Strong Convergence of a System of Generalized Mixed Equilibrium Problem , Split Variational Inclusion Problem and Fixed Point Problem in Banach Spaces

The purpose of this paper is to introduce a new algorithm to approximate a common solution for a system of generalized mixed equilibrium problems, split variational inclusion problems of a countable family of multivalued maximal monotone operators, and fixed-point problems of a countable family of left Bregman, strongly asymptotically non-expansive mappings in uniformly convex and uniformly smooth Banach spaces. A strong convergence theorem for the above problems are established. As an application, we solve a generalized mixed equilibrium problem, split Hammerstein integral equations, and a fixed-point problem, and provide a numerical example to support better findings of our result.


Introduction and Preliminaries
Let E be a real normed space with dual E * .A map B : E → E * is called: (i) monotone if, for each x, y ∈ E, η − ν, x − y ≥ 0, ∀ η ∈ Bx, ν ∈ By, where •, • denotes duality pairing, (ii) -inverse strongly monotone if there exists > 0, such that Bx − By, x − y ≥ ||Bx − By|| 2 , (iii) maximal monotone if B is monotone and the graph of B is not properly contained in the graph of any other monotone operator.We note that B is maximal monotone if, and only if it is monotone, and R(J + tB) = E * for all t > 0, J is the normalized duality map on E and R(J + tB) is the range of (J + tB) (cf.[1]).
Let H 1 and H 2 be Hilbert spaces.For the maximal monotone operators B 1 : H 1 → 2 H 1 and B 2 : H 2 → 2 H 2 , Moudafi [2] introduced the following split monotone variational inclusion: where A : H 1 → H 2 is a bounded linear operator, f : H 1 → H 1 and g : H 2 → H 2 are given operators.In 2000, Moudafi [3] proposed the viscosity approximation method, which is formulated by considering the approximate well-posed problem and combining the non-expansive mapping S with a contraction mapping f on a non-empty, closed, and convex subset C of H 1 .That is, given an arbitrary x 1 in C, a sequence {x n } defined by converges strongly to a point of F(S), the set of fixed point of S, whenever {α n } ⊂ (0, 1) such that α n → 0 as n → ∞.
In [4,5], the viscosity approximation method for split variational inclusion and the fixed point problem in a Hilbert space was presented as follows: where B 1 and B 2 are maximal monotone operators, J B 1 λ and J B 2 λ are resolvent mappings of B 1 and B 2 , respectively, f is the Meir Keeler function, T a non-expansive mapping, and A * is the adjoint of A, γ n , α n ∈ (0, 1) and λ > 0.
The algorithm introduced by Schopfer et al. [6] involves computations in terms of Bregman distance in the setting of p-uniformly convex and uniformly smooth real Banach spaces.Their iterative algorithm given below converges weakly under some suitable conditions: where Π C denotes the Bregman projection and P C denotes metric projection onto C.However, strong convergence is more useful than the weak convergence in some applications.Recently, strong convergence theorems for the split feasibility problem (SFP) have been established in the setting of p-uniformly convex and uniformly smooth real Banach spaces [7][8][9][10].Suppose that F(x, y) = f (x, y) + g(x, y) where f , g : C × C −→ R are bifunctions on a closed and convex subset C of a Banach space, which satisfy the following special properties (A 1 ) − (A 4 ), (B 1 ) − (B 3 ) and (C): (A 1 ) f (x, y) = 0, ∀x ∈ C; (A 2 ) f is maximal monotone; (A 3 ) ∀x, y, z ∈ C and t ∈ [0, 1] we have lim sup n→0 + ( f (tz + (1 − t)x, y) ≤ f (x, y)); (A 4 ) ∀x ∈ C, the function y → f (x, y)is convex and weakly lower semi-continuous; (B 1 ) g(x, x) = 0 ∀ x ∈ C; (B 2 ) g is maximal monotone, and weakly upper semi-continuous in the first variable; (B 3 ) g is convex in the second variable; (C) for fixed λ > 0 and x ∈ C, there exists a bounded set The well-known, generalized mixed equilibrium problem (GMEP) is to find an x ∈ C, such that where B is nonlinear mapping.
