## 1 Introduction  
In 1967, [PERSON] [1] found a beautiful and impressive technique named the Bregman distance function \(D_{f}\) for process designing and analyzing feasibility and optimization algorithms. This turned the research in which [PERSON]'s technique was applied towards a growing range of different ways to design and analyze iterative algorithms and to solve not only feasibility and optimization problems, but also algorithms for solving variational inequality problems, zero points of maximal monotone operators, equilibrium problems, fixed point problems for nonlinear mappings, and so on (see, e.g., [2; 3; 4] and the references therein). In recent years, many authors have constructed several iterative methods using Bregman distances for approximating fixed points (and common fixed points) of nonlinear mappings; we refer the readers to [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15] and the reference therein. In 2012, [PERSON] _et al._[7] considered strong convergence results of [PERSON]'s iteration for Bregman strongly nonexpansive mappings \(T\) in reflexive Banach spaces \(E\) as follows:  
\[x_{n+1}=\
abla f^{*}(\alpha_{n}\
abla f(u)+(1-\alpha_{n})\
abla f(Tx_{n})),\ \ \forall n\geq 0 \tag{1}\]  
where \(f\) is a strongly coercive Legendre function, which is bounded, uniformly Frechet differentiable and totally convex on bounded subsets of \(E\). They proved that the sequence \(\{x_{n}\}\) defined by Equation (1) converges strongly to a point \(p\in F(T)=\widehat{F}(T)\) under certain appropriate conditions on the parameter \(\{\alpha_{n}\}\), where \(\widehat{F}(T)\) is the set of asymptotic fixed points of \(T\). Later, [PERSON] _et al._[8] extended [PERSON]'s iteration for the Bregman strongly nonexpansive mapping \(T:E\longrightarrow E\) of [7] to Bregman strongly nonexpansive multi-valued mapping \(T:C\longrightarrow N(C)\) as follows:  
\[x_{n+1}=\
abla f^{*}(\alpha_{n}\
abla f(u)+(1-\alpha_{n})\
abla f(z_{n})),\ \ \forall n\geq 0 \tag{2}\]  
where \(z_{n}\in Tx_{n}\). They proved that the sequence \(\{x_{n}\}\) defined by Equation (2) converges strongly to a point \(p\in F(T)=\widehat{F}(T)\) under certain appropriate conditions on the parameter \(\{\alpha_{n}\}\). Very recently, [PERSON] and [PERSON] [5] introduced an iterative process for the approximation of a common fixed point of a finite family of multi-valued Bregman relatively nonexpansive mappings \(T_{i}:C\longrightarrow CB(C)\) in reflexive Banach spaces \(E\) as follows:  
\[\left\{ \begin{array}{l}w_{n}=P_{\mathrm{C}}^{f}\
abla f^{*}(\alpha_{n}\
abla f (u)+(1-\alpha_{n})\
abla f(x_{n}))\\ x_{n+1}=\
abla f^{*}(\beta_{0}\
abla f(w_{n})+\sum_{i=1}^{N}\beta_{i}\
abla f (u_{i,n})),\ \ \forall n\geq 0\end{array} \right. \tag{3}\]  
where \(u_{i,n}\in T_{i}w_{n}\) for \(i=1,2,...,N\), \(C\) is a nonempty, closed and convex subset of \(int(domf)\). Under some mild conditions on the parameters \(\{\alpha_{n}\}\) and \(\{\beta_{i,n}\}\), they prove that the sequence \(\{x_{n}\}\) defined by Equation (3) converges strongly to a point \(p\in\bigcap_{i=1}^{N}F(T_{i})\). On the other hand, [PERSON] and [PERSON] [16] introduced a multi-step iterative process by a hybrid method as follows:  
\[\left\{ \begin{array}{l}y_{n,1}=J^{-1}((1-\beta_{n,1}))x_{n}+\beta_{n,1}Jz_{ n,1})\\ y_{n,2}=J^{-1}((1-\beta_{n,2}))x_{n}+\beta_{n,2}Jz_{n,2})\\ \vdots\\ y_{n,N}=J^{-1}((1-\beta_{n,N}))x_{n}+\beta_{n,N}Jz_{n,N})\\ u_{n}\in C\ \ \text{such that}\ \ F(u_{n},y)+\frac{1}{r_{n}}(y-u_{n},Ju_{n}- Jy_{n,N})\geq 0\\ C_{n+1}=\{z\in C_{n}:\phi(z,u_{n})\leq\phi(z,x_{n})\}\\ x_{n+1}=\Pi_{C_{n+1}}x_{0},\ \ \forall n\geq 0\end{array} \right. \tag{4}\]  
where \(z_{n,1}\in T_{1}x_{n}\) and \(z_{n,i}\in T_{i}y_{n,i-1}\) for \(i=2,3...,N\), \(\Pi_{C}\) is the generalized projection from \(E\) onto \(C\), \(T_{i}\)\((i=1,2,..,N)\) is a finite family of relatively quasi-nonexpansive multi-valued mappings and \(J\) is the duality mapping on \(E\).