Strong Convergence Theorem for Generalized Mixed Equilibrium Problems and Bregman Nonexpansive Mapping in Banach Spaces

In this paper, we study an iterative method for a common fixed point of a Bregman strongly nonexpansive mapping in the frame work of reflexive real Banach spaces. Moreover, we prove the strong convergence theorem for finding common fixed points with the solutions of a generalized mixed equilibrium problem.

The set of solution of ( 4) is denoted by M V I(C, ϕ, Ψ).
The set of solutions of ( 5) is denoted by EP (Θ).This problem contains fixed point problems, includes as special cases numerous problems in physics, optimization and economics.Some methods have been proposed to solve the equilibrium problem, (see [12,14]).The above formulation (5) was shown in [2] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, Nash equilibria in noncooperative games.
Equilibrium problems which were introduced by Blum and Oettli [2] and Noor and Oettli [3] in 1994 have had a great impact and influence in the development of several branches of pure and applied sciences.It has been shown that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity and optimization.
In [26], Reich and Sabach proposed an algorithm for finding a common fixed point of finitely many Bregman strongly nonexpansive mappings T i : C → C(i = 1, 2, . . ., N ) satisfying ∩ N i=1 F (T i ) = ∅ in a reflexive Banach space E as follows: x 0 ∈ E, chosen arbitrarily, Cn∩Qn (x 0 ), ∀n ≥ 0, and where proj f C is the Bregman projection with respect to f from E onto a closed and convex subset C of E. They proved that the sequence {x n } converges strongly to a common fixed point of {T i } N i=1 .The authors of [18] introduced the following algorithm: chosen arbitrarily, where H is an equilibrium bifunction and T n is a Bregman strongly nonexpansive mapping for any n ∈ N.They proved the sequence (6) converges strongly to the point proj F (T )∩EP (H) x.
Also, in [13] the following algorithm was considered: where ϕ : C → R is a real-valued function, Θ : C × C → R is an equilibrium bifunction and T is a Bregman strongly nonexpansive mapping.It was prove that the sequence {x n } defined in (7) converges strongly to the point proj (∩ N i=1 F (T i ))∩M EP (Θ) x.In this paper, motivated by above algorithms, we present the following iterative scheme: chosen arbitrarily, where ϕ : C → R is a real-valued function, Ψ : C → E * is a continuous monotone mapping, Θ : C × C → R is an equilibrium bifunction and T is Bregman strongly nonexpansive mapping.We will prove that the sequence {x n } defined in (8) converges strongly to the point proj F (T )∩GM EP (Θ,ϕ,Ψ) x.

Preliminaries
Let f : E → (−∞, +∞] be a proper, lower semi-continuous and convex function.We denote by domf , the domain of f , that is the set {x ∈ E : Convergence Theorem for GMEP and Bregman Mapping f (x) < +∞}.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 x ∈ E}.For any x ∈ int(domf ), the right-hand derivative of f at x in the derivation y ∈ E is defined by The function f is called Gâteaux differentiable at x if lim t 0 exists for all y ∈ E. In this case, f (x, y) coincides with ∇f (x), the value of the gradient (∇f ) of f at x.The function f is called Gâteaux differentiable if it is Gâteaux differentiable for any x ∈ int(domf ) and f is called Fréchet differentiable at x if this limit is attain uniformly for all y which satisfies y = 1.The function f is uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for any 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-toweak * continuous (resp.continuous) on int(domf ) (see [5]).
When the subdifferential of f is single-valued, it coincides with the gradient ∂f = ∇f , [22].By Bauschke et al [4] the conditions (L 1 ) and (L 2 ) also yields that the function f and f * are strictly convex on the interior of their respective domains.If E is a smooth and strictly convex Banach space, then an important and interesting Legendre function is f (x) := 1 p x p (1 < p < ∞).In this case the gradient ∇f of f coincides with the generalized duality mapping of E, i.e., ∇f = J p (1 < p < ∞).In particular, ∇f = I, the identity mapping in Hilbert spaces.From now on we assume that the convex function f +∞] be a convex and Gâteaux differentiable function.The Bregman projection of x ∈ int(domf ) onto the nonempty, closed and convex subset C ⊂ domf is the necessary unique vector If E is a smooth and strictly convex Banach space and f (x) = x 2 for all x ∈ E, then we have that ∇f (x) = 2Jx for all x ∈ E, where J is the normalized duality mapping from E in to 2 E * , and hence D f (x, y) reduced to φ(x, y) = x 2 − 2 x, Jy + y 2 , for all x, y ∈ E, which is the Lyapunov function introduced by Alber [1] and Bregman projection P f C (x) reduces to the generalized projection Π C (x) which is defined by If E = H, a Hilbert space, J is the identity mapping and hence Bregman projection P f C (x) reduced to the metric projection of H onto C, P C (x). Definition 2.2.[9] Let f : E → (−∞, +∞] be a convex and Gâteaux differentiable function.f is called: (1) totally convex at x ∈ int(domf ) if its modulus of total convexity at x, that is, the function is positive whenever t > 0; (2) totally convex if it is totally convex at every point x ∈ int(domf ); (3) totally convex on bounded sets if ν 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 (2) coercive [15] if the sublevel set of f is bounded; equivalently, x = +∞; (4) sequentially consistent if for any two sequences {x n } and {y n } in E such that {x n } is bounded,  [23,28]) if C contains a sequence {x n } which converges weakly to p such that lim n→∞ x n − T x n = 0. We denote by F (T ) the set of asymptotic fixed points of T .
