Convergence results for a common solution of a finite family of equilibrium problems and quasi-Bregman nonexpansive mappings in Banach space

We introduce an iterative process for finding common fixed point of finite family of quasi-Bregman nonexpansive mappings which is a unique solution of some equilibrium problem.


Introduction
Let E be a real reflexive Banach space, C a nonempty subset of E. Let T : C → C be a map, a point x ∈ C is called a fixed point of T if T x = x, and the set of all fixed points of T is denoted by F (T ). The mapping T is called L−Lipschitzian or simply Lipschitz if there exists L > 0, such that ||T x − T y|| ≤ L||x − y||, ∀x, y ∈ C and if L = 1, then the map T is called nonexpansive.
Let g : C × C → R be a bifunction. The equilibrium problem with respect to g is to find z ∈ C such that g(z, y) ≥ 0, ∀y ∈ C. The set of solution of equilibrium problem is denoted by EP (g). Thus EP (g) := {z ∈ C : g(z, y) ≥ 0, ∀y ∈ C}.
Numerous problems in Physics, Optimization and Economics reduce to finding a solution of the equilibrium problem. Some methods have been proposed to solve equilibrium problem in Hilbert spaces; see for example Blum and Oettli [5], Combettes and Hirstoaga [12]. Recently, Tada and Takahashi [29,30] and Takahashi and Takahashi [31] obtain weak and strong convergence theorems for finding a common element of the set of solutions of an equilibrium problem and set of fixed points of nonexpansive mapping in Hilbert space. In particular, Tada and Takahashi [30] establish a strong convergence theorem for finding a common element of the two sets by using the hybrid method introduced in Nakajo and Takahashi [18]. They also proved such a strong convergence theorem in a uniformly convex and uniformly smooth Banach space.
In 1967, Bregman [7] discovered an elegant and effective technique for using socalled Bregman distance function D f see, (1.1) in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman's technique has been applied in various ways in order to design and analyze iterative algorithms for solving feasibility and optimization problems. Let f : E → (−∞, +∞] be a convex and Gâteaux differentiable function. The function D f : domf × int domf → [0, +∞) defined as follows: is called the Bregman distance with respect to f (see [10]). It is obvious from the definition of D f that We observed from (1.2), that for any y 1 , y 2 , · · · , y N ∈ E, the following holds Recall that the Bregman projection [7] of x ∈ int domf onto the nonempty closed and convex set C ⊂ domf is the necessarily unique vector P f C (x) ∈ C satisfying D f (P f C (x), x) = inf{D f (y, x) : y ∈ C}. A mapping T is said to be Bregman firmly nonexpansive [26], if for all x, y ∈ C, A point p ∈ C is said to be asymptotic fixed point of a map T , if for any sequence {x n } in C which converges weakly to p, and lim n→∞ ||x n − T x n || = 0. We denote bŷ F (T ) the set of asymptotic fixed points of T . Let f : E → R, a mapping T : C → C is said to be Bregman relatively nonexpansive [15] if F (T ) =,F (T ) = F (T ) and D f (p, T (x)) ≤ D f (p, x) for all x ∈ C and p ∈ F (T ). T is said to be quasi-Bregman relatively nonexpansive if F (T ) = ∅, and D f (p, T (x)) ≤ D f (p, x) for all x ∈ C and p ∈ F (T ).
Recently, by using the Bregman projection, in 2011 Reich and Sabach [26] proposed algorithms for finding common fixed points of finitely many Bregman firmly nonexpansive operators in a reflexive Banach space.
Under some suitable conditions, they proved that the sequence generated by (1.4) converges strongly to N i=1 F (T i ) and applied the result for the solution of convex feasibility and equilibrium problems.
In 2011, Chen et al. [11], introduced the concept of weak Bregman relatively nonexpansive mappings in a reflexive Banach space and gave an example to illustrate the existence of a weak Bregman relatively nonexpansive mapping and the difference between a weak Bregman relatively nonexpansive mapping and a Bregman relatively nonexpansive mapping. They also proved strong convergence of the sequences generated by the constructed algorithms with errors for finding a fixed point of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings under some suitable conditions. Recently in 2014, Alghamdi et al. [1] proved a strong convergence theorem for the common fixed point of finite family of quasi-Bregman nonexpansive mappings. Pang et al. [19] proved weak convergence theorems for Bregman relatively nonexpansive mappings. While, Zegeye and Shahzad in [34] and [35] proved a strong convergence theorem for the common fixed point of finite family of right Bregman strongly nonexpansive mappings and Bregman weak relatively nonexpansive mappings in reflexive Banach space respectively.
In 2015 Kumam et al. [17] introduced the following algorithm: where T n , n ∈ N, is a Bregman strongly nonexpansive mapping. They proved that the sequence {x n } which is generated by the algorithm (1.5) converges strongly to the point P f Ω x, where Ω := F (T ) ∩ EP (g). Motivated and inspired by the above works, in this paper, we prove a new strong convergence theorem for finite family of quasi-Bregman nonexpansive mapping and system of equilibrium problem in a real Banach space.

