# Hybrid Newton-like Inverse Free Algorithms for Solving Nonlinear Equations
## 2 Semi-Local Analysis  
Throughout this paper, we use the symbol \(U(x,\alpha)\) to denote the open ball centered at \(x\in T_{1}\) with radius \(\alpha>0\), and \(U[x,\alpha]\) is the closure of \(U(x,\alpha)\). The following Banach Lemma is used to prove our results. **Theorem 1** (Banach Lemma on Invertible Operators [3; 15]). : _If \(M\) is a bounded linear operator in \(T_{1}\), \(M^{-1}\) exists if and only if there is a bounded linear operator \(M_{1}\) in \(T_{1}\) such that \(M_{1}^{-1}\) exists and_  
\[\|I-M_{1}M\|<1.\]  
_If \(M^{-1}\) exists, then_  
\[M^{-1}=\sum_{n=0}^{\infty}(I-M_{1}M)^{n}M_{1}\]  
_and_  
\[\|M^{-1}\|\leq\frac{\|M_{1}\|}{1-\|I-M_{1}M\|}.\]  
Further, we use majorizing sequences to prove the semi-local convergence. Recall the definition of a majorizing sequence. **Definition 1** ([3; 5]). : _Let \(\{x_{n}\}\) be a sequence in a normed space \(X\). Then a nonnegative scalar sequence \(\{v_{n}\}\) for which_  
\[\|x_{n+1}-x_{n}\|\leq v_{n+1}-v_{n}\ \ \forall n\geq 0 \tag{6}\]  
_holds is a majorizing sequence for \(\{x_{n}\}\). Note that any majorizing sequence is necessarily nondecreasing. Moreover, if the sequence \(\{v_{n}\}\) converges, then \(\{x_{n}\}\) converges too, and for \(v^{*}=\lim_{n\longrightarrow\infty}v_{n}\)_  
\[\|x^{*}-x_{n}\|\leq v^{*}-v_{n}.\]  
_Hence, the study of the convergence of the sequence \(\{x_{n}\}\) reduces to that of \(\{v_{n}\}\)._  
Let \(S=[0,+\infty)\). Suppose:  
\((H_{1})\): There exists parameters \(\delta\geq 0,\gamma\in[0,\frac{1}{2})\), an element \(x_{0}\in\Omega\), and an invertible operator \(\Delta\) such that  
\[\|A\Delta^{-1}G(x_{0})\|\leq\delta.\]  
\((H_{2})\): There exists a function \(w_{0}:S\longrightarrow S\), which is nondecreasing as well as continuous (FNDC), such that the equation \(w_{0}(t)-1=0\) has the smallest positive solution. Denote such solution by \(\rho\) and set \(S_{0}=[0,\rho)\). \((H_{3})\): There exist (FNDC) \(w:S_{0}\longrightarrow S\) and \(w_{1}:S_{0}\longrightarrow S\). Define the scalar sequence \(\{\alpha_{n}\}\) for \(\alpha_{0}=0,\alpha_{1}=\delta\) some \(\gamma\geq 0,\tilde{b}\geq 0\), and each \(n=0,1,2,...\) by  
\[q_{n+1} = (1+\gamma+...+\gamma^{k})(\int_{0}^{1}w((1-\tau)(\alpha_{n+1}- \alpha_{n})d\tau\] \[+w_{1}(\alpha_{n})+\tilde{b}\gamma^{k+1}),\]  
and  
\[\alpha_{n+2} = \alpha_{n+1}+q_{n+1}(\alpha_{n+1}-\alpha_{n}). \tag{7}\]The sequence \(\{\alpha_{n}\}\) is shown to be majorizing for algorithm (4) in Theorem 2. But first, the convergence conditions are given for the sequence \(\{\alpha_{n}\}\). \((H_{4})\) There exists \(\rho_{0}\in[0,\rho)\) such that for each \(n=0,1,2,...\)  
\(q_{n+1}<1\) and \(\alpha_{n}\leq\rho_{0}\). This condition and the formula (7) imply that for each \(n=0,1,2,...\)  
\[0\leq\alpha_{n}\leq\alpha_{n+1}\leq\rho_{0}\]  
and there exists \(\alpha\in[0,\rho_{0})\) such that \(\lim_{n\rightarrow+\infty}=\alpha\). The parameter \(\alpha\) is the unique least upper bound of the sequence \(\{\alpha_{n}\}\). It is worth noting that this sequence can be computed a priori and relates to the initial approximation. Such conditions are weaker than the usual convergence conditions given as functions of the starting point [3; 4; 5]. Next, we relate the scalar sequences and functions \(w_{0},w\) to the operators on algorithm (4). \((H_{5})\)\(\|\Delta^{-1}(G^{\prime}(v)-\Delta)\|\leq w_{0}(\|v-x_{0}\|)\) for each \(v\in\Omega\). Set \(\Omega_{0}=U(x_{0},\rho)\cap\Omega\). \((H_{6})\)  
\[M^{-1}=\sum_{n=0}^{\infty}(I-M_{1}M)^{n}M_{1}\]0  
and  
\[M^{-1}=\sum_{n=0}^{\infty}(I-M_{1}M)^{n}M_{1}\]1  
for each \(v,v_{1},v_{2}\in\Omega_{0}\). and  
\((H_{7})\)\(U[x_{0},\alpha]\subset\Omega\). The conditions \((H_{1})\)-\((H_{5})\) and the developed notations are utilized to show the main semi-local analysis of convergence for algorithm (4). Theorem 2: Suppose that the conditions \((H_{1})\)-\((H_{5})\) hold.