## 2 Preliminaries  
Let \(E_{1}\), \(E_{2}\) be real Banach spaces and \(A\): \(E_{1}\to E_{2}\) be a bounded linear operator. The dual (adjoint) operator of \(A\), denoted by \(A^{*}\), is a bounded linear operator defined by \(A^{*}\): \(E_{2}^{*}\to E_{1}^{*}\)  
\[\langle A^{*}g,x\rangle:=\langle g,Ax\rangle \tag{8}\]  
for all \(x\in E_{1}\) and \(\bar{y}\in E_{2}^{*}\) and the equalities \(\|A^{*}\|=\|A\|\), \(\mathcal{N}(A^{*})=\mathcal{R}(A)^{\perp}\) are valid, where  
\[\mathcal{R}(A)^{\perp}:=\{x^{*}\in E_{2}^{*}:\langle x^{*},u\rangle=0,\ \forall u\in\mathcal{R}(A)\}.\]  
For more details on bounded linear operators and their duals, see [11; 12]. **Definition 1**:
1. _The duality mapping_ \(J_{E}^{p}:E\to E^{*}\) _is defined by_ \[J_{E}^{p}(x)=\{x^{*}\in E^{*}:\langle x,x^{*}\rangle=\|x\|^{p},\ \|x^{*}\|=\|x\|^{p-1}\}.\]
2. _The duality mapping_ \(J_{E}^{p}\) _is said to be weak-to-weak continuous if_ \[x_{n}\rightharpoonup x\ \Longrightarrow\ \langle J_{E}^{p}x_{n},y\rangle\to \langle J_{E}^{p}x,y\rangle\]  
_holds true for any \(y\in E\)._  
We note here that \(l_{p}\ (p>1)\) spaces has such a property, but the \(L_{p}\ (p>2)\) space does not share this property. The domain of a convex function \(f:E\to\mathbb{R}\) is defined by \(\mathrm{dom}f:=\{x\in E:f(x)<+\infty\}\). When \(\mathrm{dom}f\
eq\emptyset\), then we say that \(f\) is proper. In the sequel, we adopt the following notations in this paper: \(x_{n}\to x\) means that \(x_{n}\to x\) strongly and \(x_{n}\rightharpoonup x\) meansthat \(x_{n}\to x\) weakly. **Definition 2** ([13]). : _Let \(f\): \(E\to\mathbb{R}\) be a convex and Gateaux differentiable function. The Bregman distance with respect to \(f\) is defined by_  
\[d_{f}(x,y):=f(y)-f(x)-\langle f^{\prime}(x),y-x\rangle\]  
_for all \(x,y\in E\)._  
The duality mapping \(J_{E}^{p}\) is actually the derivative of the function \(f_{p}(x)=\frac{1}{p}\|x\|^{p}.\) If \(f=f_{p}\), then the Bregman distance with respect to \(f_{p}\) now becomes  
\[ \begin{split} d_{p}(x,y)&=\frac{1}{q}\|x\|^{p}- \langle J_{E}^{p}x,y\rangle+\frac{1}{p}\|y\|^{p}\\ &=\frac{1}{p}(\|y\|^{p}-\|x\|^{p})+\langle J_{E}^{p}x,x-y\rangle \\ &=\frac{1}{q}(\|x\|^{p}-\|y\|^{p})-\langle J_{E}^{p}x-J_{E}^{p}y,y \rangle.\end{split}  \tag{9}\]  
The Bregman distance is not symmetric and so it is not a metric, but it posses the following important properties: for all \(w,x,y\in E\),  
\[d_{p}(x,y)=d_{p}(x,w)+d_{p}(w,y)+\langle w-y,J_{E}^{p}x-J_{E}^{p}y\rangle \tag{10}\]  
and  
\[d_{p}(x,y)+d_{p}(y,x)=\langle x-y,J_{E}^{p}x-J_{E}^{p}y\rangle. \tag{11}\]  
Let \(1<q\leq 2\leq p\) with \(\frac{1}{p}+\frac{1}{q}=1.\) The modulus of smoothness of \(E\) is the function \(\rho_{E}:[0,\infty)\to[0,\infty)\) defined by  
\[\rho_{E}(t):=\sup\Big{\{}\frac{1}{2}(\|x+y\|+\|x-y\|)-1:\|x\|\leq 1,\|y\|\leq t \Big{\}}.\]  
A Banach space \(E\) is said to be uniformly smooth if  
\[\lim_{t\to 0}\frac{\rho_{E}(t)}{t}=0\]  
and, for any \(q>1\), a Banach space \(E\) is said to be \(q\)_-uniformly smooth_ if there exists \(C_{q}>0\) such that \(\rho_{E}(t)\leq C_{q}t^{q}\) for any \(t>0\). Let \(x,y\in E\) and \(q>1\). If a Banach space \(E\) is \(q\)-uniformly smooth, then there exists \(C_{q}>0\) such that  
\[\|x-y\|^{q}\leq\|x\|^{q}-q\langle J_{E}^{p}(x),y\rangle+C_{q}\|y\|^{q}. \tag{12}\]  
Let \(\dim(E)\geq 2.\) The modulus of convexity of \(E\) is the function \(\delta_{E}(\epsilon)\): \((0,2]\to[0,1]\) defined by  
\[\delta_{E}(\epsilon):=\inf\Big{\{}1-\Big{\|}\frac{x+y}{2}\Big{\|}:\|x\|=\|y\|= 1,\,\epsilon=\|x-y\|\Big{\}}.\]  
A Banach space \(E\) is said to be uniformly convex if \(\delta_{E}(\epsilon)>0\) for all \(\epsilon\in(0,2]\) and, for any \(p>1\), a Banach space \(E\) is said to be \(p\)-uniformly convex if there is \(C_{p}>0\) such that \(\delta_{E}(\epsilon)\geq C_{p}e^{p}\) for any \(\epsilon\in(0,2].\) More information concerning uniformly convex spaces can be found, for example, in the book by [PERSON] and [PERSON] [14].