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]	Piao Yongjie. Improvements of Fixed Point Theorems for Mappings with AContractions on Metric Spaces [J]. Acta mathematica scientia,Series A, 2018, 38(5): 855-863.
[2]	Zou Xia, Wu Shiliang. Traveling Waves in a Nonlocal Dispersal SIR Epidemic Model with Delay and Nonlinear Incidence [J]. Acta mathematica scientia,Series A, 2018, 38(3): 496-513.
[3]	Chang Shih-sen, Liu Zhenhai, Wen Ching-Feng, Tang Jinfang. Split Variational Inclusion Problem Involving Fixed Point for an Asymptotically Nonexpansive Semigroup with Application to Optimization Problem [J]. Acta mathematica scientia,Series A, 2018, 38(2): 231-243.
[4]	Li Haixia. Stability and Uniqueness of Positive Solutions for a Food-Chain Model [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1094-1104.
[5]	Guo Tingguang, Xu Zhiting. Existence of Traveling Waves in a Spatial Infectious Disease Model with Vaccination [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1129-1147.
[6]	Wang Chun, Xu Tianzhou. Solution and Hyers-Ulam-Rassias Stability of a Mixed Type Quadratic-Additive Functional Equation with a Parameter in Quasi-Banach Spaces [J]. Acta mathematica scientia,Series A, 2017, 37(5): 846-859.
[7]	Yao Yuwu, Chen Xiu, Niu Xin. Distributional Chaos on Complex Sectors [J]. Acta mathematica scientia,Series A, 2017, 37(5): 950-961.
[8]	Song Jiping, Wang Maofa. Common Fixed Point Results for Generalized Weak Contractions with a Rational Expression in Ordered Partial Metric Spaces [J]. Acta mathematica scientia,Series A, 2017, 37(3): 401-415.
[9]	Wang Xuewu. Convergence and Stability of Ishikawa Type Iterative Sequences for the φ-Hemi Pseudocontractive Operator Pair [J]. Acta mathematica scientia,Series A, 2017, 37(2): 248-256.
[10]	Hao Xinan, Liu Lishan. Positive Solutions for Nonlocal Boundary Value Problems of Fourth Order Differential Equation with Parameters [J]. Acta mathematica scientia,Series A, 2017, 37(2): 287-298.
[11]	Deng Weiqi. A System of Split Equilibrium Problems for Sequences of Bifunctions [J]. Acta mathematica scientia,Series A, 2016, 36(4): 763-770.
[12]	Xia Zhinan. On Existence and Asymptotic Behavior of Solutions for Functional Integral Equation of Volterra-Stieltjes Type [J]. Acta mathematica scientia,Series A, 2016, 36(1): 130-143.
[13]	Luo Ting, Zhu Chuanxi. Tripled Coincidence and Tripled Fixed Point Theorems for Compatible Mappings Under Contractive Conditions in Partially Ordered Probabilistic Metric Spaces [J]. Acta mathematica scientia,Series A, 2016, 36(1): 157-167.
[14]	Zeng Chaoying, Suyalatu. QCLkR Spaces and QCLkS Spaces [J]. Acta mathematica scientia,Series A, 2015, 35(6): 1080-1088.
[15]	Liu Yuanxing, Zhou Haiyun, Wang Peiyuan, Zhou Yu. Three Kinds of Iterative Algorithms for Solving the Constrained Split Common Fixed Point Problem in Hilbert Spaces [J]. Acta mathematica scientia,Series A, 2015, 35(6): 1115-1126.