q-一致光滑、一致凸Banach空间中关于变分不等式问题和严格伪压缩映射的不动点问题的粘性迭代算法

Abstract

Abstract:

The purpose of this paper is to investigate a viscosity iterative algorithm based on a generalized contraction for finding a common element of the set of solutions of a general variational inequality problem for two inverse-strongly accretive mappings and the set of common fixed points for infinite strict pseudo-contractions in q-uniformly smooth and uniformly convex Banach spaces. Strong convergence theorems are proved under some appropriate conditions. The results obtained in this paper improve and extend many recent ones announced by many others in this literature.

Key words: Fixed point, Variational inequality, Strong convergence, Pseudocontraction, Banach space

CLC Number:

O177.91

Cai Gang, Yekini Shehu. Viscosity Iterative Algorithm for Variational Inequality Problems and Fixed Point Problems of Strict Pseudo-Contractions in q-Uniformly Smooth and Uniformly Convex Banach Spaces[J].Acta mathematica scientia,Series A, 2016, 36(4): 623-638.

Trendmd

References

[1] Xu H K. Inequalities in Banach spaces with applications. Nonlinear Anal, 1991, 16: 1127-1138
[2] Zhang H, Su Y. Strong convergence theorems for strict pseudo-contractions in q-uniformly smooth Banach spaces. Nonlinear Anal, 2009, 70: 3236-3242
[3] Reich S. Asymptotic behavior of contractions in Banach spaces. J Math Anal Appl, 1973, 44: 57-70
[4] Kitahara S, Takahashi W. Image recovery by convex combinations of sunny nonexpansive retractions. Topological Methods in Nonlinear Anal, 1993, 2: 333-342
[5] Browder F E, Petryshyn W V. Construction of fixed points of nonlinear mappings in Hilbert spaces. J Math Anal Appl, 1967, 20: 197-228
[6] Lim T C. On characterizations of Meir-Keeler contractive maps. Nonlinear Anal, 2001, 46: 113-120
[7] Suzuki T. Moudafi's viscosity approaximations with Meir-Keeler contraction. J Math Anal Appl, 2007, 325: 342-352
[8] Ceng L C, Wang C, Yao J C. Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math Methods Oper Res, 2008, 67: 375-390
[9] Yao Y, Noor M A, Noor K I, Liou Y C. Modified extragradient methods for a system of variational inequalities in Banach spaces. Acta Appl Math, 2010, 110: 1211-1224
[10] Zegeye H, Shahzad N. Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv Fixed Point Theory, 2012, 2: 374-397
[11] Yao Y, Liou Y C, Kang S M, Yu Y. Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces. Nonlinear Anal, 2011, 74: 6024-6034
[12] Qin X, Cho S Y, Kang S M. Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications. J Comput Appl Math, 2009, 233: 231-240
[13] Qin X, Chang S S, Cho Y J. Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal: Real World Applications, 2010, 11: 2963-2972
[14] Kumama P, Petrot N, Wangkeeree R. A hybrid iterative scheme for equilibrium problems and fixed point problems of asymptotically k-strict pseudo-contractions. J Comput App Math, 2010, 233: 2013-2026
[15] Qin X, Cho S Y, Kang S M. Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications. J Comput Anal Appl, 2009, 233: 231-240
[16] Xu H K. Viscosity approximation methods for nonexpansive mappings. J Math Anal Appl, 2004, 298: 279-291
[17] Suzuki T. Strong convergence of Krasnoselskii and Manns type sequences for one parameter nonexpansive semigroups without Bochner integrals. J Math Anal Appl, 2005, 305: 227-239
[18] Xu H K. Inequalities in Banach spaces with applications. Nonlinear Anal, 1991, 16: 1127-1138
[19] Mitrinovi? D S. Analytic Inequalities. New York: Springer-Verlag, 1970
[20] Bruck R E. Properties of fixed point sets of nonexpansive mappings in Banach spaces. Trans Amer Math Soc, 1973, 179: 251-262
[21] Kamimura S, Takahashi W. Strong convergence of a proximal-type algorithm in a Banach space. SIAM J Optim, 2002, 13: 938-945
[22] Zhang H, Su Y. Strong convergence theorems for strict pseudo-contractions in q-uniformly smooth Banach spaces. Nonlinear Anal, 2009, 70: 3236-3242
[23] Chang S S. On Chidumes open questions and approximate solutions of multivalued strongly accretive mapping equations in Banach spaces. J Math Anal Appl, 1997, 216: 94-111
[24] Plubtieng S, Ungchittrakool K. Approximation of common fixed points for a countable family of relatively nonexpansive mappings in a Banach space and applications. Nonlinear Anal, 2010, 72: 2896-2908
[25] Wang Z M, Zhang X. Shrinking projection methods for systems of mixed variational inequalities of Browder type, systems of mixed equilibrium problems and fixed point problems. J Nonlinear Funct Anal, 2014, Article ID: 15
[26] Song Y L, Ceng L C. A general iteration scheme for variational inequality problem and common fixed point problems of nonexpansive mappings in q-uniformly smooth Banach spaces. J Glob Optim, 2013, 57: 1327-1348
[27] Sunthrayuth P, Kumam P. Iterative methods for variational inequality problemsand fixed point problems of a countable family of strict pseudo-contractions in a q-uniformly smooth Banach space. Fixed Point Theory and Applications, 2012, Article ID: 65
[28] Sunthrayuth P, Kumam P. Fixed point solutions for variational inequalities in image restoration over q-uniformly smooth Banach spaces. J Ineq Appl, 2014, Article ID: 473
[29] Gu F. Necessary and sufficient condition of the strong convergence for two finite families of uniformly L-Lipschitzain mappings. Acta Math Sci, 2011, 31B(5): 2058-2066
[30] Kim J K, Cho S Y, Qin X. Some results on generalized equilibrium problems involving strictly pseudocontractive mappings. Acta Math Sci, 2011, 31B(5): 2041-2057
[31] Cho S Y, Kang S M. Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math Sci, 2012, 32B(4): 1607-1618
[32] Shehu Y. Approximation of fixed points and variational solutions for pseudo-contractive mappings in Banach spaces. Acta Math Sci, 2014, 34B(2): 409-423

