## 5 Convergence Results
Assume that \(F_{ix}(\mathbb{J})\
eq\emptyset\) and that the sequence \(\{k_{n}\}\), which is iteratively generated as in (5), satisfies \(\liminf_{n\to\infty}d(k_{n},F_{ix}(\mathbb{J}))=0\). Then, this sequence converges strongly to a fixed point of \(\mathbb{J}\)._  
Now, we use condition \((I)\) for establishing another strong convergence theorem. Theorem 3: _Suppose that \(B\) is a nonempty closed convex subset of a UCBS \(X\). Assume that the multi-valued mapping \(\mathbb{J}:B\to P_{r}(B)\) satisfies condition \((I)\) and is such that \(P_{\mathbb{J}}\) is a generalized \((\alpha,\beta)\)-nonexpansive mapping. If, moreover, \(F_{ix}(\mathbb{J})\
eq\emptyset\), then any sequence which is iteratively generated as in (5) converges strongly to a fixed point of \(\mathbb{J}\)._Proof. : Let the sequence \(\{k_{n}\}\) be iteratively generated as in (5) and let \(p_{0}\in F_{ix}(\mathbb{J})\). In light of Lemma 9, since the sequence \(||k_{n}-p_{0}||\) is nonincreasing, the constant \(c\geq 0\) defined by  
\[\leq ||m_{n}-p_{0}||,\]8  
exists. It follows that for \(c=0\), the proof is trivial. Therefore, we suppose that \(c>0\). Now,  
\[\leq ||m_{n}-p_{0}||,\]9 \[||k_{n+1}-p_{0}|| = ||v_{n}-p_{0}||\]0 \[||k_{n+1}-p_{0}|| = ||v_{n}-p_{0}||\]1  
As a result, \(\lim_{n\to\infty}d(k_{n},F_{ix}(\mathbb{J}))\) exists. Additionally, according to Lemma 1, \(F_{ix}(\mathbb{J})=F_{ix}(P_{\mathbb{J}})\). From Lemma 9, together with condition (1), we then infer  
\[||k_{n+1}-p_{0}|| = ||v_{n}-p_{0}||\]2  
Due to the nonincreasing nature of \(f\) and \(f(0)=0\), we have  
\[||k_{n+1}-p_{0}|| = ||v_{n}-p_{0}||\]3  
The required result now follows from Theorem 2. **Theorem 4**. : _Suppose that \(B\) is a nonempty closed convex subset of a UCBS \(X\) which satisfies [PERSON]'s condition. Assume that \(\mathbb{J}:B\to P_{r}(B)\) is a multi-valued mapping such that \(F_{ix}(\mathbb{J})\
eq\emptyset\) and is such that \(P_{\mathbb{J}}\) is a generalized (\(\alpha,\beta\))-nonexpansive mapping. Additionally, let \(I-P_{\mathbb{J}}\) be demiclosed with respect to zero. Then, any sequence \(\{k_{n}\}\) iteratively generated as in (5) converges weakly to a fixed point of \(\mathbb{J}\)._  
Proof. : Let the sequence \(\{k_{n}\}\) be iteratively generated as in (5) and let \(p_{0}\in F_{ix}(\mathbb{J})=F_{ix}(P_{\mathbb{J}})\) (see Lemma 1). Then, according to Lemma 9, \(lim_{n\to\infty}||k_{n}-p_{0}||\) exists. Since \(X\) is uniformly convex, it is reflexive. Therefore, there must exist a subsequence \(\{k_{n_{j}}\}\) of \(\{k_{n}\}\) that converges weakly to some \(p_{1}\in B\). Additionally, \(I-P_{\mathbb{J}}\) is demiclosed at zero, therefore \(p_{1}\in F_{ix}(P_{\mathbb{J}})=F_{ix}(\mathbb{J})\). If the sequence \(k_{n}\) does not weakly converge to \(p_{1}\), then there must be a subsequence \(\{k_{n_{j}}\}\) of \(\{k_{n}\}\) such that \(k_{n_{j}}\rightharpoonup p_{2}\) where \(p_{1}\
eq p_{2}\). Clearly, \(p_{2}\in F_{ix}(P_{\mathbb{J}})=F_{ix}(\mathbb{J})\). By Opial's property, we obtain  
\[||k_{n+1}-p_{0}|| = ||v_{n}-p_{0}||\]4 \[||k_{n+1}-p_{0}|| = ||v_{n}-p_{0}||\]5 \[||k_{n+1}-p_{0}|| = ||v_{n}-p_{0}||\]6 \[||k_{n+1}-p_{0}|| = ||v_{n}-p_{0}||\]7 \[||k_{n+1}-p_{0}|| = ||v_{n}-p_{0}||\]8  
and a contradiction occurs. Thus, the sequence \(\{k_{n}\}\) converges weakly to a point in \(F_{ix}(\mathbb{J})\).