## 5 Convergence Results  
**Lemma 9**. : _Suppose that \(B\) be a nonempty closed and convex subset of UCBS \(X\). Assume that \(\mathbb{J}:B\to P_{r}(B)\) is a multi-valued mapping such that \(F_{ix}(\mathbb{J})\
eq\emptyset\). Additionally, assume that \(P_{\mathbb{J}}\) is a generalized \((\alpha,\beta)\)-nonexpansive mapping. Suppose that \(\{k_{n}\}\) is a sequence iteratively generated by \((\ref{eq:1})\) then for \(p_{{}_{0}}\in F_{ix}(\mathbb{J})\), \(\lim_{n\to\infty}||k_{n}-p_{{}_{0}}||\) exists and \(\lim_{n\to\infty}d(k_{n},P_{\mathbb{J}}(k_{n}))=0\)._  
Proof. : If \(p_{{}_{0}}\in F_{ix}(\mathbb{J})\) and \(n\in\mathbb{N}\), then, by Lemmas 2 and (5), we have  
\[||m_{n}-p_{{}_{0}}|| \leq (1-  
By the same token, we obtain  
\[||l_{n}-p_{0}|| = ||w_{n}-p_{0}|| \tag{9}\] \[\leq H\big{(}P_{\mathbb{J}}(m_{n}),P_{\mathbb{J}}(p_{0})\big{)}\] \[\leq ||m_{n}-p_{0}||,\]  
and also  
\[||k_{n+1}-p_{0}|| = ||v_{n}-p_{0}||\] \[\leq H\big{(}P_{\mathbb{J}}(n),P_{\mathbb{J}}(p_{0})\big{)}\] \[\leq ||l_{n}-p_{0}||\] \[\leq ||m_{n}-p_{0}||\] \[\leq ||k_{n}-p_{0}||.\]  
Thus, \(||k_{n}-p_{0}||\) is bounded and nonincreasing, and so \(\lim_{n\to\infty}||k_{n}-p_{0}||\) exists for \(p_{0}\in F_{ix}(\mathbb{J})\). Next, we need to prove that  
\[\lim_{n\to\infty}||k_{n}-u_{n}||=0.\]  
Suppose that  
\[\lim_{n\to\infty}||k_{n}-p_{0}||=c\;\;\mbox{where}\;p_{0}\in F_{ix}(\mathbb{J }). \tag{10}\]  
We can easily obtain the following inequalities by using (5), (9), (8) and Lemma 2:  
\[\leq H\big{(}P_{\mathbb{J}}(m_{n}),P_{\mathbb{J}}(p_{0})\big{)}\]0  
and  
\[\leq H\big{(}P_{\mathbb{J}}(m_{n}),P_{\mathbb{J}}(p_{0})\big{)}\]1  
By taking the superior limit of the extreme expressions of the above two rows of inequalities and keeping in mind (10), we infer that  
\[\leq H\big{(}P_{\mathbb{J}}(m_{n}),P_{\mathbb{J}}(p_{0})\big{)}\]2  
\[\leq H\big{(}P_{\mathbb{J}}(m_{n}),P_{\mathbb{J}}(p_{0})\big{)}\]3  
Then, it follows that  
\[\leq H\big{(}P_{\mathbb{J}}(m_{n}),P_{\mathbb{J}}(p_{0})\big{)}\]4 \[\leq H\big{(}P_{\mathbb{J}}(m_{n}),P_{\mathbb{J}}(p_{0})\big{)}\]5  
By (12) and (13), we obtain  
\[\leq H\big{(}P_{\mathbb{J}}(m_{n}),P_{\mathbb{J}}(p_{0})\big{)}\]6  
From (5) and (9) we obtain,\[\leq H\big{(}P_{\mathbb{J}}(m_{n}),P_{\mathbb{J}}(p_{0})\big{)}\]7 \[\leq H\big{(}P_{\mathbb{J}}(m_{n}),P_{\mathbb{J}}(p_{0})\big{)}\]8 \[\leq H\big{(}P_{\mathbb{J}}(m_{n}),P_{\mathbb{J}}(p_{0})\big{)}\]9 \[\leq ||m_{n}-p_{0}||,\]0 \[\leq ||m_{n}-p_{0}||,\]1 \[\leq ||m_{n}-p_{0}||,\]2  
As a result, we obtain  
\[\leq ||m_{n}-p_{0}||,\]3  
Therefore, by (11), (10), (14) and Lemma 1, we have  
\[\leq ||m_{n}-p_{0}||,\]4  
Hence,  
\[\leq ||m_{n}-p_{0}||,\]5  
This completes the proof. Next, we are to prove convergence results. Theorem 1: _Suppose that \(B\) be a nonempty compact and convex subset of UCBS \(X\). Assume that \(\mathbb{J}:B\to P_{r}(B)\) is a multi-valued mapping such that \(F_{ix}(\mathbb{J})\
eq\emptyset\). Additionally, assume that \(P_{\mathbb{J}}\) be a generalized \((\alpha,\beta)\)-nonexpansive mapping. Then, a sequence \(\{k_{n}\}\) defined as in (5) converges strongly to a fixed point of \(\mathbb{J}\)._  
Proof: We have observed in Lemma 9 that  
\[\leq ||m_{n}-p_{0}||,\]6  
Due to compactness of \(B\), there exists a subsequence \(\{k_{n_{j}}\}\) of \(\{k_{n}\}\) that converges to \(p_{0}\in B\). Since \(P_{\mathbb{J}}\) is a generalized \((\alpha,\beta)\)-nonexpansive and the sequence \(\{k_{n}\}\) satisfies (5), it follows that  
\[\leq ||m_{n}-p_{0}||,\]7  
Since \(F_{ix}(\mathbb{J})=F_{ix}(P_{\mathbb{J}})\), we obtain \(p_{0}\in\mathbb{J}(p_{0})\) as \(i\to\infty\). As a result, \(\{k_{n}\}\) converges strongly to \(p_{0}\in F_{ix}(\mathbb{J})\). Proof of the theorem below is elementary, and hence is omitted. Theorem 2: _Suppose that \(B\) is a nonempty closed convex subset of a UCBS \(X\) and let the multi-valued mapping \(\mathbb{J}:B\to P_{r}(B)\) be such that the mapping \(P_{\mathbb{J}}\) is generalized \((\alpha,\beta)\)-nonexpansive.