Viscosity Iterative Algorithm for Variational Inequality Problems and Fixed Point Problems of Strict Pseudo-Contractions in q-Uniformly Smooth and Uniformly Convex Banach Spaces

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Li Nizhou,Zhao Siyi,Zhang Jialing. Iteration of a Class of Separable Markov Mappings [J]. Acta mathematica scientia,Series A, 2025, 45(2): 321-333.
[2]	Wang Wujing, Zhu Meiling, Zhang Yongle. A New Projection Algorithm for Solving Quasimonotone Variational Inequality Problems and Fixed Point Problems [J]. Acta mathematica scientia,Series A, 2025, 45(1): 236-255.
[3]	Zhang Yuting, Gao Xinghui, Peng Jianying. Iterative Algorithms of Common Elements for the Set of Solutions of Split Feasibility Problem and the Set of Common Fixed Points of a Finite Family of Quasi-Nonexpansive Operators [J]. Acta mathematica scientia,Series A, 2025, 45(1): 256-268.
[4]	Huo Huixia, Li Yongxiang. Existence of Positive Solutions for a Bending Elastic Beam Equation [J]. Acta mathematica scientia,Series A, 2024, 44(6): 1476-1484.
[5]	Wang Ruyu, Zhao Wenling, Song Daojin. The Finite Termination of Feasible Solution Sequence for Optimization and Variational Inequality [J]. Acta mathematica scientia,Series A, 2024, 44(4): 1037-1051.
[6]	Ma Xiaojun, Chen Fu, Jia Zhifu. Research on a Strong Convergence Theorem for Proximal Split Feasibility Problems with Non-Lipschitz Stepsizes [J]. Acta mathematica scientia,Series A, 2024, 44(4): 1052-1065.
[7]	Guo Zhiyuan, Gao Yuan, Sun Jiacheng, Zhang Guowei. Positive Solutions of Nonlocal Beam Equations with Fully Nonlinear Terms [J]. Acta mathematica scientia,Series A, 2024, 44(3): 621-636.
[8]	Gao Chenghua, Ding Huanhuan, He Xingyue. Radial Solutions of Coupled Systems for a Class of Superlinear $ k$-Hessian Equations [J]. Acta mathematica scientia,Series A, 2024, 44(2): 384-395.
[9]	Zhang Guangxin, Yang Yunrui, Song Xue. Periodic Traveling Wave Solutions of Delayed SEIR Systems with Nonlocal Effects [J]. Acta mathematica scientia,Series A, 2024, 44(1): 140-159.
[10]	Zhang Zhipeng, Zhang Liang. Controllability of a Nonlinear Parabolic Systems with Drift-Diffusion Term [J]. Acta mathematica scientia,Series A, 2023, 43(6): 1744-1758.
[11]	Pan Lingrong, Wang Yuanheng. The Strong Convergence Theorem of Iterative Algorithms for the Fixed Point Problem, a System of Variational Inequalities, and a Split Equilibrium Problem in Hilbert Spaces [J]. Acta mathematica scientia,Series A, 2023, 43(6): 1880-1896.
[12]	Yang Lanxiang, Chen Yi, Ye Minglu. Inertial Projection Algorithms for Quasimonotone Variational Inequalities [J]. Acta mathematica scientia,Series A, 2023, 43(2): 593-603.
[13]	Xia Pingjing, Cai Gang. Self Adaptive Viscosity Algorithm for Solving Variational Inequality Problem in Hilbert Spaces [J]. Acta mathematica scientia,Series A, 2023, 43(2): 581-592.
[14]	Song Yuying, Fan Hongxia. Existence and Regularity of Solutions for a Class of Neutral Stochastic Evolution Equations [J]. Acta mathematica scientia,Series A, 2023, 43(1): 238-248.
[15]	Duan Jie, Xia Fuquan. A New Tseng-like Extragradient Algorithm for Common Solutions of Variational Inequalities and Fixed Point Problems [J]. Acta mathematica scientia,Series A, 2023, 43(1): 274-290.