## 2 Semilocal Convergence
: Equation (11) can be written as  
\[D_{0}=D\cap U(x_{0},r_{0}).\]1  
since, by Equation (9),  
\[D_{0}=D\cap U(x_{0},r_{0}).\]2  
and \(r\) solves Equation (11). It follows from the definition of sequence \(\{t_{k}\}\), functions \(w_{1}\), \(w_{2}\), \(v_{1}\), \(v_{2}\) and inequality (10) that  
\[D_{0}=D\cap U(x_{0},r_{0}).\]3  
\[D_{0}=D\cap U(x_{0},r_{0}).\]4  
and  
\[D_{0}=D\cap U(x_{0},r_{0}).\]5 \[D_{0}=D\cap U(x_{0},r_{0}).\]6 \[D_{0}=D\cap U(x_{0},r_{0}).\]7  
Therefore, sequences \(\{t_{k}\}\) converges to \(r^{*}\), which satisfies inequality (12). Next, we present the semilocal convergence analysis of the MMN-HSS method. Then, we also obtain that  
\[D_{0}=D\cap U(x_{0},r_{0}).\]8  
\[D_{0}=D\cap U(x_{0},r_{0}).\]9 \[\|H(x)-H(y)\|\leq w_{1}(\|x-y\|),\]0 \[\|H(x)-H(y)\|\leq w_{1}(\|x-y\|),\]1  
and  
\[\|H(x)-H(y)\|\leq w_{1}(\|x-y\|),\]2  
Hence, we get from inequality (19) that  
\[ \begin{split}\|F(x_{0}^{(m-1)})\|&\leq\int_{0}^{1}w ((s_{0}^{(m-1)}-s_{0}^{(m-2)})\xi)d\xi(s_{0}^{(m-1)}-s_{0}^{(m-2)})\\ &+w(s_{0}^{(m-2)}-t_{0})(s_{0}^{(m-1)}-s_{0}^{(m-2)})+\frac{\eta (1-\gamma\upsilon(t_{0}))}{(1+\eta)\gamma}(s_{0}^{(m)}-s_{0}^{(m-1)})\\ &\leq\frac{1-\gamma\upsilon(t_{0})}{(1+\eta)\gamma}(s_{0}^{(m)}- s_{0}^{(m-1)}).\end{split}  \tag{20}\]  
Then, we have by Equation (9) that  
\[ \begin{split}\|x_{1}-x_{0}^{(m-1)}\|&\leq\|I-T( \alpha;x_{0})_{0}^{{}^{(m)}}\|\|F^{\prime}(x_{0})^{-1}\|\|F(x_{0}^{(m-1)})\| \\ &\leq(1+((\tau+1)\theta)^{{}^{(m)}})\gamma\frac{1}{(1+\eta) \gamma}(1-\gamma\upsilon(t_{0}))(s_{0}^{(m)}-s_{0}^{(m-1)}))=t_{1}-s_{0}^{(m- 1)}\end{split} \]  
holds, and the items (17) hold for \(k=0\). Suppose that the items (17) hold for all nonnegative integers less than \(k\). Next, we prove the items (17) hold for \(k\). We get, in turn, by the induction hypotheses:  
\[ \begin{split}\|x_{k}-x_{0}\|&\leq\|x_{k}-x_{k-1}^{ (m-1)}\|+\|x_{k-1}^{(m-1)}-x_{k-1}^{(m-2)}\|+\cdots+\|x_{k-1}^{(1)}-x_{k-1}^{ (0)}\|+\|x_{k-1}-x_{0}\|,\\ &\leq(t_{k}-s_{k-1}^{(m-1)})+(s_{k-1}^{(m-1)}-s_{k-1}^{(m-2)})+ \cdots+(s_{k-1}^{(1)}-t_{k-1})+(t_{k-1}-t_{0}),\\ &=t_{k}-t_{0}<r_{*}<r.\end{split} \]  
