Construction of a common element for the set of solutions of fixed point problems and generalized equilibrium problems in Hilbert spaces

Abstract

In this paper, we propose and analyse an iterative algorithm for the approximation of a common solution for a finite family of k-strict pseudocontractions and two finite families of generalized equilibrium problems in the setting of Hilbert spaces. Strong convergence results of the proposed iterative algorithm together with some applications to solve the variational inequality problems are established in such setting. Our results generalize and improve various existing results in the current literature. 1. Introduction Throughout this paper, we work in the setting of a real Hilbert space H equipped with the inner product ⟨· , ·⟩and the induced norm ∥·∥. Let C be a nonempty subset of a real Hilbert space H and let T : C →C be a mapping. The set of fixed points of the mapping T is defined and denoted as F(T) = {x ∈ C : T(x) = x}. The mapping T is said to be (i) a contraction, if ∥Tx −Ty∥≤α∥x −y∥for all x, y ∈C and α ∈(0, 1); (ii) nonexpansive, if ∥Tx −Ty∥≤∥x −y∥for all x, y ∈C; (iii) Lipschitzian, if ∥Tx−Ty∥≤L∥x−y∥for some L > 0 and for all x, y ∈C; (iv) firmly nonexpansive, if ∥Tx −Ty∥2 ≤⟨x −y, Tx −Ty⟩ for all x, y ∈C; (1) (v) a pseudo-contraction, if ⟨x −y , Tx −Ty⟩≤∥x −y∥2 for all x, y ∈C; (2)