On Split Generalized Equilibrium and Fixed Point Problems of Bregman W-Mappings with Multiple Output Sets in Reflexive Banach Spaces

. In this paper, we introduce a Halpern iteration process for computing the common solution of split generalized equilibrium problem and fixed points of a countable family of Bregman W-mappings with multiple output sets in reflexive Banach spaces. We prove a strong convergence result for approximating the solutions of the aforementioned problems under some mild conditions. It is worth mentioning that the iterative algorithm employ in this article is designed in such a way that it does not require the prior knowledge of operator norm. We also provide some numerical


Introduction
Let Y be a reflexive Banach space with its dual Y * and D be a nonempty, closed and convex subset of Y .The Generalized Equilibrium Problem (in brief, GEP) is to find x * ∈ D such that where G : D × D → R is a bifunction and b : D × D → R is a skew matrix.If b ≡ 0, then GEP (1.1) reduces to the following Equilibrium Problem (in brief, EP) which is to find x * ∈ D such that The Equilibrium Problem is known to include many mathematical problems, for example, variational inclusion problem, complementary problem, saddle point problem, Nash equilibrium problem in non-cooperative games, minimax inequality problem, minimization problem, variational inequality problem and fixed point problem, see [6,11,14,17,19,22,33,34,37].Let D and E be nonempty, closed and convex subsets of two real Banach spaces Y 1 and Y 2 respectively.Let A : Y 1 → Y 2 be a bounded linear operator.The Split Feasibility Problem (in brief, SFP) introduced by Censor and Elfving [15] is to find a point By combining SFP (1.2) and GEP (1.1), we have the Split Generalized Equilibrium Problem (in brief, SGEP), which is to We denote by SEP (G 1 , G 2 ) the solution set of (1.5)- (1.6).The Split Generalized Equilibrium Problem is very general in the sense that it includes as particular cases, split varaitional inequality problem and split minimization problem, to mention a few, (see [1,2,3,4,23,24,30,31] ).
To solve GEP (1.1), we need the following assumptions: Let G : D × D → R. Assumption 1.3:In 2018, Phuengrattana and Lerkchayaphum [32] introduced a shrinking projection method for solving the common solution of split generalized equilibrium problem and fixed point problem of multivalued nonexpansive mappings in real Hilbert spaces.They proved that the sequence {x n } converges strongly to proj g ∆ x 0 , where ∆ := Sol(GEP (1.1) ∩ F (T ) is nonempty.Our proposed method is endowed with the following characteristics: (1) We extend the results of [1,2,32] from real Hilbert spaces to a more general space which is convex, continuous and strongly coercive Bregman function, which is bounded on bounded subsets, and uniformly convex and uniformly smooth on bounded subsets.
(2) Our method does not require computing the projection of the current iterate onto the intersection of sets C n and Q n which was used in [5,18,32].
(3) In the result of [2,18,25,32] and other related results, we were able to dispense with one of the resolvents of the EP.Using the notion of multiple output sets, we were able to generalize some related results in literature without one of the resolvents.
(4) Our method uses self-adaptive stepsizes and the implementation of our method does not require prior knowledge of the norm of the bounded linear operator A, see [32].
(5) Our result also generalizes the results of [2,18,25,32] to a type of SGEP with multiple output sets.

Preliminaries
In the sequel, we denote strong and weak convergence by "→" and "⇀", respectively.The notion of W -mapping was first introduced in 1999 by Atsushiba and Takahashi [8] and since then, it has been considered for a finite family of mappings, see ( [19, 20, 27]).The notion was extended to a Banach space by Naraghirad and Timnak [29] as follows.Let D be a nonempty, closed and convex subset of a reflexive Banach space Y .Let {S n } n∈N be an infinite family of Bregman weak relatively nonexpansive mappings of D into itself, and let {µ n,t : t, n ∈ N, 1 ≤ t ≤ n} be a sequence of real numbers such that 0 ≤ µ i,j ≤ 1 for every i, j ∈ N with i ≥ j.Then, for any n ∈ N, we define a mapping W n of D into itself as follows: for all x ∈ D, where proj g D is the Bregman projection from Y onto D. Such a mapping W n is called the Bregman W -mapping generated by S n , S n−1 , . . ., S 1 and µ n,n , µ n,n−1 , . . ., µ n,1 .
Let Y be a reflexive Banach space with Y * its dual and Q be a nonempty closed and convex subset of Y .Let g : Y → (−∞, +∞] be a proper, lower semicontinuous and convex function, then the Fenchel conjugate of g denoted as g * : Y * → (−∞, +∞] is defined as Let the domain of g be denoted as dom(g) = {x ∈ Y : g(x) < +∞}, hence for any x ∈ intdom(g) and y ∈ Y , we define the right-hand derivative of g at x in the direction of y by t .