In view of \(x_{k-1},x_{k-1}^{(1)},\cdots,x_{k-1}^{(m-1)}\in\mathbb{U}(x_{0},r)\), we have  
\[ \begin{split}\|F(x_{k})\|\leq&\|F(x_{k})-F(x_{k-1}^{ (m-1)})-F^{\prime}(x_{k-1})(x_{k}-x_{k-1}^{(m-1)})\|\\ &+\|F(x_{k-1}^{(m-1)})+F^{\prime}(x_{k-1})(x_{k}-x_{k-1}^{(m-1)} \|\\ \leq&\|F(x_{k})-F(x_{k-1}^{(m-1)})-F^{\prime}(x_{k-1}^{ (m-1)})(x_{k}-x_{k-1}^{(m-1)})\|\\ &+\|F^{\prime}(x_{k-1}^{(m-1)})-F^{\prime}(x_{k-1})\|\|(x_{k}-x_{k -1}^{(m-1)}\|+\eta\|F(x_{k-1}^{(m-1)}\|\\ \leq&\int_{0}^{1}w(\|(x_{k}-x_{k-1}^{(m-1)}\|\xi)d \xi\|\|(x_{k}-x_{k-1}^{(m-1)}\|\\ &+w(\|x_{k-1}^{(m-1)}-x_{k-1}\|)\|x_{k}-x_{k-1}^{(m-1)}\|+\frac{ \eta(1-\gamma\upsilon(t_{k-1}))}{(1+\eta)\gamma}(s_{k-1}^{(m)}-s_{k-1}^{(m-1)}) \\ \leq&\frac{(1-\gamma\upsilon(t_{k}))}{(1+\eta) \gamma}(s_{k}^{(m)}-s_{k}^{(m-1)}).\end{split}  \tag{21}\]We also get that  
\[\|x_{k}-x_{k-1}^{(m-1)}\|\leq t_{k}-s_{k-1}^{(m-1)}, \tag{22}\]  
\[\|x_{k-1}^{(m-1)}-x_{k-1}\| \leq\|x_{k-1}^{m-1}-x_{k-1}^{(m-2)}\|+\cdots+\|x_{k-1}^{(1)}-x_{k-1 }\|,\] \[\leq(s_{k-1}^{(m-1)}-s_{k-1}^{(m-2)})+\cdots+(s_{k-1}^{(1)}-t_{k-1 }), \tag{23}\] \[\leq s_{k-1}^{(m-1)}-t_{k-1}\]  
and  
\[\|F(x_{k-1}^{(m-1)})\|\leq\frac{1}{1+\eta)\gamma}(1-\gamma\upsilon(t_{k-1}))(s _{k-1}^{(m)}-s_{k-1}^{(m-1)}). \tag{24}\]  
It follows that  
\[\|x_{k}^{(1)}-x_{k}\| \leq\|I-T(\alpha;x_{k})^{i_{k}^{(1)}}\|\|F^{\prime}(x_{k})^{-1}\| \|F(x_{k})\|\] \[\leq(1+\theta^{i_{k}^{(1)}})\frac{\gamma}{1-\gamma\upsilon(t_{k}) }\frac{1-\gamma\upsilon(t_{k})}{(1+\eta)\gamma}(s_{k}^{(1)}-t_{k})\] \[\leq s_{k}^{(1)}-t_{k}.\]  
Suppose that the following items hold for any positive integers less than \(m-1\):  
\[\left\{ \begin{array}{ll}\|F(x_{0}^{(i)})\|\leq\frac{1}{(1+\eta)\gamma}(1- \gamma\upsilon(t_{k}))(s_{k}^{(i+1)}-s_{k}^{(i)}),\\ \|x_{k}^{(i+1)}-x_{k}^{(i)}\|\leq s_{k}^{(i+1)}-s_{k}^{(i)},\,i=1,2,\cdots,m-2.\end{array} \right. \tag{25}\]  
We will prove items (25) hold for \(m-1\).