In 2016, Payvand and Jahedi [11] introduced a new iterative algorithm for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, the set of common fixed points of a finite family of pseudo contraction mappings, and the set of solutions of the variational inequality for inverse strongly monotone mapping in a real Hilbert space.Their sequence is defined as follows: where g i are bifunctions, S i are − inverse strongly monotone mappings, C i are monotone and Lipschtz continuous mappings, θ i are convex and lower semicontinuous functions, A is a Φ− inverse strongly monotone mapping, and f is an ι−contraction mapping and α n , δ n , β n , λ n , γ 0 ∈ (0, 1).In this paper, inspired by the above cited works, we use a modified version of (1), ( 2) and (4) to approximate a solution of the problem proposed here.Both the iterative methods and the underlying space used here are improvements and extensions of those employed in [2,6,7,[9][10][11] and the references therein.
Let p, q ∈ (1, ∞) be conjugate exponents, that is, 1 p + 1 q = 1.For each p > 1, let g(t) = t p−1 be a gauge function where g : R + −→ R + with g(0) = 0 and lim t→∞ g(t) = ∞.We define the generalized duality map In the sequel, a ∨ b denotes max{a, b}.Lemma 1 ([12]).In a smooth Banach space E, the Bregman distance p of x to y, with respect to the convex continuous function f for all x, y ∈ E and p > 1.
A Banach space E is said to be uniformly convex if, for x, y ∈ E, 0 ρ E is continuous, convex, and nondecreasing with ρ E (0) = 0 and ρ E (r) ≤ r 2.
The function r → ρ E (r) r is nondecreasing and fulfils > 0 for all r > 0.
Definition 2 ([13]).Let E be a smooth Banach space.Let p be the Bregman distance.A mapping T : E −→ E is said to be a strongly non-expansive left Bregman with respect to the non-empty fixed point set of T, Lemma 2 ([14]).Let E be a real uniformly convex Banach space, K a non-empty closed subset of E, and T : K → K an asymptotically non-expansive mapping.Then, I − T is demi-closed at zero, if {x n } ⊂ K converges weakly to a point p ∈ K and lim n→∞ Tx n − x n = 0, then p = T p.

Lemma 3 ([12]
).In a smooth Banach space E, let x n ∈ E. Consider the following assertions: lim n→∞ x n = x and lim n→∞ J p (x n ), x = J p (x), x 3.
The implication (1) =⇒ (2) =⇒ (3) are valid.If E is also uniformly convex, then the assertions are equivalent.Lemma 4. Let E be a smooth Banach space.Let p and V p be the mappings defined by p (x, y) = 1 q x p − J p E x, y + 1 p y p for all (x, y) ∈ E × E and V p (x * , x) = 1 q x * q − x * , x + 1 p x p for all (x, x * ) ∈ E × E * .Then, p (x, y) = V p (x * , y) for all x, y ∈ E. Lemma 5 ([12]).Let E be a reflexive, strictly convex, and smooth Banach space, and J p be a duality mapping of E.Then, for every closed and convex subset C ⊂ E and x ∈ E, there exists a unique element Π p C (x) ∈ C, such that p (x, Π p C (x)) = min y∈C p (x, y); here, Π p C (x) denotes the Bregman projection of x onto C, with respect to the function f (x) = 1 p x p .Moreover, x 0 ∈ C is the Bregman projection of x onto C if J p (x 0 − x), y − x 0 ≥ 0 or equivalently p (x 0 , y) ≤ p (x, y) − p (x, x 0 ) f or every y ∈ C.
Lemma 7 ([12]).Let E be a reflexive, strictly convex, and smooth Banach space.If we write * q (x, y) = 1 p x * q − J q E * x * , y * + 1 q y * q for all (x * , y * ) ∈ E * × E * for the Bregman distance on the dual space E * with respect to the function f * q (x * ) = 1 q x * q , then we have p (x, y) = * q (x * , y * ).

Lemma 10 ([17]
).Let E be a real uniformly convex Banach space.For arbitrary r > 1, let B r (0) = {x ∈ E : x ≤ r}.Then, there exists a continuous strictly increasing convex function such that for every x, y ∈ B r (0), f x ∈ J p (x), f y ∈ J p (y) and λ ∈ [0, 1], the following inequalities hold: Lemma 11 ([18]).Suppose that ∑ ∞ n=1 sup{ T n+1 z − T n z : z ∈ C} < ∞.Then, for each y ∈ C, {T n y} converges strongly to some point of C.Moreover, let T be a mapping of C onto itself, defined by Ty = lim Lemma 12 ([19]).Let E be a reflexive, strictly convex, and smooth Banach space, and C be a non-empty, closed convex subset of E. If f , g : C × C −→ R be two bifunctions which satisfy the conditions (A 1 ) − (A 4 ), (B 1 ) − (B 3 )and(C), in (3), then for every x ∈ E and r > 0, there exists a unique point z ∈ C such that f (z, y) + g(z, y) For f (x) = 1 p x p , Reich and Sabach [20] obtained the following technical result: Lemma 13.Let E be a reflexive, strictly convex, and smooth Banach space, and C be a non-empty, closed, and convex subset of E. Let f , g : C × C −→ R be two bifunctions which satisfy the conditions (A 1 ) − (A 4 ), (B 1 ) − (B 3 )and(C), in (3).Then, for every x ∈ E and r > 0, we define a mapping S r : E −→ C as follows; Then, the following conditions hold: 1. S r is single-valued; 2.
