Approximation of common solutions for a fixed point problem of asymptotically nonexpansive mapping and a generalized equilibrium problem in Hilbert space

Abstract

In this paper, we introduce an iterative algorithm to approximate a common solution of a generalized equilibrium problem and a fixed point problem for an asymptotically nonexpansive mapping in a real Hilbert space. We prove that the sequences generated by the iterative algorithm converge strongly to a common solution of the generalized equilibrium problem and the fixed point problem for an asymptotically nonexpansive mapping. The results presented in this paper extend and generalize many previously known results in this research area. Some applications of main results are also provided.

Common solutions of fixed point and generalized equilibrium problems using asymptotically nonexpansive mapping

“…Some definitions and lemmas are as follows: Definition 2. [12] "A mapping g : D → D is known as contraction mapping if there exists η ∈ [0, 1) such that ∥g(p) − g(q)∥ ≤ ∥p − q∥, ∀p, q ∈ D." Definition 3. [4] "Let D be a closed convex subset of a Hilbert space H. A mapping T : D → D is called asymptotically regular at p if and only if lim n→∞ ∥T n p − T n+1 p∥ = 0."…”

“…(iv) EP(G) is closed and convex." [12] "Let G : D × D → R be a bifunction which satisfy (G1), (G2), (G3) and (G4). Let B : D → H be a β-inverse strong monotone mapping.…”