(3) Bregman strongly nonexpansive (see [7,26]) with respect to f and and, if whenever {x n } ⊂ C is bounded, p ∈ F (T ), and (4) Bregman firmly nonexpansive (for short BFNE) with respect to f if, for all x, y ∈ C, The existence and approximation of Bregman firmly nonexpansive mappings was studied in [23].It is also known that if T is Bregman firmly nonexpansive and f is Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subset of E, then F (T ) = F (T ) and F (T ) is closed and convex.It also follows that every Bregman firmly nonexpansive mapping is Bregman strongly nonexpansive with respect to F (T ) = F (T ).
Let C be a nonempty, closed and convex subset of E. Let f : E → R be a Gâteaux differentiable and totally convex function.Let x ∈ E it is known from [8] We also know the following: be a convex, Legendre and Gâteaux differentiable function.Following [1] and [11], we make use of the function )) for all x ∈ E and x * ∈ E * .Moreover, by the subdifferential inequality, for all x ∈ E and x * , y * ∈ E * [17].In addition, if f : E → (−∞, +∞] is a proper lower semicontinuous function, then f * : E * → (−∞, +∞] is a proper weak * lower semicontinuous and convex function (see [19]).Hence, V f is convex in the second variable.Thus, for all z ∈ E, where {x i } N i=1 ⊂ E and {t i } N i=1 ⊂ (0, 1) with N i=1 t i = 1.Lemma 2.6.[25] Let C be a nonempty, closed and convex subset of int(domf ) and T : C → C be a quasi-Bregman nonexpansive mappings with respect to f .Then F (T ) is closed and convex.
For solving the generalized mixed equilibrium problem, let us assume that the bifunction Θ : C × C → R satisfies the following conditions: (A 1 ) Θ(x, x) = 0 for all x ∈ C; (A 2 ) Θ is monotone, i.e., Θ(x, y) + Θ(y, x) ≤ 0 for any x, y ∈ C; (A 3 ) for each y ∈ C, x → Θ(x, y) is upper-hemicontinuous, i.e., for each x, y, z ∈ C, lim sup (A 4 ) for each x ∈ C, y → Θ(x, y) is convex and lower semicontinuous (see [21]).Definition 2.4.Let C be a nonempty, closed and convex subsets of a real reflexive Banach space and let ϕ be a lower semicontinuous and convex functional from C to R and Ψ : C → E * be a continuous monotone mapping.Let Θ : C × C → R be a bifunctional satisfying (A 1 )-(A 4 ).The mixed resolvent of Θ is the operator Lemma 2.7.Let f : E → (−∞, +∞] be a coercive and Gâteaux differentiable function.Let C be a closed and convex subset of E. Assume that ϕ : C → R be a lower semicontinuous and convex functional, Ψ : C → E * be a continuous monotone mapping and the bifunctional Θ : Then from [2, Theorem 1], there exists x ∈ C such that for any y ∈ C. So, we have We know that inequality (14) holds for y = tx + (1 − t)ȳ where ȳ ∈ C and t ∈ (0, 1).Therefore, for all ȳ ∈ C. By convexity of ϕ we have Since we have from (15) and (A 4 ) that for all ȳ ∈ C. From (A 1 ) we have Equivalently So, we have for all ȳ ∈ C. Since f is Gâteaux differentiable function, it follows that ∇f is norm-to-weak * continuous (see [22,Proposition 2.8].Hence, letting t → 1 −1 we then get Convergence Theorem for GMEP and Bregman Mapping Let y = p ∈ F (Res f Θ,ϕ,Ψ ), we then get Lemma 2.9.[30] Assume that {x n } is a sequence of nonnegative real numbers such that where {α n } is a sequence in (0, 1) and {β n } is a sequence such that

Main result
Theorem 3.1.Let E be a real reflexive Banach space, C be a nonempty, closed and convex subset of E. Let f : E → R be a coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let T be a Bregman strongly nonexpansive mappings with respect to f such that F (T ) = F (T ) and T is uniformly continuous.Let Θ : C × C → R satisfying conditions (A 1 )-(A 4 ) and F (T ) ∩ GM EP (Θ) is nonempty and bounded.Let {x n } be a sequence generated by where {α n } ⊂ (0, 1) satisfying lim n→∞ α n = 0 and ∞ n=1 α n = ∞.Then {x n } converges strongly to proj F (T )∩GM EP (Θ) x.
