## Preliminaries  
Throughout this present paper we get \(x_{n}\rightharpoonup x\) (resp., \(x_{n}\to x\)) to indicate that the sequence \(\{x_{n}\}\) converges weakly (resp., strongly) to \(x\). We invoke \(C\) to be a convex nonempty closed set in \(E\). Invoke that \(j(x-y)\) stands for the single-valued normalized duality for any \(x,y\in E\). A mapping \(A:C\to E\) is said to be  
1. [label=()]
2. accretive if \(\langle Ax-Ay,j(x-y)\rangle\geq 0\), \(\forall x,y\in C\). 3. \(\alpha\)-strongly accretive if \(\langle Ax-Ay,j(x-y)\rangle\geq\alpha\|x-y\|^{2}\) for some \(\alpha\in(0,1)\), \(\forall x,y\in C\). 4. \(\beta\)-inverse-strongly accretive if \(\langle Ax-Ay,j(x-y)\rangle\geq\beta\|Ax-Ay\|^{2}\) for some \(\beta>0\), \(\forall x,y\in C\). 5. \(\lambda\)-strictly pseudocontractive if \(\|x-y\|^{2}-\lambda\|(I-A)x-(I-A)y\|^{2}\geq\langle Ax-Ay,j(x-y)\rangle\) for some \(\lambda\in(0,1)\), \(\forall x,y\in C\). **Definition 1**. : _Fix a convex and closed set \(C\) in a space \(E\) and \(\{S_{n}\}_{n=0}^{\infty}:C\to C\) stands for a self-operator sequence of continuous pseudocontractive. We say \(\{S_{n}\}_{n=0}^{\infty}:C\to C\) is a family of \(\ell\)-uniformly Lipschitzian pseudocontractive operators when \(S_{n}\) is \(\ell\)-Lipschitz continuous, where \(\ell>0\) is a real constant._  
For a linear operator \(A\), we say it is strongly positive when  
\[\frac{\langle Ax,j(x)\rangle}{\tilde{\gamma}}\geq\|x\|^{2},\ \ \|BA-aI\|=\sup_{\|x\|\leq 1}| \langle(bA-aI)x,j(x)\rangle|,\ b\in[-1,1],\ a\in[0,1],\]  
where \(\tilde{\gamma}>0\) and \(j(\cdot)\) is the single-valued duality mapping. We say a Banach space \(E\) is strictly convex if \(y\
eq x\ \Rightarrow\ \|y+x\|<2\) for any \(x,y\in US_{E}\). We say it is uniformly convex if there exists \(\delta>0\) such that for any \(x,y\in US_{E}\), \(\|y-x\|\geq\epsilon\ \Rightarrow\ \|y+x\|\leq 2-2\delta\), for each \(\epsilon\in(0,2]\). one knows that a uniform space is the reflexive. Also, in the setting of reflexive spaces, \(E\) is strictly convex if and only if \(E^{*}\) is smooth as well as \(E\) is smooth if and only if \(E^{*}\) is strictly convex. A Banach space \(E\) is called to have a Gateaux differentiable norm if the limit \(\lim_{t\to\infty}t(\|\frac{1}{t}y+x\|-\|x\|)\) exists for each \(x,y\in US_{E}\) and in this case we call \(E\) is smooth; \(E\) is called to have a uniformly Gateaux differentiable norm if the above limit is achieved uniformly for any \(x\in U\). Moreover, it is called to have a uniformly Frechet differentiable norm if the above limit is attained uniformly for \(x,y\in US_{E}\) and in this case we call \(E\) uniformly smooth. The norm of \(E\) is said to be the Frechet differentiable if for each \(x\in U\), the above limit is uniformly achieved for \(y\in U\). The modulus of smoothness of \(E\) is defined by  
\[\varrho(\tau)=\sup\{\frac{\|x+y\|-2+\|x-y\|}{2}:x,y\in E,\ \|x\|=1,\ \tau=\|y\|\},\]  
where the function \(\varrho\) maps \([0,\infty)\ [0,\infty)\). Invoke that \(E\) is uniformly smooth if \(\frac{\varrho(\tau)}{\tau}\to 0\) as \(\tau\to 0\). Let \(q\) (\(q>1\) and \(q\leq 2\)) be a fixed real number. \(E\) is said to be \(q\)-uniformly smooth if \(c\tau^{q}\geq\varrho(\tau)\), \(\forall\tau>0\), \(c>0\). One has \(\|y+x\|^{q}\leq 2(\|x\|^{q}+\|xy\|^{q})-\|x-y\|^{q}\), where \(\kappa>0\) in the setting of \(q\)-uniformly smooth spaces. The best number \(\kappa\) in this inequality is named to be the \(q\)-uniformly smooth number of \(E\); see [26]. For \(q>1\), one can define the generalized duality mapping \(J_{q}\) by  
\[J_{q}(x)=\{\varrho\in E^{*}:\langle x,\varrho\rangle=\|x\|^{q},\ \|\varrho\|=\|x\|^{q-1}\},\quad \forall x\in E.\]  
It is known that \(\forall p\in E\), \(J(p)=\frac{I_{q}(p)}{\|p\|^{q-2}}\). If the setting is Hilbert, then \(J\) become \(I\), the identity. Recall that the following statements hold:1. \(J\) is norm-to-weak\({}^{*}\) continuous and single-valued in setting of smooth spaces;
2.