Abstract
Abstract The purpose of this paper is to construct a new iterative scheme and to get a strong convergence theorem for a countable family of relatively quasi-nonexpansive mappings and a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. The notion of uniformly closed mappings is presented and an example will be given which is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings. Another example shall be given which is uniformly closed but does not satisfy condition AKTT and ∗AKTT. Our results can be applied to solve a convex minimization problem. In addition, this paper clarifies an ambiguity in a useful lemma. The results of this paper modify and improve many other important recent results. MSC:47H05, 47H09, 47H10.
Highlights
Introduction and preliminaries LetE be a real Banach space and C be a nonempty closed convex subset of E
We say that a mapping T is relatively nonexpansive if the following conditions are satisfied: (I) F(T) = ∅; (II) φ(p, Tx) ≤ φ(p, x), ∀x ∈ C, p ∈ F(T); (III) F(T) = F (T)
Shehu [ ] proved strong convergence theorems for approximation of common element of set of common fixed points of countably infinite family of relatively quasinonexpansive mappings and set of common solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f -projection operator
Summary
Lemma . [ ] Let C be a nonempty, closed, and convex subset of a smooth and reflexive. Let {Tn} be a sequence of mappings from C into E, where C is a nonempty closed convex subset of a real Banach space E. Shehu [ ] proved strong convergence theorems for approximation of common element of set of common fixed points of countably infinite family of relatively quasinonexpansive mappings and set of common solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f -projection operator. In this paper we will construct a new iterative scheme and will get strong convergence theorem for a countable family of relatively quasi-nonexpansive mappings and a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f -projection operator.
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