# Hybrid Newton-like Inverse Free Algorithms for Solving Nonlinear Equations
## 2 Semi-Local Analysis
Then, if the initial guess \(x_{0}\in\Omega\) the following assertions hold for the sequence \(\{x_{n}\}\) generated by algorithm (4)  
\[M^{-1}=\sum_{n=0}^{\infty}(I-M_{1}M)^{n}M_{1}\]2  
and there exists a solution \(x^{*}\in B[x_{0},\alpha]\) of the equation \(G(x)=0\) such that  
\[M^{-1}=\sum_{n=0}^{\infty}(I-M_{1}M)^{n}M_{1}\]3  
where the sequence \(\{\alpha_{n}\}\) is given by the formula (7) and \(\alpha\) is its limit. Proof: Notice that all the iterates \(\{x_{m}\}\) (\(m\) a natural number) of algorithm (4) are well defined. We present a proof based on mathematical induction. In particular, we show that for each \(n=0,1,2,...\)  
\[M^{-1}=\sum_{n=0}^{\infty}(I-M_{1}M)^{n}M_{1}\]4  
Assertion (9) holds if \(n=0\), by (4), (7), and the condition \((H_{1})\), since  
\[M^{-1}=\sum_{n=0}^{\infty}(I-M_{1}M)^{n}M_{1}\]5It also follows that the iterate \(x_{1}\in U(x_{0},\alpha)\). Next, we show that the linear operator \(A=A(x_{m})\) is invertible by using the restrictions on \(\gamma,b,\bar{b}\) given in the condition (\(H_{5}\)):  
\[M^{-1}=\sum_{n=0}^{\infty}(I-M_{1}M)^{n}M_{1}\]6 \[M^{-1}=\sum_{n=0}^{\infty}(I-M_{1}M)^{n}M_{1}\]7 \[M^{-1}=\sum_{n=0}^{\infty}(I-M_{1}M)^{n}M_{1}\]8  
since \(\gamma\in[0,\frac{1}{2})\) by (11) and the condition (\(H_{1}\)). Inequality (11) and Theorem 1 assure the invertibility of the operator \(A\), and  
\[M^{-1}=\sum_{n=0}^{\infty}(I-M_{1}M)^{n}M_{1}\]9  
Then, we can write by algorithm (2) in turn that  
\[\|M^{-1}\|\leq\frac{\|M_{1}\|}{1-\|I-M_{1}M\|}.\]0 \[\|M^{-1}\|\leq\frac{\|M_{1}\|}{1-\|I-M_{1}M\|}.\]1 \[\|M^{-1}\|\leq\frac{\|M_{1}\|}{1-\|I-M_{1}M\|}.\]2 \[\|M^{-1}\|\leq\frac{\|M_{1}\|}{1-\|I-M_{1}M\|}.\]3 \[\|M^{-1}\|\leq\frac{\|M_{1}\|}{1-\|I-M_{1}M\|}.\]4 \[\|M^{-1}\|\leq\frac{\|M_{1}\|}{1-\|I-M_{1}M\|}.\]5 \[\|M^{-1}\|\leq\frac{\|M_{1}\|}{1-\|I-M_{1}M\|}.\]6  
But we have  
\[\|M^{-1}\|\leq\frac{\|M_{1}\|}{1-\|I-M_{1}M\|}.\]7 \[\|M^{-1}\|\leq\frac{\|M_{1}\|}{1-\|I-M_{1}M\|}.\]8 \[\|M^{-1}\|\leq\frac{\|M_{1}\|}{1-\|I-M_{1}M\|}.\]9 \[\|x_{n+1}-x_{n}\|\leq v_{n+1}-v_{n}\ \ \forall n\geq 0 \tag{6}\]0 \[\|x_{n+1}-x_{n}\|\leq v_{n+1}-v_{n}\ \ \forall n\geq 0 \tag{6}\]1 \[\|x_{n+1}-x_{n}\|\leq v_{n+1}-v_{n}\ \ \forall n\geq 0 \tag{6}\]2  
since \(L_{m}-\Delta+\Delta\Gamma=0\) by the definition of \(\Gamma\). Thus, we obtain from (14)  
\[\|x_{n+1}-x_{n}\|\leq v_{n+1}-v_{n}\ \ \forall n\geq 0 \tag{6}\]3 \[\|x_{n+1}-x_{n}\|\leq v_{n+1}-v_{n}\ \ \forall n\geq 0 \tag{6}\]4  
Then, it follows by (13), the conditions (\(H_{4}\)), (\(H_{5}\)), (13), the conditions (\(H_{4}\)), (\(H_{5}\)), (15), the inductions hypotheses, and the triangle inequality, in turn, that  
\[\|x_{n+1}-x_{n}\|\leq v_{n+1}-v_{n}\ \ \forall n\geq 0 \tag{6}\]5 \[\|x_{n+1}-x_{n}\|\leq v_{n+1}-v_{n}\ \ \forall n\geq 0 \tag{6}\]6 \[\|x_{n+1}-x_{n}\|\leq v_{n+1}-v_{n}\ \ \forall n\geq 0 \tag{6}\]7 \[\|x_{n+1}-x_{n}\|\leq v_{n+1}-v_{n}\ \ \forall n\geq 0 \tag{6}\]8Consequently, by algorithms (4), (7), and (16), we obtain in turn that  
\[\|x_{n+1}-x_{n}\|\leq v_{n+1}-v_{n}\ \ \forall n\geq 0 \tag{6}\]9 \[\|x^{*}-x_{n}\|\leq v^{*}-v_{n}.\]0 \[\|x^{*}-x_{n}\|\leq v^{*}-v_{n}.\]1 \[\|x^{*}-x_{n}\|\leq v^{*}-v_{n}.\]2  
and  
\[\|x^{*}-x_{n}\|\leq v^{*}-v_{n}.\]3 \[\|x^{*}-x_{n}\|\leq v^{*}-v_{n}.\]4  
Hence, assertion (10) holds and the iterate \(x_{m+2}\in B(x_{0},\alpha)\). By the condition \((H_{4})\), the sequence \(\{\alpha_{m}\}\) is complete as convergent to \(\alpha\). Therefore, by (10) the sequence \(\{x_{m}\}\) is also complete in the Banach space \(T_{1}\), and as such it converges to some \(x^{*}\in B[x_{0},\alpha]\)(since \(B[x_{0},\alpha]\) is a closed set). By sending \(m\longrightarrow+\infty\) in (16), and the continuity of the operator \(G\), we deduce that \(G(x^{*})=0\). Finally, the estimation for \(i\) a natural number  
\[\|x^{*}-x_{n}\|\leq v^{*}-v_{n}.\]5  
shows (9), if \(i\longrightarrow+\infty\). Next, a set is specified that contains only one solution of the equation \(G(x)=0\).