Preliminaries
Let E be a real reflexive Banach space with the norm ||.|| and E * the dual space of E. Throughout this paper, we shall assume f : E → (−∞, +∞] is a proper, lower semicontinuous and convex function. We denote by domf := {x ∈ E : f (x) < +∞} as the domain of f . Let x ∈ int domf , the subdifferential of f at x is the convex set defined by where the Fenchel conjugate of f is the function f * : E * → (−∞, +∞] defined by We know that the Young-Fenchel inequality holds: A function f on E is coercive [13] if the sublevel set of f is bounded; equivalently, A function f on E is said be strongly coercive [33] if For any x ∈ int domf and y ∈ E, the right-hand derivative of f at x in the direction y is defined by The function f is said to be Gâteaux differentiable at x if lim t→0 + f (x+ty)−f (x) t exists for any y. In this case, f • (x, y) coincides with ∇f (x), the value of the gradient ∇f of f at x. The function f is said to be Gâteaux differentiable if it is Gâeaux differentiable for any x ∈ int domf . The function f is said to be Fréchet differentiable at x if this limit is attained uniformly in ||y|| = 1. Finally, f is said to be uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for x ∈ C and ||y|| = 1. It is known that if f is Gâteaux differentiable (resp. Fréchet differentiable) on int domf , then f is continuous and its Gâteaux derivative ∇f is norm-to-weak * continuous (resp. continuous) on int domf (see also [2,6]). We will need the following results. (i) f is essentially smooth if and only if f * is essentially strictly convex (see [3], Theorem 5.4). (ii) (∂f ) −1 = ∂f * (see [6]) (iii) f is Legendre if and only if f * is Legendre, (see [3], Corollary 5.5).
Lemma 2.4. Let E be a Banach space, r > 0 be a constant, ρ r be the gauge of uniform convexity of g and g : E → R be a convex function which is uniformly convex on bounded subsets of E. Then (i) For any x, y ∈ B r and α ∈ (0, 1), If, in addition, g is bounded on bounded subsets and uniformly convex on bounded subsets of E then, for any x ∈ E, y * , z * ∈ B r and α ∈ (0, 1), [22]) Let E be a Banach space, let r > 0 be a constant and let f : E → R be a continuous and convex function which is uniformly convex on bounded subsets of E. Then We know the following two results; see [33] Theorem 2.6. Let E be a reflexive Banach space and let f : E → R be a convex function which is bounded on bounded subsets of E. Then the following assertions are equivalent: (1) f is strongly coercive and uniformly convex on bounded subsets of E; (2) domf * = E * , f * is bounded on bounded subsets and uniformly smooth on bounded subsets of E * ; (3) domf * = E * , f * is Frechet differentiable and ∇f is uniformly norm-tonorm continuous on bounded subsets of E * .
Theorem 2.7. Let E be a reflexive Banach space and let f : E → R be a continuous convex function which is strongly coercive. Then the following assertions are equivalent: (1) f is bounded on bounded subsets and uniformly smooth on bounded subsets of E; (2) f * is Frechet differentiable and f * is uniformly norm-to-norm continuous on bounded subsets of E * ; (3) domf * = E * , f * is strongly coercive and uniformly convex on bounded subsets of E * .
The following result was first proved in [8] (see also [16]).
Lemma 2.8. Let E be a reflexive Banach space, let f : E → R be a strongly coercive Bregman function and let V be the function defined by Then the following assertions hold: Examples of Legendre functions were given in [3,4]. One important and interesting Legendre function is 1 p || · || p (1 < p < ∞) when E is a smooth and strictly convex Banach space. In this case the gradient ∇f of f is coincident with the generalized duality mapping of E, i.e., ∇f = J p (1 < p < ∞). In particular, ∇f = I the identity mapping in Hilbert spaces. In the rest of this paper, we always assume that f : E → (−∞, +∞] is Legendre.
Concerning the Bregman projection, the following are well known.
The function f is called totally convex at x if v f (x, t) > 0 whenever t > 0. The function f is called totally convex if it is totally convex at any point x ∈ int domf and is said to be totally convex on bounded sets if v f (B, t) > 0 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 v Recall that the function f called sequentially consistent [8] if for any two sequence {x n } and {y n } in E such that the first one is bounded  [27] Let f : E → R be a Gâteaux differentiable and totally convex function, x 0 ∈ E and let C be a nonempty, closed and convex subset of E. Suppose that the sequence {x n } is bounded and any weak subsequential limit of {x n } belongs to C. If D f (x n , x 0 ) ≤ D f (P f C (x 0 ), x 0 ) for any n ∈ R, then {x n } converges strongly to P f C (x 0 ). Lemma 2.14. [20] Let E be a real reflexive Banach space, f : E → (−∞, +∞] be a proper lower semi-continuous function, then f * : E * → (−∞, +∞] is a proper weak * lower semi-continuous and convex function. Thus, for all z ∈ E, we have In order to solve the equilibrium problem, let us assume that a bifunction g : C × C → R satisfies the following condition [5] (A1) g(x, x) = 0, ∀x ∈ C.
(A4) The function y → g(x, y) is convex and lower semi-continuous. The resolvent of a bifunction g [12] is the operator Res f g : E → 2 C defined by From (Lemma 1, in [23]), if f : (−∞, +∞] is a strongly coercive and Gâteaux differentiable function, and g satisfies conditions (A1)-(A4), then dom(Res f g ) = E. The following lemma gives some characterization of the resolvent Res f g .

