## 1 Introduction
\(f\) _is strongly coercive and uniformly convex on bounded sets;_
2. \(f^{*}\) _is Frechet differentiable, and_ \(\
abla f^{*}\) _is uniformly norm-to-norm continuous on bounded sets of_ \(dom(f^{*})=E^{*}\)_._  
**Lemma 3**. : ([5]) _Let \(E\) be a real reflexive Banach space, and let \(f:E\longrightarrow\mathbb{R}\) be a uniformly Frechet differentiable and totally convex on bounded subsets of \(E\). Let \(C\) be a nonempty, closed and convex subset of \(int(domf)\) and \(T:C\longrightarrow CB(C)\) be a finite family of multi-valued Bregman relatively nonexpansive mappings. Then, \(F(T)\) is closed and convex._  
**Lemma 4**. : ([28]) _Let \(C\) be a nonempty, closed and convex subset of \(E\). Let \(f:E\longrightarrow\mathbb{R}\) be a Ga\(\ddot{a}\)teaux differentiable and totally convex function, and let \(x\in E\). Then:_  
1. \(z=P_{C}^{f}(x)\) _if and only if_ \(\langle\
abla f(x)-\
abla f(z),y-z\rangle\leq 0\)_,_ \(\forall y\in C\)_._
2. \(D_{f}(y,P_{C}^{f}(x))+D_{f}(P_{C}^{f}(x),x)\leq D_{f}(y,x)\)_,_ \(\forall y\in C\)_._  
**Lemma 5**. : ([30]) _Let \(E\) be a Banach space; let \(r>0\) be a constant; and let \(f:E\longrightarrow\mathbb{R}\) be a uniformly convex on bounded subsets of \(E\). Then:_  
\[f\bigg{(}\sum_{k=1}^{n}\alpha_{k}x_{k}\bigg{)}\leq\sum_{k=0}^{n}\alpha_{k}f(x_ {k})-\alpha_{i}\alpha_{j}\rho(\|x_{i}-y_{j}\|)\]  
_for all \(i,j\in\{0,1,2,...,n\}\), \(x_{k}\in B_{r}\), \(\alpha_{k}\in(0,1)\) and \(k=0,1,2,...,n\) with \(\sum_{k=0}^{n}\alpha_{k}=1\), where \(\rho_{r}\) is the gauge of uniform convexity of \(f\)._  
**Lemma 6**. : ([31]) _Let \(f:E\longrightarrow(-\infty,+\infty]\) be a proper, lower semi-continuous and convex function, then \(f^{*}:E^{*}\longrightarrow(-\infty,+\infty]\) is proper, weak\(\ddot{\text{\emph{e}}}\)lower semi-continuous and convex function. Thus, for all \(z\in E\), we have:_  
\[D_{f}\bigg{(}z,\
abla f^{*}\bigg{(}\sum_{i=1}^{N}t_{i}\
abla f(x_{i})\bigg{)} \bigg{)}\leq\sum_{i=1}^{N}t_{i}D_{f}(z,x_{i})\]  
_where \(\{x_{i}\}_{i=1}^{N}\subset E\) and \(\{t_{i}\}_{i=1}^{N}\) with \(\sum_{i=1}^{N}t_{i}=1\)._  
**Lemma 7**. : ([32]) _Let \(f:E\longrightarrow\mathbb{R}\) be a Gateaux differentiable and totally convex function. If \(x\in E\) and the sequence \(\{D_{f}(x_{n},x)\}\) is bounded, then the sequence \(\{x_{n}\}\) is also bounded._  
**Lemma 8**. : ([30]) _Let \(E\) be a Banach space, and let \(f:E\longrightarrow\mathbb{R}\) be a Gateaux differentiable function, which is totally convex on bounded subsets of \(E\). Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be bounded sequences in \(E\). Then:_  
\[\lim_{n\longrightarrow\infty}D_{f}(x_{n},y_{n})=0\iff\lim_{n\longrightarrow \infty}\|x_{n}-y_{n}\|=0\]  
The following lemma can be found in [27,33,34]. **Lemma 9**. : ([27,33,34]) _Let \(E\) be a reflexive Banach space, \(f:E\longrightarrow\mathbb{R}\) be Legendre and Gateaux differentiable function, and let \(V_{f}:E\times E^{*}\longrightarrow[0,+\infty)\) defined by:_  
\[V_{f}(x,x^{*})=f(x)-\langle x,x^{*}\rangle+f^{*}(x^{*}),\ \ \forall x\in E,\ \ x^{*}\in E^{*}\]  
_Then, the following assertions hold:_  
(1) \(D_{f}(x,\
abla f^{*}(x^{*}))=V_{f}(x,x^{*}),\ \ \forall x\in E,\ \ x^{*},y^{*}\in E^{*}.\)__  
(2) \(V_{f}(x,x^{*})+\langle y^{*},\
abla f^{*}(x^{*})-x\rangle\leq V_{f}(x,x^{*}+y ^{*})),\ \ \forall x\in E,\ \ x^{*},y^{*}\in E^{*}.\)__  
**Lemma 10**. : ([5]) _Let \(E\) be a real reflexive Banach space and \(f:E\longrightarrow\mathbb{R}\) be a Gateaux differentiable and totally convex function. Let \(C\) be a nonempty, closed and convex subset of int(dom\(f\)) and \(T_{i}:C\longrightarrow CB(C)\)\((i=1,2,...,N)\) be a finite family of multi-valued Bregman relatively nonexpansive mappings, such that \(\mathcal{F}:=\cap_{i=1}^{N}F(T_{i})\) is nonempty, closed and convex. Suppose that \(u\in C\) and \(\{x_{n}\}\) are a bounded sequence in \(C\) such, that \(\lim_{n\longrightarrow\infty}d(x_{n},T_{i}x_{n})=0\).