Strong Convergence Theorems for Quasi-Bregman Nonexpansive Mappings in Reflexive Banach Spaces

We study a strong convergence for a common fixed point of a finite family of quasi-Bregman nonexpansive mappings in the framework of real reflexive Banach spaces. As a consequence, convergence for a common fixed point of a finite family of Bergman relatively nonexpansive mappings is discussed. Furthermore, we apply our method to prove strong convergence theorems of iterative algorithms for finding a common solution of a finite family equilibrium problem and a common zero of a finite family of maximal monotone mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

Let  ∈ Dom() and  ∈ .The right-hand derivative of  at  in the direction of  is defined by  ∘ (, ) = lim  → 0 + ((+)−())/.The function  is called Gâteaux differentiable at  if lim  → 0 + (( + ) − ())/ exists for any  and hence  ∘ (, ) 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.Furthermore,  is said to be Fréchet differentiable at  if this limit is attained uniformly in ‖‖ = 1 and it is called uniformly Fréchet differentiable on a subset  of  if the limit is attained uniformly for  ∈  and ‖‖ = 1.
Let  be a nonempty and convex subset of int(dom ) and let  :  → int(dom ) be a mapping. is said to be nonexpansive if ‖ − ‖ ≤ ‖ − ‖ for all ,  ∈ , and  is said to be quasinonexpansive if () ̸ = 0 and ‖ − ‖ ≤ ‖ − ‖, for all  ∈  and  ∈ (), where () stands for the fixed point set of ; that is, () = { ∈  :  = }.A point  ∈  is said to be an asymptotic fixed point of  (see [3]) if  contains a sequence {  } which converges weakly to  such that lim  → ∞ ‖  −   ‖ = 0.The set of asymptotic fixed points of  is denoted by F().
The existence and approximation of fixed points of Bregman firmly nonexpansive mappings were studied in [5].It is also known that if  is Bregman firmly nonexpansive and  is Legendre function which is bounded, uniformly Frêchet differentiable, and totally convex on bounded subsets of , then () = F() and () is closed and convex (see [5]).It also follows that every Bregman firmly nonexpansive mapping is Bregman relatively nonexpansive and hence quasi-Bregman nonexpansive mapping.
Remark 2. But it is worth mentioning that the iteration processes ( 9) and (10) seem difficult in the sense that, at each stage of iteration, the set(s)   and/or   are/is computed and the next iterate is taken as the Bregman projection of  0 onto the intersection of   and/or   .This seems difficult to do in applications.
In this paper, we investigate an iterative scheme for finding a common fixed point of a finite family of quasi-Bregman nonexpansive mappings in reflexive Banach spaces.We prove strong convergence theorems for the sequences produced by the method.Furthermore, we apply our method to prove strong convergence theorems for finding a solution of a finite family of equilibrium problems and for finding a common zero of a finite family of maximal monotone mappings.Our results improve and generalize many known results in the current literature (see, e.g., [4,23])

Preliminaries
Let  :  → (−∞, +∞] be a Gâteaux differentiable function.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 essentially smooth and essentially strictly convex.When the subdifferential of  is single-valued, it coincides with the gradient  = ∇ (see [24]).
We note that if  is a reflexive Banach space, then we have the following.
(i)  is essentially smooth if and only if  * is essentially strictly convex (see [25,Theorem 5.4]).
When  is a smooth and strictly convex Banach space, one important and interesting example of Legendre function is In this case the gradient ∇ of  coincides with the generalized duality mapping of ; that is, ∇ =   (1 <  < ∞).In particular, ∇ = , the identity mapping in Hilbert spaces.
In the sequel, we will need the following lemmas.
A function  on  is coercive [30] if the sublevel set of  is bounded; equivalently, lim ‖‖ → ∞ () = ∞.A function  on  is said to be strongly coercive [27] Lemma 6 (see [27]).Let  be a reflexive Banach space and let  :  → R be a continuous convex function which is strongly coercive.Then the following assertions are equivalent.
Lemma 9 (see [27,32]).Let  :  → R be a strongly coercive and uniformly convex on bounded subsets of ; then  * is bounded on bounded sets and uniformly Fréchet differentiable on bounded subsets of  * .
Lemma 12 (see [36]).Let {  } be a sequence of nonnegative real numbers satisfying the following relation: where {  } ⊂ (0, 1) and {  } ⊂ R satisfying the following conditions: Lemma 13 (see [37]).Let {  } be sequences of real numbers such that there exists a subsequence {  } of {} such that    <    +1 for all  ∈ N. Then there exists an increasing sequence {  } ⊂ N such that   → ∞ and the following properties are satisfied by all (sufficiently large) numbers  ∈ N: In fact,   is the largest number  in the set {1, 2, . . ., } such that the condition   ≤  +1 holds.

Main Results
We now prove the following theorem.
The resolvent of a bifunction  :  ×  → R [39] is the operator Res We know the following lemma in [23].

Zeroes of Maximal Monotone Operators.
In this section we present an algorithm for finding a common zero of a finite family of maximal monotone mappings.