Convergence theorems for uniformly quasi-ϕ-asymptotically nonexpansive mappings, generalized equilibrium problems, and variational inequalities

Abstract

In this article, we introduce an iterative algorithm for finding a common element of the set of common fixed points of an infinite family of closed and uniformly quasi--asymptotically nonexpansive mappings, the set of the variational inequality for an a-inverse-strongly monotone operator, and the set of solutions of the generalized equilibrium problems. We obtain a strong convergence theorem for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. The results presented in this article improve and extend the recent results of

An Inertial Algorithm of Generalized f -Projection for Maximal Monotone Operators and Generalized Mixed Equilibrium Problems in Banach Spaces

“…respectively. Then, the function Γ satisfies conditions (L1) − (L4) and T r has the properties (a) − (e) of Lemma 2.16 (see [26,34,29]). Therefore iterative sequence (3.1) can be rewritten as…”

“…Existence and the uniqueness of the operator Π C follows from the properties of the functional φ(y, x) and strict monotonicity of the mapping J. If E is a real Hilbert space H, then φ(y, x) = y − x 2 and Π C become the metric projection P C of H onto C (see, for example [5,25,26] ).…”

“…Lemma 2.15. (see [7,26]) Let E be a smooth, strictly convex and reflexive Banach space, and C be a nonempty closed convex subset of E. Let Ψ : C × C −→ R be a bifunction satisfying the conditions (L 1 ) − (L 4 ). Let r > 0 be any given number and x ∈ E be any given point.…”

Extragradient methods for split feasibility problems and generalized equilibrium problems in Banach spaces

“…The extragradient algorithm is well known because of its efficiency in numerical tests. Therefore, in recent years, many authors have used extragradient method for finding a common element of solution set of a ( V I ), the set of fixed points of a nonexpansive or a relatively nonexpansive mapping and the solution set of an equilibrium problem, see for instance the literature() and the references therein. In most of these methods in Banach spaces, authors have proven weak convergence of generated sequences.…”

The hybrid block iterative algorithm for solving the system of equilibrium problems and variational inequality problems

“…In the last two decades, many papers have appeared in the literature on the existence of solutions of E P ( f ); see, for example (Blum and Oettli 1994; Combettes and Hirstoaga 2005) and references therein. Some solution methods have been proposed to solve the E P ( f ); see, for example, (Blum and Oettli 1994; Combettes and Hirstoaga 2005; Jaiboon and Kumam 2010; Katchang and Kumam 2010; Kumam 2009; Moudafi 2003; Qin et al 2009a, 2009b, 2009c; Saewan and Kumam 2010b, 2011a, 2011b, 2011c, 2011d, 2011e, 2011f, 2011g, 2012b; Zegeye et al 2010) and references therein.…”