods, World Sci., 3 (2012), 10-16
##[4]
D. Baleanu, J. A. T. Machado, A. C. J. Luo, Fractional dynamics and control, Springer, Berlin (2012)
##[5]
A. H. Bhrawy, E. H. Doha, D. Baleanu, S. S. Ezz-Eldien, A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations, J. Comput. Phys., 293 (2015), 142-156
##[6]
Y. Cao, W.-G. Ma, L.-C. Ma, Local fractional functional method for solving diffusion equations on Cantor sets, Abstr. Appl. Anal., 2014 (2014), 1-6
##[7]
A. Carpinteri, P. Cornetti, A fractional calculus approach to the description of stress and strain localization in fractal media, Chaos Solitons Fractals, 13 (2002), 85-94
##[8]
C. Cattani, H. M. Srivastava, X.-J. Yang, Fractional dynamics, De Gruyter Open, Berlin (2015)
##[9]
K. Diethelm, N. J. Ford, A. D. Freed, Y. Luchko, Algorithms for the fractional calculus: a selection of numerical methods, Comput. Methods Appl. Mech. Engrg, 194 (2005), 743-773
##[10]
A. K. Golmankhaneh, A. K. Golmankhaneh, D. Baleanu, On nonlinear fractional Klein-Gordon equation, Signal Process., 91 (2011), 446-451
##[11]
R. Goren o, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Fractals and fractional calculus in continuum mechanics, Udine, (1996), 223-276, CISM Courses and Lectures, Springer, Vienna (1997)
##[12]
R. Goren o, F. Mainardi, D. Moretti, G. Pagnini, P. Paradisi, Fractional diffusion: probability distributions and random walk models, Non extensive thermodynamics and physical applications, Villasimius, (2001), Phys. A, 305 (2002), 106-112
##[13]
J.-H. He, A tutorial review on fractal spacetime and fractional calculus, Internat. J. Theoret. Phys., 53 (2014), 3698-3718
##[14]
J. Hristov, Approximate solutions to fractional subdiffusion equations, Eur. Phys. J. Spec. Top., 193 (2013), 229-243
##[15]
H. Jafari, S. Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 1962-1969
##[16]
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 (1997)
##[17]
K. M. Kolwankar, A. D. Gangal, Local fractional Fokker-Planck equation, Phys. Rev. Lett., 80 (1998), 214-217
##[18]
C. P. Li, F. H. Zeng, The finite difference methods for fractional ordinary differential equations, Numer. Funct. Anal. Optim., 34 (2013), 149-179
##[19]
F. Liu, V. V. Anh, I. Turner, P. Zhuang, Time fractional advection-dispersion equation, J. Appl. Math. Comput., 13 (2003), 233-245
##[20]
H.-Y. Liu, J.-H. He, Z.-B. Li, Fractional calculus for nanoscale flow and heat transfer, Internat. J. Numer. Methods Heat Fluid Flow, 24 (2014), 1227-1250
##[21]
C.-F. Liu, S.-S. Kong, S.-J. Yuan, Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem, Therm. Sci., 17 (2013), 715-721
##[22]
Y. Luchko, R. Goren o, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math. Vietnam., 24 (1999), 207-233
##[23]
Y. F. Luchko, H. M. Srivastava, The exact solution of certain differential equations of fractional order by using operational calculus, Comput. Math. Appl., 29 (1995), 73-85
##[24]
C. I. Muresan, C. Ionescu, S. Folea, R. De Keyser, Fractional order control of unstable processes: the magnetic levitation study case, Nonlinear Dyn., 80 (2014), 1761-1772
##[25]
Z. Odibat, S. Momani, A generalized differential transform method for linear partial differential equations of fractional order, Appl. Math. Lett., 21 (2008), 194-199
##[26]
M. D. Ortigueira, Fractional calculus for scientists and engineers, Lecture Notes in Electrical Engineering, Springer, Dordrecht (2011)
##[27]
I. Podlubny, A. Chechkin, T. Skovranek, Y. Q. Chen, B. M. Vinagre Jara, Matrix approach to discrete fractional calculus. II. Partial fractional differential equations, J. Comput. Phys., 228 (2009), 3137-3153
##[28]
Y. A. Rossikhin, M. V. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results, Appl. Mech. Rev., 63 (2010), 1-52
##[29]
M. G. Sakar, F. Erdogan, A. Yıldırım, Variational iteration method for the time-fractional Fornberg-Whitham equation, Comput. Math. Appl., 63 (2012), 1382-1388
##[30]
S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Theory and applications, Edited and with a foreword by S. M. Nikolskiĭ, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon (1993)
##[31]
N. T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput., 131 (2002), 517-529
##[32]
S.Wei, W. Chen, Y.-C. Hon, Implicit local radial basis function method for solving two-dimensional time fractional diffusion equations, Therm. Sci., 19 (2015), 59-67
##[33]
B. J. West, P. Grigolini, R. Metzler T. F. Nonnenmacher, Fractional diffusion and Lévy stable processes, Phys. Rev. E, 55 (1997), 99-106
##[34]
X.-J. Yang, Advanced local fractional calculus and its applications, World Sci., New York (2012)
##[35]
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
##[36]
X.-J. Yang, D. Baleanu, H. M. Srivastava, Local fractional integral transforms and their applications, Elsevier/Academic Press, Amsterdam (2016)
##[37]
X.-J. Yang, J. T. Machado, J. Hristov, Nonlinear dynamics for local fractional Burgers' equation arising in fractal ow, Nonlinear Dyn., 84 (2015), 3-7
##[38]
X.-J. Yang, H. M. Srivastava, An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives, Commun. Nonlinear Sci. Numer. Simul., 29 (2015), 499-504
##[39]
A.-M. Yang, Y.-Z. Zhang, Y. Long, The Yang-Fourier transforms to heat-conduction in a semi-infinite fractal bar, Therm. Sci., 17 (2013), 707-713
##[40]
Y.-Z. Zhang, Initial boundary value problem for fractal heat equation in the semi-infinite region by Yang-Laplace transform, Therm. Sci., 18 (2014), 677-681
]
The form of solutions and periodic nature for some rational difference equations systems
The form of solutions and periodic nature for some rational difference equations systems
en
en
In this paper, we investigate the expressions of solutions and the periodic nature of the following systems
of rational difference equations with order four
\[x_{n+1} = \frac{y_{n-3} }{\pm 1\pm y_nz_{n-1}x_{n-2}y_{n-3}},
y_{n+1} = \frac{z_{n-3} }{\pm 1\pm z_nx_{n-1}y_{n-2}z_{n-3}},
z_{n+1} = \frac{x_{n-3} }{\pm 1\pm x_ny_{n-1}z_{n-2}x_{n-3}},\]
with initial conditions \(x_{-3}; x_{-2}; x_{-1}; x_0; y_{-3}; y_{-2}; y_{-1}; y_0; z_{-3}; z_{-2}; z_{-1}\) and \(z_0\) which are arbitrary real
numbers.