Lemma 2.15. [23]
Let E be a real reflexive Banach space and C be a nonempty closed convex subset of E. Let f : E → (−∞, +∞] be a Legendre function. If the bifunction g : C × C → R satisfies the conditions (A1)-(A4). Then, the followings hold: (i) Res f g is single-valued; (ii) Res f g is a Bregman firmly nonexpansive operator; (iii) F (Res f g ) = EP (g); (iv) EP (g) is closed and convex subset of C; (v) for all x ∈ E and for all q ∈ F (Res f g ), we have a m k ≤ a m k +1 and a k ≤ a m k +1 .

Main Results
We now prove the following theorem.

(3.4)
Since f is strongly coercive and uniformly convex on bounded subsets of E, f * is uniformly Fréchet differentiable on bounded sets. Moreover, f * is bounded on bounded sets, from (3.4), we obtain lim n→∞ ||z n − u j,n || = 0. (3.5) On the other hand, In view of (3) in Theorem 2.6, we know that domf * = E * and f * is strongly coercive and uniformly convex on bounded subsets. Let s = sup{||∇f (y n )||, ||∇f (T [n] y n )||} and ρ * s : E * → R be the gauge of uniform convexity of the conjugate function f * . Now from (3.1), Lemma 2.4 and 2.8, we obtain Now, we consider two cases: Case 1. Suppose that there exists n 0 ∈ N such that {D f (p, x n )} is non increasing.
Using the quasi-Bregman nonexpansivity of T (i) for each i, we obtain , we obtain the following finite table x n ) → 0 as n → ∞ then, applying Lemma 2.10 on each line above, we obtain and adding up this table, we obtain x n+N − T (n+N ) T (n+N −1) · · · T (n+1) x n → 0 as n → ∞.
as n → ∞. Then, we have from Lemma 2.10 that (3.44) Let {x ni } be a subsequence of {x n }. Since {x n } is bounded and E is reflexive, without loss of generality, we may assume that x ni ⇀ q for some q ∈ F and since x n − z n → 0 as n → ∞, then z ni ⇀ q Since the pool of mappings of T [n] is finite, passing to a further subsequence if necessary, we may further assume that, for some i ∈ {1, 2, · · · , N }, from (3.46), we get Noticing that u j,n = Res f gj (x n ) for each j = 1, 2, · · · , m, we obtain g j (u j,n , y) + y − u j,n , ∇f (u j,n ) − ∇f (x n ) ≥ 0, ∀y ∈ C Hence g j (u j,ni , y) + y − u j,ni , ∇f (u j,ni ) − ∇f (x ni ) ≥ 0, ∀y ∈ C.
Taking the limit as i → ∞ in above inequality and from (A4) and u j,ni ⇀ q, we have g j (y, q) ≤ 0 for each j = 1, 2, · · · , m. For 0 < t < 1 and y ∈ C, define y t = ty + (1 − t)q. Noticing that y, q ∈ C, we obtain y t ∈ C, which yield that g j (y t , q) ≤ 0. It follows from (A1) that 0 = g j (y t , y t ) ≤ tg j (y t , y) + (1 − t)g j (y t , q) ≤ tg j (y t , y).
It follows from Lemma 2.16 and (3.8) that D f (p, x n ) → 0 as n → ∞. Consequently, from Lemma 2.10, we obtain x n → p as n → ∞. Case 2. Suppose D f (p, x n ) is not monotone decreasing sequences, then set Φ n := D f (p, x n ) and let τ : N → N be a mapping defined for all n ≥ N 0 for some sufficiently large N 0 by τ (n) := max{k ∈ N : k ≤ n, Φ k ≤ Φ k+1 }.
This implies that lim n→∞ Φ n = 0, and hence D f (p, x n ) → 0 as n → ∞. Consequently, from Lemma 2.10, we obtain x n → p as n → ∞. Therefore from the above two cases, we conclude that {x n } converges strongly to p ∈ Ω and this complete the proof.