Hybrid projection method for generalized mixed equilibrium problem and fixed point problem of infinite family of asymptotically quasi-ϕ-nonexpansive mappings in Banach spaces

Abstract

In this paper, we consider a hybrid projection method for finding a common element in the set of fixed points of a infinite family of asymptotically quasi-ϕ-nonexpansive mappings and in the set of solutions of a generalized mixed equilibrium problem. Some strong convergence theorems of common elements are established in a uniformly smooth and strictly convex Banach space which has the Kadec–Klee property. The results presented in the paper improve and extend some recent results.

Introduction

Let E be a Banach space and let C be a closed convex subsets of E. Let f be a bifunction from C × C to R, ψ : C → R be a real-valued function and A : C → E^∗ be a nonlinear mapping. The “so-called” generalized mixed equilibrium problem is to find z ∈ C such that

$$f(z\text{,}y)+〈\mathit{Az}\text{,}y-z〉+\psi (y)-\psi (z)⩾0\text{,}\forall y\in C\text{.}$$

The set of solutions of (1.1) is denoted by GMEP, i.e.

$$\mathit{GMEP}=\{z\in C:f(z\text{,}y)+〈\mathit{Az}\text{,}y-z〉+\psi (y)-\psi (z)⩾0\text{,}\forall y\in C\}\text{.}$$

Special cases:

• If A = 0, then the problem (1.1) is equivalent to find z ∈ C such that

  $$f(z\text{,}y)+\psi (y)-\psi (z)⩾0\text{,}\forall y\in C\text{,}$$

  which is called the mixed equilibrium problem. The set of solution of (1.2) is denoted by MEP.

• If f = 0, then the problem (1.1) is equivalent to find z ∈ C such that

  $$〈\mathit{Az}\text{,}y-z〉+\psi (y)-\psi (z)⩾0\text{,}\forall y\in C\text{,}$$

  which is called the mixed variational inequality of Browder type. The set of solution of (1.3) is denoted by VI(C, A, ψ).

• If ψ = 0, then the problem (1.1) is equivalent to find z ∈ C such that

  $$f(z\text{,}y)+〈\mathit{Az}\text{,}y-z〉⩾0\text{,}\forall y\in C\text{,}$$

  which is called the generalized equilibrium problem. The set of solution of (1.4) is denoted by EP.

• If A = 0, ψ = 0, then the problem (1.1) is equivalent to find z ∈ C such that

  $$f(z\text{,}y)⩾0\text{,}\forall y\in C\text{,}$$

  which is called the equilibrium problem. The set of solution of (1.5) is denoted by EP(f).

Recently, Tada and Takahashi [1] and Takahashi and Takahashi [2] considered iterative methods for finding an element of EP(f) ∩ F(T) in Hilbert space. Very recently, Takahashi and Takahashi [3] introduced an iterative method for finding an element of EP ∩ F(T), where A : C → H is an inverse-strongly monotone mapping, T is nonexpansive mapping and then proved a strong convergence theorem. On the other hand, Takahashi and Zembayashi [4], [5] prove strong and weak convergence theorems for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping in a Banach space by using the shrinking projection method.

In this paper, motivated by Takahashi and Zembayashi [4], [5], we prove some strong convergence theorems for finding an element of $\mathit{GMEP}\bigcap {\bigcap }_{i=0}^{\infty }F(T_i)$ in a uniformly smooth and strictly convex Banach space which has the Kadec–Klee property by using a new hybrid projection method. where A : C → E^∗ is a continuous and monotone operator, {T_i} is a family of infinite asymptotically quasi-ϕ-nonexpansive mappings. The results presented this paper mainly improve the corresponding results announced in Takahashi and Zembayshi [5].