5629
5647
M. M.
El-Dessoky
Faculty of Science, Mathematics Department
Department of Mathematics, Faculty of Science
King Abdulaziz University
Mansoura University
Saudi Arabia
Egypt
dessokym@mans.edu.eg
E. M.
Elsayed
Faculty of Science, Mathematics Department
Department of Mathematics, Faculty of Science
King Abdulaziz University
Mansoura University
Saudi Arabia
Egypt
emelsayed@mans.edu.eg
E. O.
Alzahrani
Department of Mathematics, Faculty of Science
Mansoura University
Egypt
eoalzahrani@kau.edu.sa
Difference equations
recursive sequences
stability
periodic solution
system of difference equations.
Article.10.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), 1-2000
##[2]
M. Aloqeili, Dynamics of a rational difference equation, Appl. Math. Comput., 176 (2006), 768-774
##[3]
N. Battaloglu, C. Çinar, I. Yalcinkaya, The dynamics of the difference equation\( x_{n+1} = \frac{\alpha x_{n-k}}{ \beta+\gamma x^p_{n-(k+1)}}\), Ars Combin., 97 (2010), 281-288
##[4]
E. Camouzis, M. R. S. Kulenović, G. Ladas, O. Merino, Rational systems in the plane, J. Difference Equ. Appl., 15 (2009), 303-323
##[5]
C. Çinar, On the positive solutions of the difference equation system \(x_{n+1 }= \frac{1}{ y_n} ; y_{n+1} =\frac{ y_n}{ x_{n-1}y_{n-1}}\), Appl. Math. Comput., 158 (2004), 303-305
##[6]
D. Clark, M. R. S. Kulenović, A coupled system of rational difference equations, Comput. Math. Appl., 43 (2002), 849-867
##[7]
S. E. Das, M. Bayram, On a system of rational difference equations, World Appl. Sci. J., 10 (2010), 1306-1312
##[8]
Q. Din, T. F. Ibrahim, K. A. Khan, Behavior of a competitive system of second-order difference equations, Sci. World J., 2014 (2014), 1-9
##[9]
M. DiPippo, E. J. Janowski, M. R. S. Kulenović, Global asymptotic stability for quadratic fractional difference equation, Adv. Difference Equ., 2015 (2015), 1-13
##[10]
E. M. Elabbasy, S. M. Eleissawy, Asymptotic behavior of two dimensional rational system of difference equations, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 20 (2013), 221-235
##[11]
E. M. Elabbasy, H. A. El-Metwally, E. M. Elsayed, On the solutions of a class of difference equations systems, Demonstratio Math., 41 (2008), 109-122
##[12]
E. M. Elabbasy, H. A. El-Metwally, E. M. Elsayed, Global behavior of the solutions of some difference equations, Adv. Difference Equ., 2011 (2011), 1-16
##[13]
S. Elaydi, An introduction to difference equations, Third edition, Undergraduate Texts in Mathematics, Springer, New York (2005)
##[14]
M. M. El-Dessoky, The form of solutions and periodicity for some systems of third-order rational difference equations, Math. Methods Appl. Sci., 39 (2016), 1076-1092
##[15]
E. M. Elsayed, M. Mansour, M. M. El-Dessoky, Solutions of fractional systems of difference equations, Ars Combin., 110 (2013), 469-479
##[16]
M. E. Erdogan, C. Cinar, I. Yalcinkaya, On the dynamics of the recursive sequence \(x_{n+1} = \frac{x_{n-1}}{\beta+\gamma x^2_{n-2}x_{n-4}+\gamma x_{n-2}x^2_{n-4}}\), Comput. Math. Appl., 61 (2011), 553-537
##[17]
M. Garić-Demirović, M. Nurkanović, Dynamics of an anti-competitive two dimensional rational system of difference equations, Sarajevo J. Math., 7 (2011), 39-56
##[18]
E. A. Grove, G. Ladas, Periodicities in nonlinear difference equations, Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC, Boca Raton (2005)
##[19]
E. A. Grove, G. Ladas, L. C. McGrath, C. T. Teixeira, Existence and behavior of solutions of a rational system, Appl. Nonlinear Anal., 8 (2001), 1-25
##[20]
