## 1 Introduction
Under some suitable conditions, they proved that the sequence \(\{x_{n}\}\) defined by Equation (4) converges strongly to common elements of the set of common fixed points of a finite family of relatively quasi-nonexpansive multi-valued mappings and the solution set of an equilibrium problem in a real uniformly convex and uniformly smooth Banach space. Here, from the motivation of the above results, by using Bregman functions, we introduce a new multi-step iteration for approximating common fixed point of a finite family of multi-valued Bregman relatively nonexpansive mappings in the setting of reflexive Banach spaces. We derive a strong convergence theorem of the proposed iterative algorithm under appropriate situations. Furthermore, we also use our results to solving variational inequality problems and find zero points of maximal monotone operators. The results obtained in this article are new, improved and generalize many known recent results in this field. Throughout this paper, we assume that \(E\) is a real reflexive Banach space with the dual space of \(E^{*}\), and \(\langle\cdot,\cdot\rangle\) is the pairing between \(E\) and \(E^{*}\). Let \(x\in int(domf)\). The _subdifferential_ of \(f\) at \(x\) is the convex set defined by:  
\[\partial f(x)=\{x^{*}\in E^{*}:f(x)+\langle x^{*},y-x\rangle\leq f(y),\ \ \forall y\in E\}\]  
The _Fenchel conjugate_ of \(f\) is the function \(f^{*}:E^{*}\longrightarrow(-\infty,+\infty]\) defined by:  
\[f^{*}(x^{*})=\sup_{x\in E}\{\langle x^{*},x\rangle-f(x)\}\]  
We know that the _Young-Fenchel inequality_ holds, _i.e._, \(f(x)+f^{*}(x^{*})\geq\langle x^{*},x\rangle\), \(\forall x\in E\), \(x^{*}\in E^{*}\). It is also known that \(x^{*}\in\partial f(x)\) is equivalent to \(f(x)+f^{*}(x^{*})=\langle x^{*},x\rangle\) (see [17; 18]). The set \(\mathit{lev}_{\leq}^{f}(r)=\{x\in E:f(x)\leq r\}\) for some \(r\in\mathbb{R}\) is called a _sub-level_ of \(f\). A function \(f\) on \(E\) is _coercive_[19] if the sub-level set of \(f\) is bounded; equivalently,  
\[\lim_{\|x\|\longrightarrow+\infty}f(x)=+\infty\]  
A function \(f\) on \(E\) is said to be _strongly coercive_[20] if:  
\[\lim_{\|x\|\longrightarrow+\infty}\frac{f(x)}{\|x\|}=+\infty\]  
We denote by \(domf\) the domain of \(f\), _i.e._, the set \(\{x\in E:f(x)<+\infty\}\). Definition 1: ([21]) The function \(f\) is called:  
1. _Essentially smooth_ if \(f\) is both locally bounded and single-valued on its domain. 2. _Essentially strictly convex_ if \((\partial f)^{-1}\) is locally bounded on its domain and \(f\) is strictly convex on every convex subset of \(domf\). 3. _Legendre_ if it is both essentially smooth and essentially strictly convex. Remark 1: Let \(E\) be a reflexive Banach space, and let \(f\) be a Legendre function; then, we have:  
1. \(f\) is essentially smooth if and only if \(f^{*}\) is essentially strictly convex (see [21], Theorem 5.4). 2. \((\partial)^{-1}=\partial f^{*}\) (see [22]). 3. \(f\) is Legendre if and only if \(f^{*}\) is Legendre (see [22], Corollary 5.5). 4. If \(f\) is Legendre, then \(\
abla f\) is a bijection satisfying: \[\
abla f=(\
abla f^{*})^{-1},ran\
abla f=dom\
abla f^{*}=int(domf^{*})\ \ \text{and}\ \ \
abla f^{*}=dom\
abla f=int(domf)\]  
(see [22], Theorem 5.10, and [2]). Examples of Legendre functions were given in [21; 23]. One nice example of a Legendre function is \(f(x):=\frac{1}{p}\|x\|^{p}\) (\(1<p<\infty\)) when \(E\) is a smooth and strictly convex Banach space. In this case, the gradient \(\
abla f\) of \(f\) is coincident with the generalized duality mapping of \(E\), _i.e._, \(\
abla f=J_{p}\) (\(1<p<\infty\)). In particular, \(\
abla f=I\) the identity mapping in Hilbert spaces. In the rest of this article, we consider that the convex function \(f:E\longrightarrow(-\infty,+\infty]\) is Legendre.