Proof.We note from Lemma 2.6 that F (T ) is closed and convex.Let p = proj F (T )∩GM EP (Θ) x ∈ F (T ) ∩ GM EP (Θ).Then p ∈ F (T ) and p ∈ GM EP (Θ).By (18) and Lemma 2.8, we have , where ψ = f * − •, p .Since f is lower semicontinuous, f * is weak * lower semicontinuous.Hence the function ψ is coercive by Moreau-Rockafellar Theorem (see [27,Theorem 7A] and [20]).This shows that {∇f (x n )} is bounded.Since f is strongly coercive, f * is bounded on bounded sets (see [31,Lemma 3.6.1]and [4,Theorem 3.3]).Hence, ∇f * is also bounded on bounded subset of E * (see [8,Proposition 1.1.11]).Since f is a Legendre function, it follows that x n = ∇f * (∇f (x n )) is bounded for all n ∈ N. By (18) we have Since f is strongly coercive and uniformly convex on bounded subsets of E, then f * is uniformly Fréchet differentiable on bounded subsets of E * .Moreover, f * is bounded on bounded subsets.Since f is Legendre function we have (19) On the other hand, since f is uniformly Fréchet differentiable on bounded subsets of E, f is uniformly continuous on bounded subsets of E. It follows that lim From ( 11) and ( 18), we have By Lemma 2.
Since f is uniformly Fréchet differentiable, it is also uniformly continuous, we get ( 23) By Bregman distance we have for each p ∈ F (T ).By ( 20)-( 23), we obtain (24) By above equation, we have By Lemma 2.1, we have From above equation and ( 19), we can write when n → ∞.By applying the triangle inequality, we get On the other hand, since x n k − T x n k → 0 as k → ∞, we have q ∈ F (T ).It follows from the definition of the Bregman projection that (26) From (12), we obtain By Lemma 2.9 and (26), we can conclude that lim n→∞ D f (p, x n ) = 0. Therefore, by Lemma 2.1, x n → p.This completes the proof.
If in Theorem 3.1, we consider Θ ≡ 0, we have the following corollary.
Corollary 3.1.Let E be a real reflexive Banach space, C be a nonempty, closed and convex subset of E. Let f : E → R be a coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let T be a Bregman strongly nonexpansive mappings with respect to f such that F (T ) = F (T ) and T is uniformly continuous.Let F (T ) ∩ M V I(C, ϕ, Ψ) is nonempty and bounded.Let {x n } be a sequence generated by where {α n } ⊂ (0, 1) satisfying lim n→∞ α n = 0 and ∞ n=1 α n = ∞.Then {x n } converges strongly to proj F (T )∩M V I(C,ϕ,Ψ) x.
If in Theorem 3.1, we consider Ψ ≡ 0, we have the following corollary.Corollary 3.2.Let E be a real reflexive Banach space, C be a nonempty, closed and convex subset of E. Let f : E → R be a coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let T be a Bregman strongly nonexpansive mappings with respect to f such that F (T ) = F (T ) and T is uniformly continuous.where {α n } ⊂ (0, 1) satisfying lim n→∞ α n = 0 and ∞ n=1 α n = ∞.Then {x n } converges strongly to proj F (T )∩M EP (Θ,ϕ) x.
If in Theorem 3.1, we consider ϕ ≡ 0, we have the following corollary.
Corollary 3.3.Let E be a real reflexive Banach space, C be a nonempty, closed and convex subset of E. Let f : E → R be a coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let T be a Bregman strongly nonexpansive mappings with respect to f such that F (T ) = F (T ) and T is uniformly continuous.Let Θ : C × C → R satisfying conditions (A 1 )-(A 4 ) and F (T ) ∩ GEP (Θ, Ψ) is nonempty and bounded.Let {x n } be a sequence generated by where {α n } ⊂ (0, 1) satisfying lim n→∞ α n = 0 and ∞ n=1 α n = ∞.Then {x n } converges strongly to proj F (T )∩GEP (Θ,Ψ) x.
If in Theorem 3.1, we assume that E is a uniformly smooth and uniformly convex Banach space and f (x) := 1 p x p (1 < p < ∞), we have that ∇f = J p , where J p is the generalization duality mapping from E onto E * .Thus, we get the following corollary.
The function f is totally convex on bounded subsets if and only if it is sequentially consistent.Lemma 2.2.[26, Proposition 2.3] If f : E → (−∞, +∞] is Fréchet differentiable and totally convex, then f is cofinite. [8]ma 2.3.[8]Letf: E → (−∞, +∞] be a convex function whose domain contains at least two points.Then the following statements hold:(1) f is sequentially consistent if and only if it is totally convex on bounded sets; (2) If f is lower semicontinuous, then f is sequentially consistent if and only if it is uniformly convex on bounded sets; (3) If f is uniformly strictly convex on bounded sets, then it is sequentially consistent and the converse implication holds when f is lower semicontinuous, Fréchet differentiable on its domain and Fréchet derivative ∇f is uniformly continuous on bounded sets.Lemma 2.4.[24, Proposition 2.1] Let f : E → R be uniformly Fréchet differentiable and bounded on bounded subsets of E. Then ∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E * .Lemma 2.5.[26, Lemma 3.1] Let f : E → R be a Gâteaux differentiable and totally convex function.If x 0 ∈ E and the sequence {D f (x n , x 0 )} is bounded, then the sequence {x n } is also bounded.Let T : C → C be a nonlinear mapping.The fixed points set of T is denoted by