T. F. Ibrahim, Periodicity and analytic solution of a recursive sequence with numerical examples, J. Interdiscip. Math., 12 (2009), 701-708
##[21]
V. L. Kocić, G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (1993)
##[22]
M. R. S. Kulenović, Z. Nurkanović, Global behavior of a three-dimensional linear fractional system of difference equations, J. Math. Anal. Appl., 310 (2005), 673-689
##[23]
A. S. Kurbanli , On the behavior of solutions of the system of rational dierence equations \(x_{n+1} =\frac{ x_{n-1}}{\gamma _nx_{n-1}-1} ; \gamma _{n+1} = \frac{\gamma _{n-1}}{ x_n\gamma_{ n-1}-1} ; z_{n+1} =\frac{ 1}{ \gamma _nz_n}\), Adv. Difference Equ., 2011 (2011), 1-8
##[24]
A. S. Kurbanli, C. Çinar, I. Yalçinkaya, On the behavior of positive solutions of the system of rational difference equations \(x_{n+1} =\frac{ x_{n-1}}{ y_nx_{n-1}+1} , y_{n+1} =\frac{ y_{n-1}}{ x_ny_{n-1}+1}\), Math. Comput. Modelling, 53 (2011), 1261-1267
##[25]
A. Y. Özban, On the system of rational difference equations \(x_{n+1} = \frac{a}{y_{n-3}}; y_{n+1} = \frac{by_{n-3}}{x_{n-q}y_{n-q}}\), Appl. Math. Comput., 188 (2007), 833-837
##[26]
O. Özkan, A. S. Kurbanli, On a system of difference equations, Discrete Dyn. Nat. Soc., 2013 (2013), 1-7
##[27]
N. Touafek, E. M. Elsayed, On the periodicity of some systems of nonlinear difference equations, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 55 (2012), 217-224
##[28]
C.-Y. Wang, S. Wang, W. Wang, Global asymptotic stability of equilibrium point for a family of rational difference equations, Appl. Math. Lett., 24 (2011), 714-718
##[29]
I. Yalcinkaya, On the global asymptotic behavior of a system of two nonlinear difference equations, Ars Combin., 95 (2010), 151-159
##[30]
Y. Yang, L. Chen, Y.-G. Shi, On solutions of a system of rational difference equations, Acta Math. Univ. Comenian. (N.S.), 80 (2011), 63-70
##[31]
X. F. Yang, Y. X. Liu, S. Bai, On the system of high order rational difference equations \(x_n =\frac{ a}{ y_{n-p}} ; y_n =\frac{ by_{n-p}}{ x_{n-q}y_{n-q}}\), Appl. Math. Comput., 171 (2005), 853-856
##[32]
Z. H. Yuan, L. H. Huang, All solutions of a class of discrete-time systems are eventually periodic, Appl. Math. Comput., 158 (2004), 537-546
##[33]
Q. H. Zhang, J. Z. Liu, Z. G. Luo, Dynamical behavior of a system of third-order rational difference equation, Discrete Dyn. Nat. Soc., 2015 (2015), 1-6
##[34]
Y. Zhang, X. F. Yang, G. M. Megson, D. J. Evans, On the system of rational difference equations \(x_n = A + \frac{1}{ y_{n-p}} ; y_n = A +\frac{ y_{n-1}}{ x_{n-r}y_{n-s}}\), Appl. Math. Comput., 176 (2006), 403-408
]
New Hermite-Hadamard inequalities for twice differentiable \(\phi\)-MT-preinvex functions
New Hermite-Hadamard inequalities for twice differentiable \(\phi\)-MT-preinvex functions
en
en
New Hermite-Hadamard-type integral inequalities for \(\phi\)-MT-preinvex functions are obtained. Our results
in special cases yield some of those results proved in recent articles concerning with the differentiable MTconvex
functions. Some applications to special means and the trapezoidal formula are also considered,
respectively.
5648
5660
Sheng
Zheng
College of Science
China Three Gorges University
P. R. China
zhengshengctgu@gmail.com
Ting-Song
Du
College of Science
China Three Gorges University
P. R. China
tingsongdu@ctgu.edu.cn
Sha-Sha
Zhao
College of Science
China Three Gorges University
P. R. China
zhaoshashactgu@gmail.com
Lian-Zi
Chen
College of Science
China Three Gorges University
P. R. China
chenlianzictgu@gmail.com
\(\phi\)-MT-preinvex functions
Hermite-Hadamard's integral inequality
Hölder's inequality.