Let g : Y → (−∞, +∞] be a function, then g is said to be: (i) Gâteaux differentiable at x if lim t→0 + g(x+ty)−g(x) t exists for any y.In this case, g 0 (x, y) coincides with ∇g(x) (the value of the gradient ∇g of g at x); (ii) Gâteaux differentiable, if it is Gâteaux differentiable for any x ∈ intdomg; (iii) Fréchet differentiable at x, if its limit is attained uniformly in ∥y∥ = 1; (iv) Uniformly Fréchet differentiable on a subset Q of Y , if the above limit is attained uniformly for x ∈ Q and ∥y∥ = 1.(v) Essentially smooth, if the subdifferential of g denoted as ∂g is both locally bounded and single-valued on its domain, where ∂g(x) = {w ∈ Y : g(x)−g(y) ≥ ⟨w, y − x⟩, y ∈ Y }; (vi) Essentially strictly convex, if (∂g) −1 is locally bounded on its domain and g is strictly convex on every convex subset of dom ∂g; (vii) Legendre, if it is both essentially smooth and essentially strictly convex.See [9,10] for more details on Legendre functions.Alternatively, a function g is said to be Legendre if it satisfies the following conditions: (i) The intdom(g) is nonempty, g is Gâteaux differentiable on intdom(g) and dom∇g = intdom(g); (ii) The intdomg * is nonempty, g * is Gâteaux differentiable on intdomg * and dom∇g * = intdom(g).Let E be a Banach space and B s := {z ∈ Y : ∥z∥ ≤ s} for all s > 0.Then, a function g : Y → R is said to be uniformly convex on bounded subsets of Y , [see pp.203 and 221] [39] if ρ s t > 0 for all s, t > 0, where ρ s : [0, +∞) → [0, ∞] is defined by x,y∈Bs,∥x−y∥=t,α∈(0,1) for all t ≥ 0, with ρ s denoting the gauge of uniform convexity of g.
where ∇ g Y is the gradient function of Y dependent on g.It is clear that D g (x, y) ≥ 0 for all x, y ∈ Y .
It is well-known that Bregman distance D g does not satisfy all the properties of a metric function because D g fail to satisfy the symmetric and triangular inequality property.However, the Bregman distance satisfies the following socalled three point identity: for any x ∈ dom(g) and y, z ∈ intdom(g), In particular, The relationship between D g and ∥.∥ is guaranteed when g is strongly convex with strong convexity constant ρ > 0, i.e., Let g : Y → R be a strictly convex and Gâteaux differentiable function and We denote by F (T ) the set of all fixed points of T .Furthermore, a point p ∈ Q is called an asymptotic fixed point of T if Q contains a sequence {x n } which converges weakly to p such that lim n→∞ ∥T x n −x n ∥ = 0. We denote by F (T ) the set of asymptotic fixed points of T .A point p ∈ Q is called a strong asymptotic fixed point of T if C contains a sequence {x n } which converges strongly to p such that lim n→∞ ∥T x n − x n ∥ = 0. We denote the set of strong asymptotic fixed points of T by F (T ).It follows from the definition that Let Q be a nonempty closed and convex subset of int(dom g), then we define an operator T : Q → int(domg) to be: (i) Bregman relatively nonexpansive, if F (T ) ̸ = ∅, and (ii) Bregman weak relatively nonexpansive, if F (T ) ̸ = ∅, and (iii) Bregman quasi-nonexpansive mapping if F (T ) ̸ = ∅ and (iv) Bregman firmly nonexpansive (BFNE), if We define a countable family S j : H → H by for all j ≥ 1 and n ≥ 0. It is clear that F (S j ) = {0} for all j ≥ 1.
It can be shown that T and S j are Bregman quasi-nonexpansive, precisely Bregman weak relatively nonexpansive (see [16,28]).Definition 2. [21] Let Q be a nonempty, closed and convex subset of a reflexive Banach space Y and g : Y → (−∞, +∞] be a strongly coercive Bregman function.Let β and γ be real numbers with β ∈ (−∞, 1) and γ ∈ [0, ∞), respectively.Then a mapping T : Lemma 1. [38] Let Y be a Banach space, s > 0 be a constant, ρ s be the gauge of uniform convexity of g and g : Y → R be a strongly coercive Bregman function.Then, (i) For any x, y ∈ B s and α ∈ (0, 1), we have Lemma 2. [13] Let Y be a reflexive Banach space, g : Y → R be a strongly coercive Bregman function and V be a function defined by The following assertions also hold: for all x ∈ Y and x * , y * ∈ Y * .Also, following a similar approach as in Lemma 2 and for any x ∈ Y, y * , z * ∈ B r and α ∈ (0, 1), we have The resolvent of G : D×D → R with respect to b is the operator res g G,b : Y → 2 D defined as follows: We obtain some properties of the resolvent operator res g G,b .Lemma 3.
