Strong convergence theorem for the modified generalized equilibrium problem and fixed point problem of strictly pseudo-contractive mappings

Abstract

The purpose of this paper is to modify the generalized equilibrium problem introduced by Ceng et al. (J. Glob. Optim. 43:487-502, 2012) and to introduce the K-mapping generated by a finite family of strictly pseudo-contractive mappings and finite real numbers modifying the results of Kangtunyakarn and Suantai (Nonlinear Anal. 71:4448-4460, 2009). Then we prove the strong convergence theorem for finding a common element of the set of fixed points of a finite family of strictly pseudo-contractive mappings and a finite family of the set of solutions of the modified generalized equilibrium problem. Moreover, using our main result, we obtain the additional results related to the generalized equilibrium problem.

Strong Convergence of New Viscosity Rules of Nonexpansive Mappings

“…In 2014, Suwannaut and Kangtunyakarn [8] established the following result for K-mapping generated by T 1 , T 2 , T 3 , ..., T N and λ 1 , λ 2 , λ 3 , ..., λ N . Lemma 3.9.…”

“…Lemma 3.9. [8] Let C be a nonempty closed convex subset of a real Hilbert space H. For i = 1, 2, 3, ..., N, let {T i } N i=1 be a finite family of K i -strictly pseudocontractive mapping of C into itself with K i ≤ ω i and ∩ N i=1 F (T i ) ̸ = ∅, namely there exist constants K i ∈ [0, 1) such that…”