## Preliminaries
\(J\) is norm-to-norm uniformly continuous on bounded sets and single-valued in setting of uniformly smooth spaces;
3. \(2\)-uniformly smooth spaces are: Hilbert spaces, the Sobolev spaces \(W^{p}_{m}\), and \(L^{p}\) spaces, where \(p\geq 2\);
4. \(L^{p}\) with \(p>1\), is uniformly convex and uniformly smooth. **Proposition 1** ([27]). : _Fix a convex nonempty closed set \(C\) in a Banach space \(E\). Let \(S_{0},S_{1},...:C\to C\) be a self-mapping sequence. Suppose that \(\sum_{n=1}^{\infty}\sup\{\|(S_{n}-S_{n-1})x\|:x\in C\}\) is a fixed real number. Then \(\{S_{n}y\}\), where \(y\in C\), converges in norm. If the mapping is defined by \(Sy=\lim_{n\to\infty}S_{n}y\) for all \(y\in C\), then \(\sup\{\|Sx-S_{n}x\|:x\in C\}\to 0\) as \(n\to\infty\)._  
**Proposition 2** ([28]). : _Fix a convex nonempty closed set \(C\) in a Banach space \(E\). Every continuous and strong pseudocontraction self-mapping \(T:C\to C\) has a fixed point. Furthermore, it has only a unique fixed point._  
Fix a set \(D\) in the set \(C\) and let projective mapping \(\Pi\) map \(C\) into \(D\). Then \(\Pi\) is named as a sunny if \(\Pi[t(x-\Pi(x))+\Pi(x)]=\Pi(x)\), whenever \(t(x-\Pi(x))+\Pi(x)\in C\) for all \(x\in C\) and for all \(t\geq 0\). A projective mapping \(\Pi\) of \(C\) into itself is named a retraction once \(\Pi\Pi x=\Pi x\). Set \(D\) is called is called a sunny nonexpansive retract of set \(C\) if there is a \(C\)-\(D^{\prime}\)s sunny nonexpansive retraction. In a setting of smooth Banach spaces, \(J\) is called a weakly sequentially continuous mapping if \(\{j(x_{n})\}\) converges weakly\({}^{*}\) to \(j(x)\) for each \(\{x_{n}\}\subset E\), here, \(x_{n}\rightharpoonup x\). From [29], any space with a weakly continuous duality mapping satisfies the Opial inequality. In addition, if a space with the Opial inequality and a uniformly Gateaux differentiable norm, then it enjoys duality mappings are weakly continuous. **Proposition 3**. : _Fix a convex nonempty and closed set \(C\) in a smooth Banach space \(E\), set \(D\) as a nonempty subset of \(C\). If \(\Pi:C\to D\) be a retraction, one has (i) \(\Pi\) is both nonexpansive and sunny; (ii) \(\langle x-y,j(\Pi(x)-\Pi(y))\rangle\geq\|\Pi(x)-\Pi(y)\|^{2}\), \(\forall x,y\in C\); (iii) for any \(x\in C,y\in D\), \(\langle x-\Pi(x),j(y-\Pi(x))\rangle\leq 0\)._  
One knows that any sunny nonexpansive retraction coincides with the metric projection in setting of a Hilbert space. **Proposition 4**. : _Fix a convex nonempty and closed set \(C\) in a reflexive Banach space \(E\). If \(E\) has a uniformly Gateaux differentiable norm and the set \(D\) is a nonexpansive retract in \(C\), it is a sunny nonexpansive retract of \(C\)._  
**Proposition 5** ([30]). : _In the setting of \(2\)-uniformly smooth Banach space,_  
\[\|x+y\|^{2}-2\langle y,j(x)\rangle\leq\|x\|^{2}+2\|xy\|^{2},\]  
_where \(\kappa\) is the \(2\)-uniformly smooth constant of \(E\)._  
In particular, if the duality pairing \(\langle\cdot,\cdot\rangle\) become the inner product, \(j=I\) the identity mapping of \(E\), and \(\kappa=\sqrt{2}/2\) in setting of Hilbert space. For obtaining our theorem, we need to invoke some lemmas in the sequel. The following lemma is obvious. **Lemma 1**. : _Let \(j\) be the normalized single-valued duality mapping on a real Banach space \(E\). For any given \(x,y\in E\), it holds \(|x\|^{2}+2\langle y,j(x+y)\rangle\geq\|x+y\|^{2}\)._  
**Lemma 2** ([31]). : _Fix a convex nonempty and closed set \(C\) in a smooth Banach space \(E\). Let \(\Pi_{C}:E\to C\) be a nonexpansive sunny retraction and \(B:C\to E\) be an accretive mapping.