## 2 Main Results  
_Theorem 1_.: Let \(E\) be a real reflexive Banach space and \(f:E\longrightarrow\mathbb{R}\) be a strongly coercive Legendre function, which is bounded, uniformly Fr\(\ddot{e}\)chet differentiable and totally convex on bounded subsets of \(E\). Let \(C\) be a nonempty, closed and convex subset of int(dom\(f\)) and \(T_{i}:C\longrightarrow\mathbb{R}\) be a\(CB(C)\)\((i=1,2,...,N)\) be a finite family of multi-valued Bregman relatively nonexpansive mapping, such that \(\mathcal{F}:=\bigcap_{i=1}^{N}F(T_{i})\
eq\emptyset\). For \(u,x_{0}\in C\), let \(\{x_{n}\}\) be a sequence generated by:  
\[\left\{\begin{array}{l}y_{n,1}=\
abla f^{*}(\  
Bregman relatively nonexpansive mapping, such that: \(\mathcal{F}:=\bigcap_{i=1}^{N}F(T_{i})\
eq\emptyset\). For \(u,x_{0}\in C\), let \(\{x_{n}\}\) be a sequence generated by:_  
\[\left\{\begin{array}{l}y_{n,1}=\
abla f^{*}(\beta_{n,1}\
abla f(x_{n})+(1- \beta_{n,1})\
abla f(T_{1}x_{n}))\\ y_{n,2}=\
abla f^{*}(\beta_{n,2}\
abla f(x_{n})+(1-\beta_{n,2})\
abla f(T_{2} y_{n,1}))\\ \vdots\\ y_{n,N}=\
abla f^{*}(\beta_{n,N}\
abla f(x_{n})+(1-\beta_{n,m})\
abla f(T_{N} y_{n,N-1}))\\ x_{n+1}=\
abla f^{*}(\alpha_{n}\
abla f(u)+(1-\alpha_{n})\
abla f(y_{n,N})),\ \ \forall n\geq 0\end{array}\right. \tag{20}\]  
_Suppose that \(\{\alpha_{n}\}\) and \(\{\beta_{n,i}\}_{i=1}^{N}\) are as in Theorem 1. Then, \(\{x_{n}\}\) converges strongly to \(p=P_{\mathcal{F}}^{f}(u)\), where \(P_{\mathcal{F}}^{f}\) is the Bregman projection of \(C\) onto \(\mathcal{F}\)._  
If we take \(E\) to be a uniformly smooth and uniformly convex Banach space and \(f(x)=\|x\|^{2}\) for all \(x\in E\) in Theorem 1, then we get the following result:  
Corollary 4: _Let \(E\) be a uniformly smooth and uniformly convex Banach space. Let \(C\) be a nonempty, closed and convex subset of \(E\) and \(T_{i}:C\longrightarrow CB(C)\)\((i=1,2,...,N)\) be a finite family of multi-valued relatively nonexpansive mapping, such that \(\mathcal{F}:=\bigcap_{i=1}^{N}F(T_{i})\
eq\emptyset\). For \(u,x_{0}\in C\), let \(\{x_{n}\}\) be a sequence generated by:_  
\[\left\{\begin{array}{l}y_{n,1}=J^{-1}(\beta_{n,1}Jx_{n}+(1-\beta_{n,1})Jz_{n,1})\\ y_{n,2}=J^{-1}(\beta_{n,2}Jx_{n}+(1-\beta_{n,2})Jz_{n,2})\\ \vdots\\ y_{n,N}=J^{-1}(\beta_{n,N}Jx_{n}+(1-\beta_{n,N})Jz_{n,N})\\ x_{n+1}=J^{-1}(\alpha_{n}Ju+(1-\alpha_{n})Jy_{n,N}),\ \ \forall n\geq 0\end{array}\right. \tag{21}\]  
_where \(z_{n,1}\in T_{1}x_{n}\), \(z_{n,i}\in T_{i}y_{n,i-1}\) for \(i=2,3,...,N\). Suppose that \(\{\alpha_{n}\}\) and \(\{\beta_{n,i}\}_{i=1}^{N}\) are as in Theorem 1. Then, \(\{x_{n}\}\) converges strongly to \(p=\Pi_{\mathcal{F}}(u)\), where \(\Pi_{\mathcal{F}}\) is the generalized projection of \(C\) onto \(\mathcal{F}\)._  
In Theorem 1, if we take \(E=H\) to be a real Hilbert space, then \(J=I\) is the identity mapping. Thus, we obtain the following corollary:  
Corollary 5: _Let \(H\) be a real Hilbert space, and let \(C\) be a nonempty, closed and convex subset of \(H\). Let \(T_{i}:C\longrightarrow CB(C)\)\((i=1,2,...,N)\) be a finite family of multi-valued relatively nonexpansive mapping, such that \(\mathcal{F}:=\bigcap_{i=1}^{N}F(T_{i})\
eq\emptyset\). For \(u,x_{0}\in C\), let \(\{x_{n}\}\) be a sequence generated by:_  
\[\left\{\begin{array}{l}y_{n,1}=\beta_{n,1}x_{n}+(1-\beta_{n,1})z_{n,1}\\ y_{n,2}=\beta_{n,2}x_{n}+(1-\beta_{n,2})z_{n,2}\\ \vdots\\ y_{n,N}=\beta_{n,N}x_{n}+(1-\beta_{n,N})z_{n,N}\\ x_{n+1}=\alpha_{n}u+(1-\alpha_{n})y_{n,N},\ \ \forall n\geq 0\end{array}\right. \tag{22}\]  
_where \(z_{n,1}\in T_{1}x_{n}\), \(z_{n,i}\in T_{i}y_{n,i-1}\) for \(i=2,3,...,N\). Suppose that \(\{\alpha_{n}\}\) and \(\{\beta_{n,i}\}_{i=1}^{N}\) are as in Theorem 1. Then, \(\{x_{n}\}\) converges strongly to \(p=P_{\mathrm{C}}(u)\), where \(P_{\mathrm{C}}\) is the metric projection of \(C\) onto \(\mathcal{F}\)._