## 3 Generalized (\(\alpha\), \(\beta\))-Nonexpansive Multi-Valued Mapping
Now  
\[\frac{1}{2}d(k_{n},\mathbb{J}k_{n})=0\leq||k_{n}-p_{0}||.\]Using Definition 6, we have  
\[H(\mathbb{J}a^{\prime},\mathbb{J}b^{\prime}) \leq \alpha d(a^{\prime},\mathbb{J}b^{\prime})+\alpha d(b^{\prime}, \mathbb{J}a^{\prime})+\beta d(a^{\prime},\mathbb{J}a^{\prime})+\beta d(b^{ \prime},\mathbb{J}b^{\prime})+(1-\]0 \[H(\mathbb{J}a^{\prime},\mathbb{J}b^{\prime}) \leq \alpha d(a^{\prime},\mathbb{J}b^{\prime})+\alpha d(b^{\prime}, \mathbb{J}a^{\prime})+\beta d(a^{\prime},\mathbb{J}a^{\prime})+\beta d(b^{ \prime},\mathbb{J}b^{\prime})+(1-\]1 \[H(\mathbb{J}a^{\prime},\mathbb{J}b^{\prime}) \leq \alpha d(a^{\prime},\mathbb{J}b^{\prime})+\alpha d(b^{\prime}, \mathbb{J}a^{\prime})+\beta d(a^{\prime},\mathbb{J}a^{\prime})+\beta d(b^{ \prime},\mathbb{J}b^{\prime})+(1-\]2 \[H(\mathbb{J}a^{\prime},\mathbb{J}b^{\prime}) \leq \alpha d(a^{\prime},\mathbb{J}b^{\prime})+\alpha d(b^{\prime}, \mathbb{J}a^{\prime})+\beta d(a^{\prime},\mathbb{J}a^{\prime})+\beta d(b^{ \prime},\mathbb{J}b^{\prime})+(1-\]3 \[H(\mathbb{J}a^{\prime},\mathbb{J}b^{\prime}) \leq \alpha d(a^{\prime},\mathbb{J}b^{\prime})+\alpha d(b^{\prime}, \mathbb{J}a^{\prime})+\beta d(a^{\prime},\mathbb{J}a^{\prime})+\beta d(b^{ \prime},\mathbb{J}b^{\prime})+(1-\]4 \[H(\mathbb{J}a^{\prime},\mathbb{J}b^{\prime}) \leq \alpha d(a^{\prime},\mathbb{J}b^{\prime})+\alpha d(b^{\prime}, \mathbb{J}a^{\prime})+\beta d(a^{\prime},\mathbb{J}a^{\prime})+\beta d(b^{ \prime},\mathbb{J}b^{\prime})+(1-\]5 \[H(\mathbb{J}a^{\prime},\mathbb{J}b^{\prime}) \leq \alpha d(a^{\prime},\mathbb{J}b^{\prime})+\alpha d(b^{\prime}, \mathbb{J}a^{\prime})+\beta d(a^{\prime},\mathbb{J}a^{\prime})+\beta d(b^{ \prime},\mathbb{J}b^{\prime})+(1-\]6  
The previous inequalities imply  
\[H(\mathbb{J}a^{\prime},\mathbb{J}b^{\prime}) \leq \alpha d(a^{\prime},\mathbb{J}b^{\prime})+\alpha d(b^{\prime}, \mathbb{J}a^{\prime})+\beta d(a^{\prime},\mathbb{J}a^{\prime})+\beta d(b^{ \prime},\mathbb{J}b^{\prime})+(1-\]7  
because \(0<(1-\alpha-\beta)<1\) and \(\alpha,\beta\in(0,1)\). So \(p_{{}_{0}}\in\mathbb{J}(p_{{}_{0}})\) and hence \(p_{{}_{0}}\in F_{ix}(\mathbb{J})\). Thus, \(F_{ix}(\mathbb{J})\) is closed. \(\square\)  
**4. Fixed Point Existence Results**  
**Lemma 4. ** _Let \(B\) be a nonempty subset of a Banach space \(X\) and let \(\mathbb{J}:B\to S_{cb}(B)\) be a generalized \((\alpha,\beta)\)-nonexpansive multi-valued mapping. In addition, let \(e\) and \(r^{\prime}\) be elements in \(B\) for which \(e\in\mathbb{J}r^{\prime}\). Then, the following inequalities holds:_  
_(a)_ \(d(e,\mathbb{J}e)\leq||r^{\prime}-e||\)_,_  
_(b)_ _for every_ \(s^{\prime}\in B\) _we have either_ \(\frac{1}{2}d(r^{\prime},\mathbb{J}r^{\prime})\leq||r^{\prime}-s^{\prime}||\) _or_ \(\frac{1}{2}d(e,\mathbb{J}e)\leq||e-s^{\prime}||\)_._  
**Proof. ** (\(a\)) Since \(\frac{1}{2}d(r^{\prime},\mathbb{J}r^{\prime})\leq||r^{\prime}-e||\) for any \(e\in\mathbb{J}r^{\prime}\), we obtain  
\[ \begin{array}{lll}H(\mathbb{J}r^{\prime},\mathbb{J}e)&\leq&\alpha d(r^{\prime },\mathbb{J}e)+\alpha d(e,\mathbb{J}r^{\prime})+\beta d(r^{\prime},\mathbb{J} r^{\prime})+\beta d(e,\mathbb{J}e)+(1-\\ &&2\alpha-2\beta)||r^{\prime}-e||\\ &=&\alpha d(r^{\prime},\mathbb{J}e)+\beta d(r^{\prime},\mathbb{J}r^{\prime})+ \beta d(e,\mathbb{J}e)+(1-2\alpha-2\beta)||r^{\prime}-e||\\ &\leq&\alpha||r^{\prime}-e||+\alpha d(e,\mathbb{J}e)+\beta||r^{\prime}-e||+ \beta d(e,\mathbb{J}r^{\prime})+\beta d(e,\mathbb{J}e)\\ &&+(1-2\alpha-2\beta)||r^{\prime}-e||\\ &\leq&(\alpha+\beta)d(e,\mathbb{J}e)+(1-\alpha-\beta)||r^{\prime}-e||\\ \end{array} \]  
Since \(e\in\mathbb{J}r^{\prime}\) and \((1-\alpha-\beta)>0\), we can write,  
\[ \begin{array}{lll}d(e,\mathbb{J}e)&\leq&(\alpha+\beta)d(e,\mathbb{J}e)+(1- \alpha-\beta)||r^{\prime}-e||\\ d(e,\mathbb{J}e)&\leq&||r^{\prime}-e||,\end{array} \]  
for any \(e\in\mathbb{J}r^{\prime}\). (\(b\)) Suppose to the contrary that for any \(r^{\prime},s^{\prime}\in B\) and \(e\in\mathbb{J}r^{\prime}\), we have  
\[