Article.11.pdf
[
[1]
T. Antczak, Mean value in invexity analysis, Nonlinear Anal., 60 (2005), 1473-1484
##[2]
S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math., 33 (2002), 55-65
##[3]
N. Eftekhari, Some remarks on (s;m)-convexity in the second sense, J. Math. Inequal., 8 (2014), 489-495
##[4]
H. K. Fok, S. W. Vong, Generalizations of some Hermite-Hadamard-type inequalities, Indian J. Pure Appl. Math., 46 (2015), 359-370
##[5]
A. G. Ghazanfari, A. Barani, Some Hermite-Hadamard type inequalities for the product of two operator preinvex functions, Banach J. Math. Anal., 9 (2015), 9-20
##[6]
R. Jakšić, L. Kvesić, J. Pečarić, On weighted generalization of the Hermite-Hadamard inequality,, Math. Inequal. Appl., 18 (2015), 649-665
##[7]
M. A. Latif, S. S. Dragomir, Generalization of Hermite-Hadamard type inequalities for n-times differentiable functions which are s-preinvex in the second sense with applications, Hacet. J. Math. Stat., 44 (2015), 839-853
##[8]
M. A. Latif, S. S. Dragomir, New integral inequalities of Hermite-Hadamard type for n-times differentiable s- logarithmically convex functions with applications, Miskolc Math. Notes, 16 (2015), 219-235
##[9]
M. A. Latif, S. S. Dragomir, On Hermite-Hadamard type integral inequalities for n-times differentiable log-preinvex functions, Filomat, 29 (2015), 1651-1661
##[10]
Y. J. Li, T. S. Du, On Simpson type inequalities for functions whose derivatives are extended (s;m)-GA-convex functions, Pure Appl. Math. China, 31 (2015), 487-497
##[11]
Y. J. Li, T. S. Du, Some improvements on Hermite-Hadamard's inequalities for s-convex functions, J. Math. Study, 49 (2016), 82-92
##[12]
Y. J. Li, T. S. Du, Some Simpson type integral inequalities for functions whose third derivatives are (\(\alpha,m\))-GA- convex functions, J. Egyptian Math. Soc., 24 (2016), 175-180
##[13]
Y. J. Li, T. S. Du, A generalization of Simpson type inequality via differentiable functions using extended \((s;m)_\phi\)- preinvex functions, J. Comput. Anal. Appl., 22 (2017), 613-632
##[14]
W. J. Liu, Ostrowski type fractional integral inequalities for MT-convex functions, Miskolc Math. Notes, 16 (2015), 249-256
##[15]
M. A. Noor, Some new classes of nonconvex functions, Nonlinear Funct. Anal. Appl., 11 (2006), 165-171
##[16]
M. A. Noor, K. I. Noor, Generalized preinvex functions and their properties, J. Appl. Math. Stoch. Anal., 2006 (2006), 1-13
##[17]
K. I. Noor, M. A. Noor, Relaxed strongly nonconvex functions, Appl. Math. E-Notes, 6 (2006), 259-267
##[18]
M. E. Özdemir, M. Avci, H. Kavurmaci, Hermite-Hadamard-type inequalities via (\(\alpha,m\))-convexity, Comput. Math. Appl., 61 (2011), 2614-2620
##[19]
M. E. Özdemir, M. Avci, E. Set, On some inequalities of Hermite-Hadamard type via m-convexity, Appl. Math. Lett., 23 (2010), 1065-1070
##[20]
J. K. Park, Hermite-Hadamard-like Type Inequalities for Twice Differentiable MT-Convex Functions, Appl. Math. Sci., 9 (2015), 5235-5250
##[21]
J. K. Park, Some Hermite-Hadamard Type Inequalities for MT-Convex Functions via Classical and Riemann- Liouville Fractional Integrals, Appl. Math. Sci., 9 (2015), 5011-5026
##[22]
Z. Pavić, Improvements of the Hermite-Hadamard inequality,, J. Inequal. Appl., 2015 (2015), 1-11
##[23]
R. Pini, Invexity and generalized convexity, Optimization, 22 (1991), 513-525
##[24]
M. H. Qu, W. J. Liu, J. Park, Some new Hermite-Hadamard-type inequalities for geometric-arithmetically s- convex functions, WSEAS Trans. Math., 13 (2014), 452-461
##[25]
M. Z. Sarikaya, M. E. Kiris, Some new inequalities of Hermite-Hadamard type for s-convex functions, Miskolc Math. Notes, 16 (2015), 491-501
##[26]
M. Tunç, On some integral inequalities via h-convexity, Miskolc Math. Notes, 14 (2013), 1041-1057
##[27]
M. Tunç, Ostrowski type inequalities for functions whose derivatives are MT-convex, J. Comput. Anal. Appl., 17 (2014), 691-696
##[28]
M. Tunç, Y. Şubaş, I. Karabayir, On some Hadamard type inequalities for MT-convex functions, Int. J. Open Problems Comput. Math., 6 (2013), 102-113
##[29]
M. Tunç, H. Yildirim, On MT-convexity, arXiv preprint, 2012 (2012), 1-7
##[30]
B.-Y. Xi, F. Qi, Inequalities of Hermite-Hadamard type for extended s-convex functions and applications to means, J. Nonlinear Convex Anal., 16 (2015), 873-890
##[31]
Z. Q. Yang, Y. J. Li, T. S. Du, A generalization of Simpson type inequality via differentiable functions using (s;m)-convex functions, Ital. J. Pure Appl. Math., 35 (2015), 327-338
##[32]
X. M. Yang, X. Q. Yang, K. L. Teo, Generalized invexity and generalized invariant monotonicity, J. Optim. Theory Appl., 117 (2003), 607-625
]
Some coupled fixed point results on cone metric spaces over Banach algebras and applications
Some coupled fixed point results on cone metric spaces over Banach algebras and applications
en
en
Our purpose in this work is to present several coupled fixed point results for different contraction mappings on cone metric spaces over Banach algebras by virtue of the properties of spectral radiuses. Also as
an application, we give a simple example at the end of the paper.
