## 1 Introduction
Let \(C\) be a nonempty, closed and convex subset of a Banach space \(E\), and let \(N(C)\) and \(CB(C)\) denote the family of nonempty subsets and nonempty closed bounded subsets of \(C\), respectively. Let \(H\) be the Hausdorff metric on \(CB(C)\) defined by:  
\[f^{*}(x^{*})=\sup_{x\in E}\{\langle x^{*},x\rangle-f(x)\}\]4  
for all \(A,B\in CB(C)\), where \(d(a,B)=\inf_{b\in B}\{\|a-b\|\}\) is the distance from the point ato the subset \(B\). Let \(T:C\longrightarrow CB(C)\) be a multi-valued mapping. A mapping \(T\) is said to be _nonexpansive_ if:  
\[f^{*}(x^{*})=\sup_{x\in E}\{\langle x^{*},x\rangle-f(x)\}\]5  
We denote the set of fixed points of \(T\) by \(F(T)\), that is \(F(T)=\{p\in C:p\in Tp\}\). A point \(p\in C\) is called an _asymptotic fixed point_ of \(T\) if there exists a sequence \(\{x_{n}\}\) in \(C\) that converges weakly to \(p\), such that \(\lim_{n\longrightarrow\infty}d(x_{n},Tx_{n})=0\). We denote by \(\widehat{F}(T)\) for the set of asymptotic fixed points of \(T\). Now, we give some definitions for class of multi-valued Bregman mappings. **Definition 5**. : A multi-valued mapping \(T:C\longrightarrow CB(C)\) with \(F(T)\
eq\emptyset\) is called:  
1. _Relatively quasi-nonexpansive_ if, \[f^{*}(x^{*})=\sup_{x\in E}\{\langle x^{*},x\rangle-f(x)\}\]6
2. _Relatively nonexpansive_ if \(T\) is relatively quasi-nonexpansive and \(\widehat{F}(T)=F(T)\);
3. _Bregman relatively quasi-nonexpansive_ if, \[f^{*}(x^{*})=\sup_{x\in E}\{\langle x^{*},x\rangle-f(x)\}\]7
4. _Bregman relatively nonexpansive_ if \(T\) is Bregman relatively quasi-nonexpansive and \(\widehat{F}(T)=F(T)\). We remark that the class of single-valued Bregman relatively nonexpansive mappings is contained in the class of multi-valued Bregman relatively nonexpansive mappings. Hence, the class of multi-valued Bregman relatively nonexpansive mappings is more general than class single-valued Bregman relatively nonexpansive mappings. The example of multi-valued Bregman relatively nonexpansive mapping given by [5] is shown below:  
**Example 1**. : _Let \(I=[0,1]\), \(X=L^{p}(I)\), \(1<p<\infty\) and \(C=\{f\in X:f(x)\geq 0,\forall x\in I\}\). Let \(T:C\longrightarrow CB(C)\) be defined by:_  
\[T(f)=\left\{ \begin{array}{l}\{h\in C:f(x)-\frac{1}{2}\leq h(x)\leq f(x)-\frac {1}{4},\ \forall x\in I\}\text{ if }f(x)>1,\forall x\in I\\ \{0\},\ \text{ otherwise}\end{array} \right. \tag{6}\]  
_It is clear in [5] that \(T\) defined by Equation (6) is a multi-valued Bregman relatively nonexpansive mapping._Let us take \(E\) as a reflexive real Banach space and \(E^{*}\) as its dual. Let \(f:E\longrightarrow(-\infty,+\infty]\) be a G\(\ddot{a}\)teaux differentiable mapping. The modulus of total convexity of \(f\) at \(x\in domf\) is the function \(v_{f}(x,\cdot):[0,+\infty)\longrightarrow[0,+\infty]\) defined by:  
\[v_{f}(x,\cdot):=\inf\{D_{f}(y,x):y\in domf,\|y-x\|=t\}\]  
The function \(f\) is said to be _totally convex_ at \(x\) if \(v_{f}(x,t)>0\), whenever \(t>0\). Any function \(f\) is called _totally convex_ if it is totally convex at any point \(x\in int(domf)\) and is called _totally convex on bounded sets_ if \(v_{f}(B,t)>0\) 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 \(v_{f}:int(domf)\times[0,+\infty)\longrightarrow[0,+\infty]\) defined by:  
\[v_{f}(B,t):=\inf\{v_{f}(x,t):x\in B\cap domf\}\]  
It is well known that \(f\) is totally convex on bounded sets if and only if it is uniformly convex on bounded sets (see [28], Theorem 2.10). **Lemma 1**. : ([29]) _If \(f:E\longrightarrow\mathbb{R}\) is uniformly Frechet differentiable and bounded on bounded subsets of \(E\), then \(\
abla f\) is uniformly continuous on bounded subsets of \(E\) from the strong topology of \(E\) to the strong topology of \(E^{*}\)._  
**Lemma 2**. : ([20]) _Let \(E\) be a reflexive Banach space, and let \(f:E\longrightarrow\mathbb{R}\) be a convex function that is bounded on bounded sets. Then, the following assertions are equivalent:_  
1.