A novel hybrid method for equilibrium problem and a countable family of generalized nonexpansive-type maps with applications

Abstract

Let C be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space E with dual space E * . A novel hybrid method for finding a solution of an equilibrium problem and a common element of fixed points for a family of a general class of nonlinear nonexpansive maps is constructed. The sequence of the method is proved to converge strongly to a common element of the family and a solution of the equilibrium problem. Finally, an application of our theorem complements, generalizes and extends some recent important results (Takahashi et al., Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,

A hybrid scheme for fixed points of a countable family of generalized nonexpansive-type maps and finite families of variational inequality and equilibrium problems, with applications

“…Definition 2.8. (Generalized J * nonexpansive) [20] A map T : C → E * will be called generalized J * −nonexpansive if F J (T ) = ∅, and φ(p, (J * oT )x) ≤ φ(p, x) for all x ∈ C and for all p ∈ F J (T ).…”

“…Remark 2.2. Exampes of generalized J * −nonexpansive maps in Hilbert and more general Banach spaces were given in [4,20].…”

“…Lemma 2.8. [20] Let C be a nonempty closed subset of a smooth, strictly convex and reflexive Banach space E. Let A : C → E * be a continuous monotone mapping. For r > 0 and let x ∈ E, define a mapping F r (x) : E → C as follows:…”

“…Lemma 2.9. [20] Let E be a uniformly convex and uniformly smooth real Banach space with dual space E * and let C be a closed subset of E such that JC is closed and convex. Let T be a generalized J * −nonexpansive map from C to E * such that F J (T ) = ∅, then F J (T ) and JF J (T ) are closed.…”

“…In this paper, inspired by the above results especially the works in [4,20,24], we present an algorithm for finding a common element of the fixed point of an infinite family of generalized J * −nonexpansive maps, the solution set of the variational inequality problem of a finite family of continuous monotone maps and the solution set of the equilibrium point of a finite family of bifunctions satisfying some given conditions. Our results complement, generalize and extend results in [13,16,17,24] (see and other recent results in this direction.…”