# Estimating the Local Radius of Convergence for Picard Iteration
## 3 Local Convergence  
The following theorem is the basis of our approach in estimating local radius of convergence. Its proof is similar to the proof of Theorem 1[22], except in some details. We present the detailed proof here for completeness. Theorem 3.1: _Let \(T:C\rightarrow\mathcal{H}\) be a (nonlinear) mapping with nonempty set of fixed points, where \(C\) is an open set of a real Hilbert space \(\mathcal{H}\). Let \(p_{0}\) be a fixed point and let \(r_{0}\) be such that \(B_{r_{0}}\subset C\). We shall suppose that \(Fix(T)\subset B_{r_{0}}\) (if \(Fix(T)\) is bounded and \(C=\mathcal{H}\), then this condition is satisfied for any \(p_{0}\in Fix(T)\) and suitable \(r_{0}\)). Suppose further that_  
1. \(I-T\) _is demicclosed at zero on_ \(C\)_,_
2. \(T\) _is demicontractive with_ \(\lambda>0.5\) _on_ \(B_{r_{0}}\)_,_ _then the sequence_ \(\{x_{n}\}\) _given by Picard iteration,_ \(x_{n+1}=T(x_{n}),\ x_{0}\in B_{r_{0}}\) _remains in_ \(B_{r_{0}}\) _and converges weakly to some fixed point of_ \(T\)_. If, in addition,_
3. \(T\) _is quasi-expansive on_ \(B_{r_{0}}\)_,_ _then_ \(p_{0}\) _is the unique fixed point of_ \(T\) _in_ \(B_{r_{0}}\) _and the sequence_ \(\{x_{n}\}\) _converges strongly to_ \(p_{0}\)Proof. : Let \(p\) be a fixed point of \(T\). If \(x_{n}\in B_{r_{0}}\) then, using (ii), we have  
\[ \begin{array}{ll}\|x_{n+1}-p\|^{2}&=\|(x_{n}-p)-(x_{n}-T(x_{n}))\|^{2}\\ &=\|x_{n}-p\|^{2}-2\langle x_{n}-T(x_{n}),x_{n}-p\rangle+\|x_{n}-T(x_{n})\|^{2 }\\ &\leq\|x_{n}-p\|^{2}-(2\lambda-1)\|x_{n}-T(x_{n})\|^{2}.\end{array} \]  
Therefore, for any \(p\in B_{r_{0}}\), including \(p_{0}\), \(\|x_{n+1}-p\|\leq\|x_{n}-p\|\) and \(x_{n+1}\in B_{r_{0}}\), i.e., \(\{x_{n}\}\subset B_{r_{0}}\). Moreover, \(\|x_{n}-p\|\to\varrho_{p}\), so that  
\[\|x_{n}-T(x_{n})\|^{2}\leq\frac{1}{2\lambda-1}\left(\|x_{n}-p\|^{2}-\|x_{n+1}- p\|^{2}\right)\to 0.\]  
As \(B_{r_{0}}\) is weakly compact and \(\{x_{n}\}\) is bounded, there exists a subsequence \(\{x_{n_{j}}\}\) which converges weakly to \(q\). Then \(x_{n_{j}}-T(x_{n_{j}})\to 0\) and (i) implies \(q\in Fix(T)\). Suppose now that there exist two subsequences, say \(\{u_{k}\}\) and \(\{v_{k}\}\), which converge weakly to \(u\) and \(v\) respectively. As above, \(u,v\in Fix(T)\) and \(\|x_{n}-u\|\to\varrho_{u}\), \(\|x_{n}-v\|\to\varrho_{v}\). Therefore  
\[ \begin{array}{ll}\|u_{k}-u\|\to\varrho_{u},\ \|v_{k}-u\|\to\varrho_{u},\\ \|u_{k}-v\|\to\varrho_{v},\ \|v_{k}-v\|\to\varrho_{v}.\end{array} \]  
Let \(e_{k}\) defined by  
\[e_{k}=\|u_{k}-u\|^{2}-\|v_{k}-u\|^{2}-\|u_{k}-v\|^{2}+\|v_{k}-v\|^{2}.\]  
We have \(e_{k}\to 0\) as \(n\to\infty\). On the other hand, \(\frac{1}{2}e_{k}=-\langle u_{k}-v_{k},u-v\rangle\to-\|u-v\|^{2}\) which entails \(u=v\). Therefore \(x_{n}\rightharpoonup q\). Finally, from (iii) it results that \(p_{0}\) is the unique fixed point of \(T\) in \(B_{r_{0}}\) and  
\[\|x_{n}-p_{0}\|\leq\beta\|n_{n}-T(x_{n})\|\to 0.\]  
so that \(x_{n}\to p_{0}\). A demicontractive mapping is weak contractive, and a quasi-expansive mapping is weak expansive. The next theorem shows that the demicontractivity and the quasi-expansivity are not contradictory, there exists a relatively large class of mappings which are both demicontractive and quasi-expansive. **Theorem 2**. : _Let \(T:C\to C\) be a (nonlinear) mapping, where \(C\) is an open convex subset of \(\mathcal{H}\). Suppose that \(Fix(T)\
eq\emptyset\), that \(T\) is differentiable on \(C\) and \(\|T^{\prime}(x)\|\leq\eta<\sqrt{5}-2,\ \forall x\in C\). Then \(T\) is both demicontractive and quasi-expansive on \(C\)._  
Proof. : (I) \(T\) is quasi-expansive on \(C\) (therefore \(Fix(T)\) consists of a unique element). Let \(p\) be a fixed point of \(T\). Using mean value theorem, for any \(x\in C\), we have  
\[x-T(x)=(x-p)-(T(x)-T(p))=(x-p)-\left(\int_{0}^{1}T^{\prime}(p+t(x-p))dt\right)( x-p).\]  
Using the notation \(D_{x,p}=\int_{0}^{1}T^{\prime}(p+t(x-p))dt\), we have \(x-T(x)=(I-D_{x,p})(x-p)\) and \(\|D_{x,p}\|\leq\eta\).