Abstract
Abstract The purpose of this article is first to introduce the concept of total quasi-ϕ-asymptotically nonexpansive multi-valued mapping which contains many kinds of mappings as its special cases, and then by using the hybrid shrinking technique to propose an iterative algorithm for finding a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of common fixed points for a countable family of multi-valued total quasi-ϕ-asymptotically nonexpansive mappings in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the article not only generalize some recent results from single-valued mappings to multi-valued mappings, but also improve and extend the main results of Homaeipour and Razani. 2000 AMS Subject Classification: 47J06; 47J25.
Highlights
Throughout this article, we always assume that X is a real Banach space with the dual X*, C is a nonempty closed convex subset of X, and J : X ® 2X is the normalized duality mapping defined by
We use F(T ) to denote the set of fixed points of a mapping T, and use R and R+ to denote the set of all real numbers and the set of all nonnegative real numbers, respectively
Motivated and inspired by the researches going on in this direction, the purpose of this article is first to introduce the concept of total quasi-j-asymptotically nonexpansive multi-valued mapping which contains multi-valued relatively nonexpansive mappings and many other kinds of mappings as its special cases, and by using the hybrid shirking iterative algorithm for finding a common element of the set of solutions for a generalized mixed equilibrium problem (MEP), the set of solutions for variational inequality problems, and the set of common fixed points for a countable family of multi-valued total quasi-j-asymptotically nonexpansive mappings in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property
Summary
Throughout this article, we always assume that X is a real Banach space with the dual X*, C is a nonempty closed convex subset of X, and J : X ® 2X is the normalized duality mapping defined by. Motivated and inspired by the researches going on in this direction, the purpose of this article is first to introduce the concept of total quasi-j-asymptotically nonexpansive multi-valued mapping which contains multi-valued relatively nonexpansive mappings and many other kinds of mappings as its special cases, and by using the hybrid shirking iterative algorithm for finding a common element of the set of solutions for a generalized MEP, the set of solutions for variational inequality problems, and the set of common fixed points for a countable family of multi-valued total quasi-j-asymptotically nonexpansive mappings in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property.
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