## 2 Preliminaries
It is known that a Banach space \(E\) is \(p\)-uniformly convex and uniformly smooth if and only if its dual \(E^{*}\) is \(q\)-uniformly smooth and uniformly convex. It is also well known that the duality \(J_{E}^{p}\) is one-to-one, single valued and satisfies \(J_{E}^{p}=(J_{E^{*}}^{q})^{-1}\), where \(J_{E}^{q}\) is the duality mapping of \(E^{*}\). For any \(p\)-uniformly convex Banach space \(E\), the metric and the Bregman distance have the following relation :  
\[\tau\|x-y\|^{p}\leq d_{p}(x,y)\leq\langle x-y,J_{E}^{p}x-J_{E}^{p}y\rangle, \tag{13}\]  
where \(\tau>0\) is a fixed number. Let \(C\) be a nonempty closed convex subset of \(E\). The metric projection  
\[P_{C}x:=\arg\min_{y\in C}\|x-y\|\]  
for all \(x\in E\) is the unique minimizer of the norm distance, which can be characterized by a variational inequality  
\[\langle J_{E}^{p}(x-P_{C}x),z-P_{C}x\rangle\leq 0 \tag{14}\]  
for all \(z\in C\). Similarly, the Bregman projection is defined as follows:  
\[\Pi_{C}x=\arg\min_{y\in C}\,d_{p}(x,y)\]  
for all \(x\in E\), which is the unique minimizer of the Bregman distance. In addition, the Bregman projection can also be characterized by a variational inequality  
\[d_{p}(x,y)+d_{p}(y,x)=\langle x-y,J_{E}^{p}x-J_{E}^{p}y\rangle. \tag{11}\]0  
for all \(z\in C\), from which one has  
\[d_{p}(x,y)+d_{p}(y,x)=\langle x-y,J_{E}^{p}x-J_{E}^{p}y\rangle. \tag{11}\]1  
for all \(z\in C\). Following [15; 16], we will make use of the function \(V_{p}\): \(E^{*}\times E\to[0,+\infty)\) associated with \(f_{p}\), which is defined by  
\[d_{p}(x,y)+d_{p}(y,x)=\langle x-y,J_{E}^{p}x-J_{E}^{p}y\rangle. \tag{11}\]2  
for all \(x\in E\) and \(\bar{x}\in E^{*}\). Then \(V_{p}\) is nonnegative and  
\[d_{p}(x,y)+d_{p}(y,x)=\langle x-y,J_{E}^{p}x-J_{E}^{p}y\rangle. \tag{11}\]3  
for all \(x\in E\) and \(\bar{x}\in E^{*}\). Moreover, by the subdifferential inequality, we have  
\[d_{p}(x,y)+d_{p}(y,x)=\langle x-y,J_{E}^{p}x-J_{E}^{p}y\rangle. \tag{11}\]4  
for all \(x\in E\) and \(\bar{x},\bar{y}\in E^{*}\) (see also [17; 18]). In addition, \(V_{p}\) is convex in the first variable. Thus, for all \(z\in E\),  
\[d_{p}(x,y)+d_{p}(y,x)=\langle x-y,J_{E}^{p}x-J_{E}^{p}y\rangle. \tag{11}\]5  
where \(\{x_{i}\}\subset E\) and \(\{t_{i}\}\subset(0,1)\) with \(\sum\limits_{i=1}^{N}=1\). For more details, see [19; 20]. Let \(C\) be a nonempty, closed and convex subset of \(E\). A mapping \(T\): \(C\to C\) is said to be _nonexpansive_ if  
\[d_{p}(x,y)+d_{p}(y,x)=\langle x-y,J_{E}^{p}x-J_{E}^{p}y\rangle. \tag{11}\]6  
for all \(x,y\in C\). We denote by \(F(T)\) the set of fixed points of \(T\), that is, \(F(T)=\{x\in C:Tx=x\}\). Let \(C\) be a convex subset of int (dom \(f_{p}\)), where \(f_{p}(x)=(\frac{1}{p})\|x\|^{p}\), \(2\leq p<\infty\), and \(T\) be a self-mapping of \(C\). A point \(\widehat{x}\in C\) is called an asymptotic fixed point of \(T\) if \(C\) contains a sequence \(\{x_{n}\}\) which converges weakly to \(\widehat{x}\) and \(\lim\limits_{n\to\infty}\|x_{n}-Tx_{n}\|=0\). The set of asymptotic fixed point of \(T\) is denoted by \(\widehat{F}(T)\) (see [21]). In general, the Bregman distance is not a metric due to the absence of symmetry, but it has some distance-like properties. Definition 3: A nonlinear mapping \(T\) with a nonempty asymptotic fixed point set is said to be:  
1. \(T\) is called right Bregman quasi-nonexpansive (shortly, \(R\)-BQNE) (see [22]) if \(F(T)\
eq\emptyset\) and \[d_{p}(x,y)+d_{p}(y,x)=\langle x-y,J_{E}^{p}x-J_{E}^{p}y\rangle. \tag{11}\]7 for all \(x\in C\) and \(\bar{x}\in F(T)\). 2.