S r is a Bregman firmly non-expansive-type mapping, that is, or equivalently p (S r x, S r y) + p (S r y, S r x) + p (S r x, x) + p (S r y, y) ≤ p (S r x, y) + p (S r y, x); 3.
for all x ∈ E and for all v ∈ F(S r ), p (v, S r x) + p (S r x, x) ≤ p (v, x).

Main Results
Let E 1 and E 2 be uniformly convex and uniformly smooth Banach spaces and E * 1 and E * 2 be their duals, respectively.For i ∈ I, let a bounded and linear operator, A * denotes the adjoint of A and AK be closed and convex.For each i ∈ I, let S i : E 1 → E 1 be a uniformly continuous Bregman asymptotically non-expansive operator with the sequences {k n 1 a firmly non-expansive mapping.Suppose that, for i ∈ I, θ i : K → R are convex and lower semicontinuous functions, G i : K → E 1 are ε− inverse strongly monotone mappings and C i : K → E 1 , are monotone and Lipschitz continuous mappings.Let f : be the solution set of a system of generalized mixed equilibrium problems, and = {x * ∈ ∩ ∞ i=1 F(S i )} be the common fixed-point set of S i for each i ∈ I. Let the sequence {x n } be defined as follows: where . We shall strictly employ the above terminology in the sequel.Lemma 14. Suppose that σq is the function (5) in Lemma 6 for the characteristic inequality of the uniformly smooth dual E * 1 .For the sequence {x n } ⊂ E 1 defined by (7), Let , for λ n,i > 0 and r n,i > 0, i ∈ I be defined by , and Then for µ n,i = where G q is the constant defined in Lemma 6 and ρ E * 1 is the modulus of smoothness of E * 1 .
Lemma 15.For the sequence {x n } ⊂ E 1 , defined by (7) 1 , and λ n > 0 and r n,i > 0, i ∈ I, be defined by and where ι, γ ∈ (0, 1) and µ n,i = 1 x n p−1 are chosen such that and Then, for all v ∈ Γ, we get and Proof.By Lemmas 13, 4 and 6, for each i ∈ I, we get that , and hence it follows that By Lemmas 6 and 14, we have Substituting ( 24) into ( 23), we have, by Lemma 4 Substituting ( 18) and ( 20) into (25), we have Thus, (22) holds.
Now, for each i ∈ I, let v = B U i γ v and Av = B T i γ Av.By Lemma 4, we have where, As AK is closed and convex, by Lemma 5 and the variational inequality for the Bregman projection of zero onto AK − ∑ ∞ i=0 β n,i Au n,i , we arrive at By Lemma 6, 14 and (27), we get Substituting ( 17) and ( 19) into (28), we have Thus, ( 21) holds as desired.
We now prove our main result.
Proof.For x, y ∈ K and i ∈ I, let Φ i (x, y) = g i (x, y) + J p E 1 C i x, y − x + θ i (y) − θ i (x).Since g i are bi-functions satisfying (A1) − (A4) in (3) and C i are monotone and Lipschitz continuous mappings, and θ i are convex and lower semicontinuous functions, therefore Φ i (i ∈ I) satisfy the conditions (A1) − (A4) in (3), and hence the algorithm (29) can be written as follows: We will divide the proof into four steps.
