## 1 Introduction
Then:  
(i) \(\mathbb{J}\) is called nonexpensive if  
\[H(\mathbb{J}a^{\prime},\mathbb{J}b^{\prime})\leq||a^{\prime}-b^{\prime}||, \forall\ a^{\prime},b^{\prime}\in B.\] (ii) \(\mathbb{J}\) is called quasi-nonexpensive if \(F_{ix}(\mathbb{J})\
eq\emptyset\) and  
\[H(\mathbb{J}a^{\prime},\mathbb{J}p_{{}_{0}})\leq||a^{\prime}-p_{{}_{0}}||, \forall\ \ p_{{}_{0}}\in F_{ix}\ and\ \forall\ a^{\prime}\in B(\mathbb{J}).\] (iii) \(\mathbb{J}\) is called Suzuki generalized nonexpansive or endowed with condition \((C)\) if  
\[\frac{1}{2}d(a^{\prime},\mathbb{J}a^{\prime})\leq||a^{\prime}-b^{\prime}||\ \ \ \Rightarrow\ H(\mathbb{J}a^{\prime},\mathbb{J}b^{\prime})\leq||a^{\prime}-b^{ \prime}||,\ \ \forall\ a^{\prime},b^{\prime}\in B.\] (iv) \(\mathbb{J}\) is called \(\alpha\)-nonexpensive if there exists \(\alpha\in[0,1)\) such that \(\forall\ a^{\prime},b^{\prime}\in B\),  
\[H^{2}(\mathbb{J}a^{\prime},\mathbb{J}b^{\prime})\leq\alpha d^{2}(a^{\prime}, \mathbb{J}b^{\prime})+\alpha d^{2}(b^{\prime},\mathbb{J}a^{\prime})+(1-2\alpha )||a^{\prime}-b^{\prime}||^{2}.\] (v) \(\mathbb{J}\) is called generalized \(\alpha\)-nonexpansive if \(\alpha\in[0,1)\) exists such that \(\forall\ a^{\prime},b^{\prime}\in B\),  
\[\frac{1}{2}d(a,\mathbb{J}a^{\prime})\leq||a^{\prime}-b^{\prime}||\ \ \Longrightarrow\]\[H(\mathbb{J}a^{\prime},\mathbb{J}b^{\prime})\leq\alpha d(a^{\prime},\mathbb{J}b^{ \prime})+\alpha d(b^{\prime},\mathbb{J}a^{\prime})+(1-2\alpha)||a^{\prime}-b^{ \prime}||.\]
5. \(\mathbb{J}\) is called \(\beta\)-Reich-Suzuki type nonexpansive if \(\beta\in[0,1)\) exists such that \(\forall\ a^{\prime},b^{\prime}\in B\), \[\frac{1}{2}d(a,\mathbb{J}a^{\prime})\leq||a^{\prime}-b^{\prime}||\implies\] \[H(\mathbb{J}a^{\prime},\mathbb{J}a^{\prime})\leq\beta d(a^{\prime},\mathbb{J}a^{ \prime})+\beta d(b^{\prime},\mathbb{J}b^{\prime})+(1-2\beta)||a^{\prime}-b^{ \prime}||.\] The single-valued example of a generalized \((\alpha,\beta)\)-nonexpansive selfmaps is the following. **Example 1**. : _If \(\mathcal{B}=[0,\infty)\), then we may set_  
\[\mathbb{J}r^{\prime}=\left\{ \begin{array}{ll}0&\text{if }\ r^{\prime}\leq \frac{1}{2}\\ \frac{r^{\prime}}{2}&\text{if }\ r^{\prime}>\frac{1}{2}.\end{array} \right.\]  
_If one chooses \(\alpha=\beta=\frac{1}{2}\), then is easy to observe that \(\mathbb{J}\) exceeds the class of generalized \(\alpha\)-nonexpansive and \(\beta\)-Reich-Suzuki type selfmaps; however, with the help of triangular inequality, one can show that it is generalized \((\alpha,\beta)\)-nonexpansive._  
**Proposition 1** ([31]). : _For a multi-valued mapping \(\mathbb{J}:B\to P(B)\) and \(P_{\mathbb{J}}(a^{\prime})=\{b^{\prime}\in\mathbb{J}a^{\prime}:d(a^{\prime}, \mathbb{J}a^{\prime})=||a^{\prime}-b^{\prime}||\}\). Then, the following conditions are equivalent:_  
1. \(p_{\alpha}\in F_{ix}(\mathbb{J})\)_,_
2. \(P_{\mathbb{J}}(p_{\alpha})=\{p_{\alpha}\}\)_,_
3. \(p_{\alpha}\in F_{ix}(P_{\mathbb{J}})\)_._ _Further_ \(F_{ix}(\mathbb{J})=F_{ix}(P_{\mathbb{J}})\)_._  
**Definition 1**. : _Let \(\{k_{n}\}\) be a bounded sequence in \(X\) and \(B\) be a subset of \(X\), then:_  
1. _Asymptotic radius of_ \(\{k_{n}\}\) _at a point_ \(x\in X\) _is defined as_ \[r(x,\{k_{n}\})=\limsup_{n\to\infty}||k_{n}-x||.\]
2. _Asymptotic radius of_ \(\{k_{n}\}\) _with respect to_ \(B\) _is defined as_ \[r(x,B)=\inf\{r(x,k_{n}):x\in B\}.\]
3. _The asymptotic center of_ \(\{k_{n}\}\) _with respect to_ \(B\) _is defined as_ \[A(B,\{k_{n}\})=\{x\in B;r(x,\{k_{n}\})=r(B,\{k_{n}\})\}.\]  
**Lemma 1** ([35]). : _Suppose that \(X\) is a UCBS. Consider a real sequence named \(\{t_{n}\}\) in such a manner that \(0<\inf_{n\in\mathbb{N}}t_{n}\leq\sup_{n\in\mathbb{N}}t_{n}<1\). Now assume \(\{a^{\prime}_{n}\}\) and \(\{b^{\prime}_{n}\}\) are two sequences in \(X\) with the property that there exists a positive real number \(h\) such that \(\limsup_{n\to\infty}||a^{\prime}_{n}||\leq h\), \(\limsup_{n\to\infty}||b^{\prime}_{n}||\leq h\) and \(\lim_{n\to\infty}||(1-t_{n})a^{\prime}_{n}+t_{n}b^{\prime}_{n}||=h\).