Strong Convergence Iterative Algorithms for Equilibrium Problems and Fixed Point Problems in Banach Spaces

and Applied Analysis 3 Recall that the Bregman projection [13] of x ∈ int domf onto the nonempty, closed, and convex subset C of domf is the necessarily unique vector projf C (x) ∈ C satisfying D f (projf C (x) , x) = inf{D f (y, x) : y ∈ C} . (12) Let f : E → (−∞, +∞] be a convex and Gâteaux differentiable function. The function f is said to be totally convex at x ∈ int domf if its modulus of total convexity at x, that is, the function ] f : int domf × [0, +∞) → [0, +∞] defined by ] f (x, t) := inf{D f (y, x) : y ∈ domf, 󵄩󵄩󵄩󵄩y − x 󵄩󵄩󵄩󵄩 = t} , (13) is positive whenever t > 0. The function f is said to be totally convex when it is totally convex at every point x ∈ int domf. In addition, the function f is said to be totally convex on bounded sets if ] f (B, t) is positive for any nonempty bounded subset B of E and t > 0, where the modulus of total convexity of the function f on the set B is the function ] f : int domf × [0, +∞) → [0, +∞] defined by ] f (B, t) := inf {] f (x, t) : x ∈ B ∩ domf} . (14) Some examples of the totally convex functions can be found in [14, 15]. Recall that the function f is said to be sequentially consistent [15] if, for any two sequences {x n } and {y n } in E such that the first is bounded, lim n→∞ D f (y n , x n ) = 0 󳨐⇒ lim n→∞ 󵄩󵄩󵄩󵄩yn − xn 󵄩󵄩󵄩󵄩 = 0. (15) Let C be a nonempty, closed, and convex subset of E and g : C × C → R a bifunction that satisfies the following conditions: (C1) g(x, x) = 0 for all x ∈ C; (C2) g is monotone, that is, g(x, y) + g(y, x) ≤ 0 for all x, y ∈ C; (C3) lim sup t↓0 g(tz + (1 − t)x, y) ≤ g(x, y) for all x, y, z ∈ C; (C4) for all x ∈ C, g(x, ⋅) is convex and lower semicontinuous. The equilibrium problem with respect to g is as follows: find x ∈ C such that g (x, y) ≥ 0, ∀y ∈ C. (16) The set of all solutions of (16) is denoted by EP(g). The resolvent of a bifunction g : C × C → R [16] is the operator Res g : E → 2 C denoted by Resf g (x) = {z ∈ C : g (z, y) + ⟨∇f (z) − ∇f (x) , y − z⟩ ≥ 0, ∀y ∈ C} . (17) For any x ∈ E, there exists z ∈ C such that z = Res g (x); see [3]. Let K be a convex subset of int domf and T : K → K a mapping. A point p in the closure of K is said to be an asymptotic fixed point of T [17, 18] if K contains a sequence {x n } which converges weakly to p such that the strong lim n→∞ (x n − Tx n ) = 0. The set of asymptotic fixed points of T will be denoted by F(T). The mapping T is called Bregman quasi-nonexpansive if F(T) ̸ = 0 and D f (V, x) ≤ D f (V, x) , ∀V ∈ F (T) , x ∈ K. (18) T is said to be Bregman (quasi)-strongly nonexpansive [6]with respcet to a nonempty F(T) if D f (p, Tx) ≤ D f (p, x) , (19) for all p ∈ F(T) and x ∈ K, and if whenever {x n } ⊂ K is bounded, p ∈ F(T), and lim n→∞ (D f (p, x n ) − D f (p, Tx n )) = 0, (20) it follows that lim n→∞ D f (Tx n , x n ) = 0. (21) The mapping T is called Bregman firmly nonexpansive if ⟨∇f (Tx) − ∇f (Ty) , Tx − Ty⟩ ≤ ⟨∇f (x) − ∇f (y) , Tx − Ty⟩ (22) for all x, y ∈ K. Next, we introduce a new mapping that is called Bregman asymptotically quasinonexpansive mapping which is a natural extension of Bregman quasinonexpansive mapping introduced by Reich and Sabach [3]. The mapping T : K → K is said to be Bregman asymptotically quasi-nonexpansive if there exists a sequence {k n } ⊂ [1,∞) satisfying lim n→∞ k n = 1 such that, for every n ≥ 1, D f (V, T n x) ≤ k n D f (V, x) , ∀V ∈ F (T) , x ∈ K. (23) Obviously, every Bregman quasinonexpansive mapping is a Bregman asymptotically quasi-nonexpansive one with k n = 1. Let E be a Banach space and C a nonempty subset of E. The mapping T : C → C is said to be uniformly asymptotically regular on C if