5661
5671
Pinghua
Yan
Department of Mathematics
Nanchang University
P. R. China
mathyph@163.com
Jiandong
Yin
Department of Mathematics
Nanchang University
P. R. China
yjdaxf@163.com
Qianqian
Leng
Department of Mathematics
Nanchang University
P. R. China
13517914026@163.com
Cone metric spaces over Banach algebras
coupled fixed points
contractions
spectral radiuses.
Article.12.pdf
[
[1]
M. Abbas, M. Ali Khan, S. Radenović, Common coupled fixed point theorems in cone metric spaces for w- compatible mappings, Appl. Math. Comput., 217 (2010), 195-202
##[2]
M. Abbas, G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl., 341 (2008), 416-420
##[3]
M. Abbas, B. E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl. Math. Lett., 22 (2009), 511-515
##[4]
T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379-1393
##[5]
L. G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468-1476
##[6]
V. Lakshmikantham, L. Ćirić, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70 (2009), 4341-4349
##[7]
H. Liu, S. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory and Appl., 2013 (2013), 1-10
##[8]
S. K. Malhotra, S. Shukla, R. Sen, Some coincidence and common fixed point theorems in cone metric spaces, Bull. Math. Anal. Appl., 2 (2012), 64-71
##[9]
S. Radenović, Common fixed points under contractive conditions in cone metric spaces, Comput. Math. Appl., 58 (2009), 1273-1278
##[10]
S. Radenović, B. E. Rhoades, Fixed point theorem for two non-self mappings in cone metric spaces, Comput. Math. Appl., 57 (2009), 1701-1707
##[11]
H. Rahimi, P. Vetro, G. Soleimani Rad, Coupled fixed-point results for T-contractions on cone metric spaces with applications, Math. Notes, 98 (2015), 158-167
##[12]
M. Rangamma, K. Prudhvi, Common fixed points under contractive conditions for three maps in cone metric spaces, Bull. Math. Anal. Appl., 4 (2012), 174-180
##[13]
W. Rudin, Functional analysis, Second edition, International Series in Pure and Applied Mathematics, McGraw- Hill, Inc., New York (1991)
##[14]
F. Sabetghadam, H. Masiha, A. H. Sanatpour, Some coupled fixed point theorems in cone metric spaces, Fixed Point Theory and Appl., 2009 (2009), 1-8
##[15]
W. Shatanawi, On w-compatible mappings and common coupled coincidence point in cone metric spaces, Appl. Math. Lett., 25 (2012), 925-931
##[16]
S. Y. Xu, S. Radenović, Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality, Fixed Point Theory and Appl., 2014 (2014), 1-12
]
General viscosity iterative method for a sequence of quasi-nonexpansive mappings
General viscosity iterative method for a sequence of quasi-nonexpansive mappings
en
en
In this paper, we study a general viscosity iterative method due to Aoyama and Kohsaka for the fixed
point problem of quasi-nonexpansive mappings in Hilbert space. First, we obtain a strong convergence
theorem for a sequence of quasi-nonexpansive mappings. Then we give two applications about variational
inequality problem to encourage our main theorem. Moreover, we give a numerical example to illustrate our
main theorem.
5672
5682
Cuijie
Zhang
College of Science
Civil Aviation University of China
China
cuijie_zhang@126.com
Yinan
Wang
College of Science
Civil Aviation University of China
China
yinan_wang@163.com
Quasi-nonexpansive mapping
variational inequality
fixed point
viscosity iterative method.
Article.13.pdf
[
[1]
K. Aoyama, F. Kohsaka, Viscosity approximation process for a sequence of quasinonexpansive mappings, Fixed Point Theory and Appl., 2014 (2014), 1-11
##[2]
A. Cegielski, R. Zalas, Methods for variational inequality problem over the intersection of fixed point sets of quasi-nonexpansive operators, Numer. Funct. Anal. Optim., 34 (2013), 255-283
##[3]
W. G. Doson, Fixed points of quasi-nonexpansive mappings, J. Austral. Math. Soc., 13 (1972), 167-170
##[4]
M. K. Ghosh, L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings, J. Math. Anal. Appl., 207 (1997), 96-103
##[5]
S. N. He, C. P. Yang, Solving the variational inequality problem defined on intersection of finite level sets, Abstr. Appl. Anal., 2013 (2013), 1-8
##[6]
G. E. Kim, Weak and strong convergence for quasi-nonexpansive mappings in Banach spaces, Bull. Korean. Math. Soc., 49 (2012), 799-813
##[7]
D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London (1980)
##[8]
R. Li, Z. H. He, A new iterative algorithm for split solution problems of quasi-nonexpansive mappings, J. Inequal. Appl., 2015 (2015), 1-12
##[9]
P. E. Maingé, The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces, Comput. Math. Appl., 59 (2010), 74-79
##[10]
G. Marino, H.-K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318 (2006), 43-52
##[11]
A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55
##[12]
W. V. Petryshyn, T. E. Williamson, Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, J. Math. Anal. Appl., 43 (1973), 459-497
##[13]
M. Tian, X. Jin, A general iterative method for quasi-nonexpansive mappings in Hilbert space, J. Inequal. Appl., 2012 (2012), 1-8
##[14]
I. Yamada, N. Ogura, Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings, Numer. Funct. Anal. Optim., 25 (2006), 619-655
##[15]
J. Zhao, S. N. He, Strong convergence of the viscosity approximation process for the split common fixed-point problem of quasi-nonexpansive mappings, J. Appl. Math., 2012 (2012), 1-12
]
Hybrid projection algorithm concerning split equality fixed point problem for quasi-pseudo-contractive mappings with application to optimization problem
Hybrid projection algorithm concerning split equality fixed point problem for quasi-pseudo-contractive mappings with application to optimization problem
en
en
The purpose of this paper is by using the shrinking projection method to study the split equality fixed
point problem for a class of quasi-pseudo-contractive mappings in the setting of Hilbert spaces. Under
suitable conditions, some strong convergence theorems are obtained. As applications, we utilize the results
presented in the paper to study the existence problem of solutions to the split equality variational inequality
problem and the split equality convex minimization problem. The results presented in our paper extend
and improve some recent results.
