## 1 Introduction  
Throughout this article, \(E\) will be supposed to be a real Banach space and \(E^{*}\) stands for its topological dual. Recall the normalized duality mapping \(J:E\to 2^{E^{*}}\) as follows  
\[Jx:=\{\varphi\in E^{*}:\langle x,\varphi\rangle=\|x\|^{2},\ \|\varphi\|=\|x\|\}\ \forall x\in E.\]  
The duality \(J\) will be reduced to the identical operator in setting of Hilbert space. Set \(US_{E}=\{x\in E:\|x\|=1\}\). \(E\) is said to be a smooth space if \(\lim_{t\to 0}(|x/t+y\|-\frac{\|x\|}{t})\) exists for all \(x,y\in US_{E}\). If \(E\) is smooth, then normalized duality is a single-valued operator. For the rest of this paper, we shall invoke \(j\) the normalized duality with the single-valued. \(E\) is said to be a uniformly smooth Banach space if the above limit is uniformly achieved for all \(x,y\in US_{E}\). Let \(C\) be a closed nonempty and convex set in \(E\). Invoke that the operator \(f\) is said to be \(\delta\)-Lipschitzian continuous in the set \(C\) if  
\[\|f(x)-f(y)\|\leq\delta\|x-y\|,\quad\forall x,y\in C,\]  
where \(\delta\) is a real number in \([0,+\infty)\). The existence of solutions of many abstract problems is equivalent to the existence of a fixed point of a Lipschitzian continuous mapping. Therefore, existence results of Lipschitzian continuous mappings are significant in mathematical sciences. If \(f\) is \(\delta\)-Lipschitzian continuous with \(\delta<1\), then \(f\) is called a \(\delta\)-contraction. Each contractive operator equation \(f:C\to C\) has a unique solution in \(C\) according to the famous Banach contractive mapping principal. Invoke that the mapping \(f\) is said to be a nonexpansive mapping when \(\delta=1\). For the rest of the paper, we use Fix\((f)\) to denote the set of fixed points of \(f\). Invoke that the mapping \(T\) is said to be asymptotically nonexpansive [1] when  
\[\|T^{n}x-T^{n}y\|\leq(\theta_{n}+1)\|x-y\|,\quad\forall x,y\in C,n\geq 0.\]  
Here \(\{\theta_{n}\}\) is in the interval \([0,+\infty)\) with \(\lim_{n\to\infty}\theta_{n}=0\). If the operator \(T\) is continuous and satisfies  
\[\limsup_{n\to\infty}\sup_{x,y\in C}\left(\|T^{n}x-T^{n}y\|-\|x-y\|\right)\leq 0, \tag{1}\]  
then the operator \(T\) is called an asymptotically nonexpansive operator, which is in the intermediate sense; see [2]. It is easy to conclude that the family of asymptotically nonexpansive mappings, which is in the intermediate sense, as special cases, includes the one of contractive operators, the one of nonexpansive operators and the one of asymptotically nonexpansive operators. For the rest this paper, set  
\[c_{n}:=\max\{\sup_{x,y\in C}\left(\|T^{n}x-T^{n}y\|-\|x-y\|\right),0\}. \tag{2}\]  
Then \(c_{n}\geq 0\ \forall n\geq 0,\ c_{n}\to 0\ (n\to\infty)\), and (1) become the inequality  
\[\|T^{n}x-T^{n}y\|\leq\|x-y\|+c_{n},\quad\forall x,y\in C,\ \forall n\geq 0.\]  
Solutions of nonlinear equations containing nonexpansive operators, asymptotically nonexpansive operators, and asymptotically nonexpansive operators in the intermediate sense recently attracted much attention from many authors; see [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14] and the references therein. Invoke that an operator \(T\) with range \(R(T)\) and domain \(D(T)\) is called a pseudocontractive operator if  
\[\|x-y\|\leq\|x-y+r((I-T)x-(I-T)y)\|,\quad\forall x,y\in D(T),\ r>0.\]  
From the result of [PERSON] [15], we know that the definition of pseudocontractive operators is equivalent to: for each \(x,y\in D(T)\), \(\langle Tx-Ty,j(x-y)\rangle\leq\|x-y\|^{2}\). Here \(j(x-y)\in J(x-y)\) is the single-valued normalized duality. One can clearly see the situation the family of pseudocontractive operators is an important and significant generation of nonexpansive operators. Moreover, methodology shed on pseudocontractive operator is from their counterpart: accretivity (monotonicity in Hilbert spaces); see [16; 17; 18; 19; 20; 21; 22] and the references therein. Let the Banach space \(E\) be smooth. Let \(B_{1},B_{2}:C\to E\) be two mappings.