## 1 Introduction
For any \(x\in int(domf)\) and \(y\in E\), we denote by \(f^{\circ}(x,y)\) the right-hand derivative of \(f\) at \(x\) in the direction \(y\), that is:  
\[f^{\circ}(x,y):=\lim_{t\longrightarrow 0^{+}}\frac{f(x+ty)-f(x)}{t} \tag{5}\]  
The function \(f\) is called _Gateaux differentiable_ at \(x\), if limit Equation (5) exists for any \(y\). In this case, the gradient of \(f\) at \(x\) is the function \(\
abla f:E\longrightarrow E^{*}\) defined by \(\langle\
abla f(x),y\rangle=f^{\circ}(x,y)\) for all \(y\in E\). The function \(f\) is said to be G\(\ddot{a}\)teaux differentiable if it is G\(\ddot{a}\)teaux differentiable at each \(x\in int(domf)\). If the limit Equation (5) is attained uniformly in \(\|y\|=1\), then the function \(f\) is called _Frechet differentiable_ at \(x\), if limit Equation (5) is attained uniformly in \(\|y\|=1\), and \(f\) is said to be _uniformly Frechet differentiable_ on a subset \(C\) of \(E\), if limit Equation (5) is attained uniformly for \(x\in C\) and \(\|y\|=1\). It is known that if \(f\) is G\(\ddot{a}\)teaux differentiable (resp. Frechet differentiable) on \(int(domf)\), then \(f\) is continuous, and its G\(\ddot{a}\)teaux derivative \(\
abla f\) is norm-to-weak\({}^{*}\)continuous (resp. continuous) on \(int(domf)\) (see [22; 24]). Definition 2: ([1]) Let \(f:E\longrightarrow(-\infty,+\infty]\) be a G\(\ddot{a}\)teaux differentiable function. The function \(D_{f}:domf\times int(domf)\longrightarrow[0,+\infty)\) defined by:  
\[D_{f}(y,x):=f(y)-f(x)-\langle\
abla f(x),y-x\rangle\]  
is called the _Bregman distance with respect to \(f\)_. We remark that the Bregman distance \(D_{f}\) does not satisfy the well-known properties of a metric because \(D_{f}\) is not symmetric and does not satisfy the triangle inequality. The Bregman distance has the following important properties (see [25]):  
(1) (_The three point identity_): for each \(x\in domf\) and \(y,z\in int(domf)\),  
\[D_{f}(x,y)+D_{f}(y,z)-D_{f}(z,x)=\langle\
abla f(z)-\
abla f(y),x-y\rangle\]  
(2) (_The four point identity_): for each \(y,\omega\in domf\) and \(x,z\in int(domf)\). \[D_{f}(y,x)-D_{f}(y,z)-D_{f}(\omega,x)+D_{f}(\omega,z)=\langle\
abla f(z)- \
abla f(x),y-\omega\rangle\]  
Definition 3: ([1]) A Bregman projection of \(x\in int(domf)\) onto the nonempty, closed and convex set \(C\subset domf\) is the unique vector \(P_{C}^{f}(x)\in C\) satisfying:  
\[D_{f}(P_{C}^{f}(x),x)=\inf\{D_{f}(y,x):y\in C\}\]  
If \(E\) is a smooth Banach space, and setting \(f(x)=\|x\|^{2}\) for any \(x\in E\), we get \(\
abla f(x)=2 Jx\) for all \(x\in E\), where \(J\) is the normalized duality mapping from \(E\) onto \(2^{E^{*}}\); then, the Bregman distance reduces to \(\phi(x,y)=\|x\|^{2}-2\langle x,Jy\rangle+\|y\|^{2}\) for all \(x,y\in E\), where \(\phi\) is called the _Lyapunov function_ introduced by [PERSON] [26; 27]; and the Bregman projection reduces to the generalized projection \(\Pi_{C}\) defined by \(\phi\big{(}\Pi_{C}(x),x\big{)}=\min_{y\in C}\phi(y,x)\). If \(E:=H\) is a Hilbert space, then the Bregman distance reduces to \(\phi(x,y)=\|x-y\|^{2}\) for all \(x,y\in H\), and the Bregman projection reduces to the metric projection \(P_{C}\) from \(E\) onto \(C\). Definition 4: Let \(C\) be a nonempty and convex subset of \(int(domf)\). A mapping \(T:C\longrightarrow int(domf)\) with \(F(T)\
eq\O\) is called:  
(1) _Relatively quasi-nonexpansive_ if  
\[f^{*}(x^{*})=\sup_{x\in E}\{\langle x^{*},x\rangle-f(x)\}\]0  
(2) _Relatively nonexpansive_ if \(\widehat{F}(T)=F(T)\),  
\[f^{*}(x^{*})=\sup_{x\in E}\{\langle x^{*},x\rangle-f(x)\}\]1  
(3) _Bregman relatively quasi-nonexpansive_ if,  
\[f^{*}(x^{*})=\sup_{x\in E}\{\langle x^{*},x\rangle-f(x)\}\]2_Bregman relatively nonexpansive_ if, \(\widehat{F}(T)=F(T)\),  
\[f^{*}(x^{*})=\sup_{x\in E}\{\langle x^{*},x\rangle-f(x)\}\]3  
**Remark 2**. : The class of relatively nonexpansive mappings is contained in a class of Bregman relatively nonexpansive mappings with \(f(x)=\|x\|^{2}\).