5683
5694
Jin-Fang
Tang
College of Mathematics
Yibin University
China
Shih-Sen
Chang
Center for General Education
China Medical University
Taiwan
changss2013@163.com
Ching-Feng
Wen
Center for Fundamental Science
Kaohsiung Medical University
Taiwan
Jian
Dong
College of Mathematics
Yibin University
China
Split equality fixed point problem
quasi-pseudo-contractive mapping
hybrid projection algorithm
strong convergence theorem.
Article.14.pdf
[
[1]
E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145
##[2]
C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441-453
##[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]
Y. Censor, A. Motova, A. Segal, Perturbed projections and subgradient projections for the multiple-sets split feasibility problem, J. Math. Anal. Appl., 327 (2007), 1244-1256
##[7]
S.-S. Chang, R. P. Agarwal, Strong convergence theorems of general split equality problems for quasi-nonexpansive mappings, J. Inequal. Appl., 2014 (2014), 1-14
##[8]
S.-S. Chang, L. Wang, L.-J. Qin, Split equality fixed point problem for quasi-pseudo-contractive mappings with applications, Fixed Point Theory and Appl., 2015 (2015), 1-12
##[9]
S.-S. Chang, L. Wang, Y. K. Tang, G. Wang, Moudafi's open question and simultaneous iterative algorithm for general split equality variational inclusion problems and general split equality optimization problems, Fixed Point Theory and Appl., 2014 (2014), 1-17
##[10]
H. T. Che, M. X. Li, A simultaneous iterative method for split equality problems of two finite families of strictly pseudononspreading mappings without prior knowledge of operator norms, Fixed Point Theory and Appl., 2015 (2015), 1-14
##[11]
Z. H. He, W.-S. Du, On split common solution problems: new nonlinear feasible algorithms, strong convergence results and their applications, Fixed Point Theory and Appl., 2014 (2014), 1-16
##[12]
A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl., 150 (2011), 275-283
##[13]
A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Anal., 79 (2013), 117-121
##[14]
A. Moudafi, E. Al-Shemas, Simultaneous iterative methods for split equality problem, Trans. Math. Program. Appl., 1 (2013), 1-11
##[15]
J. F. Tang, S.-S. Chang, Strong convergence theorem of two-step iterative algorithm for split feasibility problems, J. Inequal. Appl., 2014 (2014), 1-13
##[16]
J. F. Tang, S.-S. Chang, F. Yuan, A strong convergence theorem for equilibrium problems and split feasibility problems in Hilbert spaces, Fixed Point Theory Appl., 2014 (2014), 1-16
##[17]
Y. H. Yao, Y.-C. Liou, J.-C. Yao, Split common fixed point problem for two quasi-pseudo-contractive operators and its algorithm construction, Fixed Point Theory and Appl., 2015 (2015), 1-19
##[18]
J. L. Zhao, Q. Z. Yang, Several solution methods for the split feasibility problem, Inverse Problems, 21 (2005), 1791-1799
]
Strong convergence of a general iterative algorithm for asymptotically nonexpansive semigroups in Banach spaces
Strong convergence of a general iterative algorithm for asymptotically nonexpansive semigroups in Banach spaces
en
en
In this paper, we study a general iterative process strongly converging to a common fixed point of an
asymptotically nonexpansive semigroup \(\{T(t) : t \in \mathbb{R }^+\}\) in the framework of reflexive and strictly convex
spaces with a uniformly Gáteaux differentiable norm. The process also solves some variational inequalities.
Our results generalize and extend many existing results in the research field.
5695
5711
Lishan
Liu
School of Mathematical Sciences
Department of Mathematics and Statistics
Qufu Normal University
Curtin University
China
Australia
mathlls@163.com
Chun
Liu
School of Mathematical Sciences
Qufu Normal University
China
liuchunjn@aliyun.com
Fang
Wang
School of Mathematical Sciences
Qufu Normal University
China
Yonghong
Wu
Department of Mathematics and Statistics
Curtin University
Australia
Asymptotically nonexpansive semigroups
variational inequality
strong convergence
reflexive and strictly convex Banach spaces
fixed point.