(where ran(A) denotes the range of (A).Then for any So, given any real numbers ξ 1 and ξ 2 , the mapping and are well-defined, where γ is any nonnegative real number.Moreover, for any (x, p) ∈ E 1 × K, we have , where Remark 1.It is easy to see from [21] that res g G,b is (0, 1)-demigeneralized.Therefore, we conclude from (2.3) that D g1 (q, y)≤D g1 (q, x)−(γL where T = res g G,b .Lemma 5. [13] Let Y be a Banach space and g : Y → R a Gâteaux differentiable function which is uniformly convex on bounded subsets of Y .Let {x n } n∈N and {y n } n∈N be bounded sequences in Y .Then, Lemma 6. [29] Let Y be a Banach space and g : Y → R a Gâteaux differentiable function which is uniformly convex on bounded subsets of Y .Let D be a nonempty, closed and convex subset of Y and S 1 , S 2 , . . ., S n be Bregman weak relatively nonexpansive mappings of D into itself such that Γ : ∩ N i=1 F (S i ) ̸ = ∅.Let {µ n,t : t, n ∈ N, 1 ≤ t ≤ n} be a sequence of real numbers such that 0 < µ n,1 ≤ 1 and 0 < µ n,i < 1 for every i = 2, 3, . . ., n.Let W n be the Bregman W -maping generated by S n , S n−1 , . . ., S 1 and µ n,n , µ n,n−1 , . . ., µ n,1 .Then, the following assertions holds: x), (iii) for every n ∈ N, W n is a Bregman weak relatively nonexpansive mapping.Lemma 7. [36] Let g : Y → R be a Gâteaux differentiable and totally convex function.If x 0 ∈ Y and the sequence {D g (x n , x 0 )} is bounded, then the sequence {x n } is also bounded.Definition 5. Let Q be a nonempty closed and convex subset of a reflexive Banach space Y and g : Y → (−∞, +∞] be a strongly coercive Bregman function.A Bregman projection of x ∈ int(dom(g) x) : y ∈ Q}.Lemma 8. [35] Let Q be a nonempty closed and convex subset of a reflexive Banach space Y and x ∈ Y .Let g : Y → R be a strongly coercive Bregman function.Then, [7,26] Let {a n } be a sequence of non-negative real numbers, {γ n } be a sequence of real numbers in (0, 1) with conditions ∞ n=1 γ n = ∞ and {d n } be a sequence of real numbers.Assume that If lim sup k→∞ d n k ≤ 0 for every subsequence {a n k } of {a n } satisfying the condition: lim sup k→∞ (a n k − a n k +1 ) ≤ 0, then lim n→∞ a n = 0.
(4) Assume that is nonempty.Let γ > 0 be a real number and {α n } n∈N , {θ j,m } be two sequences in (0, 1) with m j=0 θ j,m = 1 satisfying the following control conditions: Algorithm 1. Define a sequence {x n } ∞ n=1 generated arbitrarily by chosen x 1 ∈ E and any fixed u ∈ E, such that Let the sequences {ξ 1,n } n∈N and {ξ 2,n } n∈N satisfy the following condition: there exists a positive real number ρ such that where ((res and (res ((res , if Then, the sequence {x n } generated iteratively by Algorithm 1 converges strongly to z = proj g Ω u, where proj g Ω is the Bregman projection of Y onto Ω.
We proceed with the following two steps.
Step 1: Boundedness of the iterative method.
Using Lemma 7, then we conclude that {x n } is bounded.Consequently, {u n } and {z n } are bounded.
From (2.1) and (3.1), we obtain that Also, for all n ∈ N. Since ▽ g Y is uniformly norm-to-norm continuous on bounded subsets of Y , by following the same approach as in (3.7) and applying (3.9), (3.13), we obtain From (3.13) and (3.16), we deduce that (3.17) On the other hand, we obtain ∀t ∈ N with k ≥ 2. In view of (3.17), we get From (3.13) and (3.17), we get This together with (3.18) implies that Since {x n k } is bounded, there exists a subsequence {x n k j } of {x n k } such that {x n k } converges weakly to z ∈ Ω.Also, from (3.14) and (3.15), there exist {u n k j } of {u n k } and {z n k j } of {z n k } which converge weakly to z respectively.Thus, for each j = 0, 1, 2, . . .m, K j is a bounded linear operator, then it follows that K j x n k ⇀ K j z ∈ Y j as k → ∞.
The results of this experiment are reported in Figure 1 below.expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS.