Strong Convergence Theorems for a Common Fixed Point of a Finite Family of Bregman Weak Relativity Nonexpansive Mappings in Reflexive Banach Spaces

We introduce an iterative process for finding an element of a common fixed point of a finite family of Bregman weak relatively nonexpansive mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.


Introduction
Throughout this paper, : → (−∞, +∞] is a proper, lower semicontinuous, and convex function, where is a real reflexive Banach space with * as its dual.
(3) Remark 1. If is a smooth Banach space, setting ( ) = ‖ ‖ 2 , for all ∈ we have ∇ ( ) = 2 , for all ∈ , where is the normalized duality mapping from onto * , and hence we have the following.
(ii) The Bregman projection ( ) reduces to the generalized projection Π ( ) (see, e.g., [3]) which is defined by Let be a nonempty and convex subset of int(dom ) and let : → int(dom ) be a mapping. A mapping : → is said to be nonexpansive if ‖ − ‖ ≤ ‖ − ‖, for all , ∈ . is said to be quasi-nonexpansive if ( ) ̸ = 0 and ‖ − ‖ ≤ ‖ − ‖, for all ∈ and ∈ ( ), where ( ) stands for the fixed point set of ; that is, ( ) = { ∈ : = }. A point ∈ is called an asymptotic fixed point of (see [4]) 2 The Scientific World Journal if contains a sequence { } which converges weakly to such that lim → ∞ ‖ − ‖ = 0. We denote bŷ( ) the set of asymptotic fixed points of . A point ∈ is called a strong asymptotic fixed point of if there exists a sequence { } in which converges strongly to and lim → ∞ ‖ − ‖ = 0. We denote the set of all strong asymptotic fixed points of bỹ( ).

(9)
Remark 2. We observe from the above definitions that every relatively nonexpansive mapping is Bregman relatively nonexpansive mapping with respect to ( ) = ‖ ‖ 2 , for all ∈ , where is called relatively nonexpansive mapping if the following conditions are satisfied: If = , a real Hilbert space, then relatively nonexpansive mappings are demiclosed quasi-nonexpansive mappings which include the class of nonexpansive mappings with fixed point nonempty.

Remark 3.
It is shown in [6] that if is Bregman firmly nonexpansive then̂( ) =̃( ) = ( ) and hence it is Bregman relatively nonexpansive provided that the Legendre function is uniformly Fréchet differentiable and bounded on bounded sets of .

Remark 4.
We observe from the above facts that the class of Bregman weak relatively nonexpansive mappings includes the class of Bregman relatively nonexpansive mappings and hence the class of Bregman firmly nonexpansive mappings. In addition, we also have that every continuous Bregman quasinonexpansive is Bregman weak relatively nonexpansive mapping.
The following example by Chen et al. [7] shows that the inclusion is proper.
Let : → be defined by Then, it is shown in [7] that is Bregman weak relatively nonexpansive but not Bregman relatively nonexpansive.
Construction of fixed points of nonexpansive mappings and relatively nonexpansive mappings and their generalizations is an important subject in nonlinear operator theory and its applications, in particular, in image recovery and signal processing (see, e.g., [8][9][10][11][12][13][14] and the references therein). Mann [15] and Ishikawa [16] iteration process for approximating fixed point iteration process for nonexpansive mappings and relatively nonexpansive mappings in Hilbert spaces and Banach spaces have been studied extensively by many authors to solve nonlinear operator equations as well as variational inequalities (see, e.g., [15][16][17][18][19][20][21]). However, both iteration processes have only weak convergence even in Hilbert spaces (see, e.g., [15,16]). Some attempts to modify the Mann iteration method so that strong convergence is guaranteed have been made.
In 2003, Nakajo and Takahashi [22] proposed the following modification of the Mann iteration method for a nonexpansive mapping : → in a Hilbert space : 0 ∈ , chosen arbitrarily, where { } ⊂ [0, 1] and denotes the metric projection from onto a closed convex subset of . They proved that the sequence { } defined by (13) where { } ⊂ [0, 1] and ( , ) is as shown in Remark 1. They showed that { } generated by (14) converges strongly to a fixed point of under some suitable assumptions.
In [24], Reich and Sabach proposed an algorithm for finding a common fixed point of finitely many Bregman firmly nonexpansive mappings : → ( = 1, 2, . . . , ) satisfying ⋂ =1 ( ) ̸ = 0 in a reflexive Banach space as follows: 0 ∈ , chosen arbitrarily, They proved that, under suitable conditions, the sequence { } generated by (15) converges strongly to ⋂ =1 ( ) and applied it to the solution of convex feasibility and equilibrium problems. You may also see [5,25]. Inspired and motivated by the above works, Chen et al. [7] proposed an algorithm for finding a fixed point of Bregman weak relatively nonexpansive mapping : → ( = 1, 2, . . . , ) satisfying F := ⋂ =1 ( ) ̸ = 0 in a reflexive Banach space as follows: is the Bregman projection of onto ( ). Moreover, in [26], Naraghirad and Yao introduced a hybrid iteration algorithm for finding a common fixed point of infinite family of Bregman weak relatively nonexpansive mapping : , in a reflexive Banach space . They proved that, under suitable conditions, their hybrid iteration algorithm converges strongly to F. Remark 6. It is worth mentioning that all the iteration algorithms introduced and used above seem cumbersome and complicated in the sense that at each stage of the iteration computations of the set(s) and/or are required and the next iterate is taken as the Bergman projection of 0 on the intersection of and/or . This seems difficult to do in applications.
It is our purpose in this paper to introduce an iterative algorithm for finding a common fixed point of a finite family of Bregman weak relatively nonexpansive mappings in reflexive Banach spaces. We prove strong convergence theorem for the sequence produced by the method. Our scheme does not require computations of the set or at each stage of iterates. We prove strong convergence theorems for the sequences produced by the method. Our results improve and generalize many known results in the current literature (see, e.g., [7,23,24]).

Preliminaries
Let : → (−∞, +∞] be a function. For any ∈ Dom( ) and ∈ , the right-hand derivative of at in the direction of is defined by ∘ ( , ) = lim → 0 + ( ( + ) − ( ))/ . The function is said to be Gâteaux differentiable at if lim → 0 + ( ( + ) − ( ))/ exists for any . In this case, ∘ ( , ) coincides with ∇ ( ), the value of the gradient ∇ of at . The function is said to be Gâteaux differentiable if it is Gâteaux differentiable for any ∈ int(dom ). The function is said to be Fréchet differentiable at if this limit is attained uniformly in ‖ ‖ = 1 and is said to be uniformly Fréchet differentiable on a subset of if the limit is attained uniformly for ∈ and ‖ ‖ = 1.
The function is said to be essentially smooth if is both locally bounded and single-valued on its domain. It is called essentially strictly convex, if ( ) −1 is locally bounded on its domain and is strictly convex on every convex subset of dom . is said to be a Legendre, if it is both 4 The Scientific World Journal essentially smooth and essentially strictly convex. When the subdifferential of is single-valued, it coincides with the gradient = ∇ (see [27]). For a Legendre function the following properties are known.
(i) is essentially smooth if and only if * is essentially strictly convex (see [28,Theorem 5.4]).
When is a smooth and strictly convex Banach space, one important and interesting example of Legendre function is ( ) := (1/ )‖ ‖ (1 < < ∞). In this case the gradient ∇ of coincides with the generalized duality mapping of ; that is, ∇ = (1 < < ∞). In particular, ∇ = , the identity mapping in Hilbert spaces.
A function on is coercive [30] if the sublevel set of is bounded; equivalently, lim ‖ ‖ → ∞ ( ) = ∞. A function on is said to be strongly coercive [31] if lim ‖ ‖ → ∞ ( )/‖ ‖ = ∞. Let for all ≥ 0. The function is called the gauge of uniform convexity of . The function is also said to be uniformly smooth on bounded subsets of if and only if * is uniformly convex on bounded subsets of .
In the sequel, we will need the following lemmas.
We know that is totally convex on bounded sets if and only if is uniformly convex on bounded sets (see [32,Theorem 2.10]). The next lemma will be useful in the proof of our main results.
The Scientific World Journal 5

Main Result
In the sequel, we will need the following lemma.
We now prove the following theorem.

Corollary 14.
Let : → R be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of . Let be a nonempty, closed, and convex subset of int(dom ) and let : → , for = 1, 2, . . . , , be a finite family of Bregman relatively nonexpansive mappings. Assume that the interior of F := ⋂ =1 ( ) is nonempty. For 0 ∈ let { } be a sequence generated by (28). Then, the sequence { } ∈N converges strongly to an element of F.
If, in Theorem 13, we take = 1, then we have the following corollary.

Remark 17.
Our results are new even if the convex function is chosen to be ( ) = (1/ )‖ ‖ (1 < < ∞) in uniformly smooth and uniformly convex spaces.
Remark 18. Our theorems improve and unify most of the results that have been proved for these important classes of nonlinear operators. In particular, Theorem 3.2 extends Theorem 3.1 of [23], Theorem 1 of [24], and Theorem 3.1 of [7] in the sense that either our theorem is applicable to a more general class of a finite family of Bregman weak relatively nonexpansive mappings or our scheme does not require the computation of or for each ≥ 0 provided that the interior of F is nonempty.
Moreover, we observe that Theorem 3.2 extends Theorem 3.1 of [26] in the sense that our scheme does not require the computations of for each ≥ 0 when we consider finite family of Bregman weak relatively nonexpansive mappings and the interior of F is nonempty.