Article.15.pdf
[
[1]
K. Aoyama, F. Kohsaka, Viscosity approximation process for a sequence of quasinonexpansive mappings, Fixed Point Theory and Appl., 2014 (2014), 1-11
##[2]
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120
##[3]
G. Cai, S. Q. Bu, A viscosity approximation scheme for finite mixed equilibrium problems and variational inequality problems and fixed point problems, Comput. Math. Appl., 62 (2011), 440-454
##[4]
G. Cai, S. Q. Bu, Convergence analysis for variational inequality problems and fixed point problems in 2-uniformly smooth and uniformly convex Banach spaces, Math. Comput. Modelling, 55 (2012), 538-546
##[5]
G. Cai, S. Q. Bu, A viscosity scheme for mixed equilibrium problems, variational inequality problems and fixed point problems, Math. Comput. Modelling, 57 (2013), 1212-1226
##[6]
K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174
##[7]
K. S. Ha, J. S. Jung, On generators and nonlinear semigroups in Banach spaces, J. Korean Math. Soc., 25 (1988), 245-257
##[8]
B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957-961
##[9]
I. Inchan, S. Plubtieng, Strong convergence theorems of hybrid methods for two asymptotically nonexpansive mappings in Hilbert spaces, Nonlinear Anal. Hybrid Syst., 2 (2008), 1125-1135
##[10]
A. R. Khan, H. Fukhar-ud-din, A. Kalsoom, B. S. Lee, Convergence of a general algorithm of asymptotically nonexpansive maps in uniformly convex hyperbolic spaces, Appl. Math. Comput., 238 (2014), 547-556
##[11]
X. N. Li, J. S. Gu, Strong convergence of modified Ishikawa iteration for a nonexpansive semigroup in Banach spaces, Nonlinear Anal., 73 (2010), 1058-1092
##[12]
L. S. Liu, Fixed points of local strictly pseudo-contractive mappings using Mann and Ishikawa iteration with errors, Indian J. Pure Appl. Math., 26 (1995), 649-659
##[13]
L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl., 194 (1995), 114-125
##[14]
L. S. Liu, Ishikawa-type and Mann-type iterative processes with errors for constructing solutions of nonlinear equations involving m-accretive operators in Banach spaces, Nonlinear Anal., 34 (1998), 307-317
##[15]
J. Lou, L.-J. Zhang, Z. He, Viscosity approximation methods for asymptotically nonexpansive mappings, Appl. Math. Comput., 203 (2008), 171-177
##[16]
G. Marino, V. Colao, X. L. Qin, S. M. Kang, Strong convergence of the modified Mann iterative method for strict pseudo-contractions, Comput. Math. Appl., 57 (2009), 455-465
##[17]
G. Marino, H.-K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318 (2006), 43-52
##[18]
A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55
##[19]
C. I. Podilchuk, R. J. Mammone, Image recovery by convex projections using a least-squares constrain, J. Optical Soc. Amer. A, 7 (1990), 517-521
##[20]
X. L. Qin, M. J. Shang, S. M. Kang, Strong convergence theorems of modified Mann iterative process for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 70 (2009), 1257-1264
##[21]
G. S. Saluja, M. Postolache, A. Kurdi, Convergence of three-step iterations for nearly asymptotically nonexpansive mappings in CAT(k) spaces, J. Inequal. Appl., 2015 (2015), 1-18
##[22]
M. I. Sezan, H. Stark, Applications of convex projection theory to image recovery in tomography and related areas, H. Stark (Ed.), Image Recovery: Theory and Applications, Academic, 1987 (1987), 415-462
##[23]
Y. S. Song, S. M. Xu, Strong convergence theorems for nonexpansive semigroup in Banach spaces, J. Math. Anal. Appl., 338 (2008), 152-161
##[24]
Y. F. Su, S. H. Li, Strong convergence theorems on two iterative method for non-expansive mappings, Appl. Math. Comput., 181 (2006), 331-341
##[25]
B. S. Thakur, R. Dewangan, M. Postolache, Strong convergence of new iteration process for a strongly continuous semigroup of asymptotically pseudocontractive mappings, Numer. Funct. Anal. Optim., 34 (2013), 1418-1431
##[26]
D. Thakur, B. S. Thakur, M. Postolache, New iteration scheme for numerical reckoning fixed points of nonexpansive mappings, J. Inequal. Appl., 2014 (2014), 1-15
##[27]
B. S. Thakur, D. Thakur, M. Postolache, A new iteration scheme for approximating fixed points of nonexpansive mappings, Filomat, ((in press)), -
##[28]
S. Thianwan, Common fixed points of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space, J. Comput. Appl. Math., 224 (2009), 688-695
##[29]
Y. X. Tian, S. S. Chang, J. L. Huang, On the approximation problem of common fixed points for a finite family of non-self asymptotically quasi-nonexpansive-type mappings in Banach spaces, Comput. Math. Appl., 53 (2007), 1847-1853
##[30]
H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256
##[31]
H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279-291
##[32]
L.-P. Yang, The general iterative scheme for semigroups of nonexpansive mappings and variational inequalities with applications, Math. Comput. Modelling, 57 (2013), 1289-1297
##[33]
L.-P. Yang, W.-M. Kong, A general iterative algorithm for semigroups of nonexpansive mappings with generalized contractive mapping, Appl. Math. Comput., 222 (2013), 671-679
##[34]
Y. H. Yao, M. Postolache, Iterative methods for pseudomonotone variational inequalities and fixed-point problems, J. Optim. Theory Appl., 155 (2012), 273-287
##[35]
Y. H. Yao, M. Postolache, Y.-C. Liou, Strong convergence of a self-adaptive method for the split feasibility problem, Fixed Point Theory and Appl., 2013 (2013), 1-12
##[36]
Y. H. Yao, M. Postolache, Y.-C. Liou, Variant extragradient-type method for monotone variational inequalities, Fixed Point Theory and Appl., 2013 (2013), 1-15
##[37]
D. C. Youla, Mathematical theory of image restoration by the method of convex projections, H. Stark (Ed.), Image Recovery: Theory and Applications, Academic, New York, 1987 (1987), 29-77
##[38]
D. C. Youla, On deterministic convergence of iterations of relaxed projection operators, J. Vis. Commun. Image Represent., 1 (1990), 12-20
##[39]
H. Zegeye, N. Shahzad, Convergence theorems for strongly continuous semi-groups of asymptotically nonexpansive mappings, Nonlinear Anal., 71 (2009), 2308-2315
##[40]
H. Zegeye, N. Shahzad, Convergence of Mann's type iteration method for generalized asymptotically nonexpansive mappings, Comput. Math. Appl., 62 (2011), 4007-4014
]
On the existence of solutions of generalized equilibrium problems with \(\alpha-\beta-\eta\)-monotone mappings
On the existence of solutions of generalized equilibrium problems with \(\alpha-\beta-\eta\)-monotone mappings
en
en
The present paper is concerned with the new concept of relaxed \(\alpha-\beta-\eta\)-monotonicity and relaxed \(\alpha-\beta-\eta\)-pseudomonotonicity in Banach space which is applied to prove the existence of solutions of generalized
equilibrium problem and classic equilibrium problem. In this regard, we use the well-known KKM-theory
to obtain solutions of mentioned problems.