Step One: x n = 0.Then, by (32), we have By (33) and Lemma 13, for each i ∈ I, we have that . By Lemma 4 and for v ∈ Γ and v = Υ r n,i v, we have In addition, for each i ∈ I, let v = B U i γ v.By Lemma 4 and for v ∈ Γ, we have Then by (32), we have that By (36) and Lemma 13, for each i ∈ I, we have (22) in Lemma 15, we get In addition, for each i ∈ I, v ∈ Γ, (21) in Lemma 15 gives Let u n,i = 0.By Lemma 1, we have and by ( 27), (39), Lemmas 4 and 15, we have However, by ( 30) and ( 40), we have This implies that By ( 42) and (37), we get In addition, it follows from the assumption η n,0 ≤ ∑ ∞ i=1 η n,i , (43), Definition 3, Lemmas 9 and 4 By (44), we conclude that {x n } is bounded, and hence, from (42), ( , {y n } and {u n,i } are also bounded. Step Two: We show that lim m→∞ p (x n+1 , x n ) = 0.By Lemmas 1, 4, 10, and 7, we have, by the convexity of p in the first argument and for where In view of the assumption ∑ ∞ n=1 ∑ ∞ i=1 η n−1,i M < ∞ and (45), Lemmas 11 and 8 imply Step Three: We show that lim n→∞ p (S n,i y n , y n ) = 0.For each i ∈ I, we have By (47) and Definition 2, we get By uniform continuity of S, we have Step Four: We show that x * for some x * ∈ E 1 .It follows from Lemmas 2, 3, and (48) that x * = S i x * , for each i ∈ I.This means that x * ∈ .
In addition, by (31) and the fact that u n,i → x * as n → ∞, we arrive at By ( 21), we have and by (41), we have From ( 53), (54), and (52), by passing n to infinity in (52), we have that 0 ∈ ∑ ∞ i=0 α n,i U i (x * ).This implies that x * ∈ SOLV IP(U i ).In addition, by (48), we have Ay n Ax * .Thus, by ( 53), ( 54) and an application of the demi-closeness of ).This means that x * ∈ Ω.Now, we show that x * ∈ (∩ ∞ i=1 GMEP(θ i , C i , G i , g i ).By (32), we have Since Φ i , for each i ∈ I, are monotone, that is, for all y ∈ K, By the lower semicontinuity of Φ i , for each i ∈ I, the weak upper semicontinuity of G, and the facts that, for each i ∈ I, u n,i → x * as n → ∞ and J p is norm − to − weak * uniformly continuous on a bounded subset of E 1 , we have From (57), by the lower semicontinuity of Φ i , for each i ∈ I, we have for y t → x * as t → 0 Therefore, by (58) we can conclude that x * ∈ (∩ ∞ i=1 GMEP(θ i , C i , G i , g i ).This means that x * ∈ ω.Hence, x * ∈ Γ.
Finally, we show that x n → x * , as n → ∞.By Definition 3, we have By (59) and Lemma 8, we have that The proof is completed.
In Theorem 1, i = 0 leads to the following new result.
Corollary 1.Let g : K × K → R be bifunctions satisfying (A1) − (A4) in (3).Let (I − Π p AK B T δ ) be demiclosed at zero.Suppose that x 1 ∈ E 1 is chosen arbitrarily and the sequence {x n } is defined as follows: , and and ι ∈ (0, 1) and τ n = 1 u n p−1 are chosen such that and lim n→∞ When U = D and T = D in Corollary 1, the algorithm (60) becomes and its strong convergence is guaranteed, which solves the problem of a common solution of a system of generalized mixed equilibrium problems, split Hammerstein integral equations, and fixed-point problems for the mappings involved in this algorithm.

Definition 3 .
Let E be a smooth Banach space.Let p be the Bregman distance.A mapping T : E −→ E is said to be a strongly asymptotically non-expansive left Bregman with {k n } ⊂ [1, ∞) if there exists non-negative real sequences {k n } with lim n→∞ where {β n } is a sequence in (0, 1) and {δ n } is a sequence in R, such that Therefore, by (51) and the boundedness of {y n }, and since by (46), {x n } is Cauchy, we can assume without loss of generality that y n
4. Let C ⊂ R n be bounded.Let k : C × C → R and f : C × R → R be measurable real-valued functions.An integral equation of Hammerstien-type has the form u(x) + C k(x, y) f (y, u(y))dy = w(x), where the unknown function u and non-homogeneous function w lies in a Banach space E of measurable real-valued functions.By transforming the above equation, we have that u + KFu = w, and therefore, without loss of generality, we have u + KFu = 0. (63) The split Hammerstein integral equations problem is formulated as finding x * ∈ E 1 and y * ∈ E * 1 such that x * + KFx * = 0 with Fx * = y * and Ky * + x * = 0 and Ax *