Approximating common solutions of variational inequalities by iterative algorithms with applications

In this paper, we introduce an iterative scheme for a general variational inequality. Strong convergence theorems of common solutions of two variational inequalities are established in a uniformly convex and 2-uniformly smooth Banach space. As applications, we, still in Banach spaces, consider the convex feasibility problem.

[1]  Tomonari Suzuki Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals , 2005 .

[2]  Wataru Takahashi,et al.  Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings , 2005 .

[3]  Hong-Kun Xu Iterative Algorithms for Nonlinear Operators , 2002 .

[4]  S. Reich Strong convergence theorems for resolvents of accretive operators in Banach spaces , 1980 .

[5]  F. Browder,et al.  FIXED-POINT THEOREMS FOR NONCOMPACT MAPPINGS IN HILBERT SPACE. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

[6]  Ronald E. Bruck Properties of fixed-point sets of nonexpansive mappings in Banach spaces , 1973 .

[7]  Hong-Kun Xu Inequalities in Banach spaces with applications , 1991 .

[8]  F. Browder,et al.  Construction of fixed points of nonlinear mappings in Hilbert space , 1967 .

[9]  Wataru Takahashi,et al.  Image recovery by convex combinations of sunny nonexpansive retractions , 1993 .

[10]  W. Takahashi Nonlinear Functional Analysis , 2000 .

[11]  Jen-Chih Yao,et al.  On modified iterative method for nonexpansive mappings and monotone mappings , 2007, Appl. Math. Comput..

[12]  Simeon Reich,et al.  Asymptotic behavior of contractions in Banach spaces , 1973 .

[13]  K. Deimling Nonlinear functional analysis , 1985 .

[14]  Y. Censor,et al.  Parallel Optimization: Theory, Algorithms, and Applications , 1997 .

[15]  P. L. Combettes,et al.  The Convex Feasibility Problem in Image Recovery , 1996 .

[16]  Yongfu Su,et al.  Approximation of a zero point of accretive operator in Banach spaces , 2007 .

[17]  J Shamir,et al.  Image restoration by a novel method of parallel projection onto constraint sets. , 1995, Optics letters.

[18]  Jen-Chih Yao,et al.  Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities , 2008, Math. Methods Oper. Res..

[19]  M. Shang,et al.  Strong convergence of the composite Halpern iteration , 2008 .

[20]  Wataru Takahashi,et al.  Weak convergence of an iterative sequence for accretive operators in Banach spaces , 2006 .

[21]  Yan Hao,et al.  Strong Convergence of an Iterative Method for Inverse Strongly Accretive Operators , 2008 .