5712
5719
A.
Farajzadeh
Department of Mathematics
Razi University
Iran
farajzadehali@gmail.com
S.
Plubtieng
Department of Mathematics, Faculty of Science
Center of Excellence in Nonlinear Analysis and Optimization, Faculty of Science
Naresuan University
Naresuan University
Thailand
Thailand
Somyotp@nu.ac.th
A.
Hosseinpour
Department of Mathematics, Faculty of Science
Center of Excellence in Nonlinear Analysis and Optimization, Faculty of Science
Naresuan University
Naresuan University
Thailand
Thailand
h.mathematical@gmail.com
KKM-mappings
hemicontinuity
\(\alpha-\beta-\eta\)-monotonicity
\(\alpha-\beta-\eta\)-pseudomonotonicity
semicontinuous mappings
Banach space.
Article.16.pdf
[
[1]
C. L. Ballard, D. Fullerton, J. B. Shoven, J. Whalley, A general equilibrium model for tax policy evaluation, University of Chicago Press, Chicago (1985)
##[2]
M. Binachi, S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl., 90 (1996), 31-43
##[3]
G. Bonanno, General equilibrium theory with imperfect competition, J. Econ. Surv., 4 (1990), 297-328
##[4]
L.-C. Ceng, C.-M. Chen, C.-F. Wen, C. T. Pang, Relaxed iterative algorithms for generalized mixed equilibrium problems with constraints of variational inequalities and variational inclusions, Abstr. Appl. Anal., 2014 (2014), 1-25
##[5]
O. Chadli, Z. Chbani, H. Riahi, Equilibrium problem with generalized monotone bifunctions and applications to variational inequalities, J. Optim. Theory Appl., 105 (2000), 299-323
##[6]
Y. J. Cho, N. Petrot, An optimization problem related to generalized equilibrium and fixed point problems with applications, Fixed Point Theory, 11 (2010), 237-250
##[7]
X.-P. Ding, Iterative algorithm of solutions for generalized mixed implicit equilibrium-like problems, Appl. Math. Comput., 162 (2005), 799-809
##[8]
X.-P. Ding, J.-C. Yao, L.-J. Lin, Solutions of system of generalized vector quasi-equilibrium problems in locally G-convex uniform spaces, J. Math. Anal. Appl., 298 (2004), 398-410
##[9]
J. N. Ezeora, Y. Shehu, An iterative method for mixed point problems of nonexpansive and monotone mappings and generalized equilibrium problems, Common. Math. Anal., 12 (2012), 76-95
##[10]
K. Fan, A generalization of Tychonoff's fixed point theorem, Math. Ann., 142 (1961), 305-310
##[11]
Y. F. Ke, C. F. Ma, A new relaxed extragradient-like algorithm for approaching common solutions of generalized mixed equilibrium problems, a more general system of variational inequalities and a fixed point problem, Fixed Point Theory Appl., 2013 (2013), 1-21
##[12]
N. K. Mahato, C. Nahak, Mixed equilibrium problems with relaxed \(\alpha\)-monotone mapping in Banach spaces, Rend. Circ. Mat. Palermo, 62 (2013), 207-213
##[13]
N. K. Mahato, C. Nahak, Equilibrium problems with generalized relaxed monotonicities in Banach spaces, Opsearch, 51 (2014), 257-269
##[14]
M. A. Noor, Auxiliary principle technique for equilibrium problems, J. Optim. Theory Appl., 122 (2004), 371-386
##[15]
N. Onjai-uea, C. Jaiboon, P. Kumam, A relaxed hybrid steepest descent method for common solutions of generalized mixed equilibrium problems and fixed point problems, Fixed Point Theory Appl., 2011 (2011), 1-20
##[16]
H. A. Rizvi, A. Kılıçman, R. Ahmad, Generalized equilibrium problem with mixed relaxed monotonicity, Sci.World J., 2014 (2014), 1-4
##[17]
W. Takahashi, K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal., 70 (2009), 45-57
##[18]
S. H. Wang, G. Marino, F. H. Wang, Strong convergence theorems for a generalized equilibrium problem with a relaxed monotone mapping and a countable family of nonexpansive mappings in a Hilbert space, Fixed Point Theory and Appl., 2010 (2010), 1-22
##[19]
X.-Y. Zang, L. Deng, Iterative algorithm of solutions for multivalued general mixed implicit equilibrium-like problems, Appl. Math. Mech. (English Ed.), 29 (2008), 477-484
]