A modified homotopy method for solving nonconvex fixed points problems (Q2863568)

In this paper, the authors consider the general non-convex set \(\Omega=\{x \in \mathbb{R}^{n}:g_{i}(x)\leq 0,\;i=1, \dots, m,\;h_{j}(x)=0,\;j=1, \dots, l\}\), where \(g_{i}(x), h_{j}(x):\mathbb{R}^{n} \to \mathbb{R}\) are all \(C^{2}\) functions. Let \(\Omega^{0}=\{x \in \mathbb{R}^{n}:g_{i}(x)<0,\;i=1,2, \dots,m,\;h_{j}(x)=0,\;j=1,2, \dots{}, l\}\) and \(\partial \Omega=\Omega\setminus\Omega^{0}\) be the interior and the boundary of \(\Omega\), respectively. Let \(\Omega_{i}=\{x \in \mathbb{R}^{n}:g_{i}(x)\leq 0\}\), \(\hat{\Omega}_{j}=\{x \in \mathbb{R}^{n}:h_{j}(x)=0\}\), \(I(x)=\{i \in \{1,2, \dots{}, m\}:g_{i}(x)=0\}\).NEWLINENEWLINEThe following assumptions are made.NEWLINENEWLINEAssumption 2.1 \(\Omega^{0}\) is nonempty.NEWLINENEWLINEAssumption 2.2 For any \(x\in \Omega\), the matrix \(\nabla h(x)\) is of full column rank.NEWLINENEWLINEAssumption 2.3 (The pseudo cone condition on the set \(\Omega\)) There exist mappings \(\eta_{i}(x,y_{i}):\mathbb{R}^{n}\times \mathbb{R} \to \mathbb{R}^{n}, i=1,2, \dots, m\) and \(\zeta_{j}(x,z_{j}):\mathbb{R}^{n}\times \mathbb{R} \to \mathbb{R}^{n}\), \(j=1,2, \dots, l\), called hair mappings of \(\Omega_{i}\) and \(\hat{\Omega}_{j}\), respectively, such thatNEWLINENEWLINE{\parindent=8mmNEWLINE\begin{itemize}\item[(i)] \(\eta_{i}(x,y_{i})=0\) iff \(y_{i}=0\) (\(1\leq i \leq m)\), \(\zeta_{j}(x,z_{j})=0\) iff \(z_{j}=0\) (\(1 \leq j \leq l\)).NEWLINENEWLINE\item[(ii)] \(\| \eta_{i}(x,y_{i})\| \to \infty\) when \(y_{i} \to + \infty\) (\(1\leq i \leq m\)), \(\| \zeta_{j}(x,z_{j}) \| \to \infty\) when \(|z_{j}| \to + \infty\) (\(1 \leq j \leq l\)).NEWLINENEWLINE\item[(iii)] (Positive independence) For all \(x \in \partial \Omega\), \(\sum_{i \in I(x)} \eta_{i}(x,y_{i})+ \sum^{l}_{j=1} \zeta_{j}(x,z_{j})=0\), \(0 \leq y_{i}, z_{j} \in \mathbb{R}\) implies that \(y_{i}=0\), \(i \in I(x)\) and \(z_{j}=0\), \( j=1,2, \dots{}, l\).NEWLINENEWLINE\item[(iv)] For any \(x \in \Omega\), the matrix \(\nabla_{z}(\sum^{l}_{j=1}\zeta_{j}(x,z_{j}))\) is of full row rank.NEWLINENEWLINE\item[(v)] For all \(x \in \Omega\),NEWLINENEWLINENEWLINE\[NEWLINE \left\{x+ \sum_{i \in I(x)} \eta_{i}(x,y_{i})+ \sum^{l}_{j=1} \zeta_{j}(x,z_{j}): y_{i} \geq 0,\;i \in I(x), z_{j} \in \mathbb{R}\right\}\cap \Omega= \{x\}.NEWLINE\]NEWLINENEWLINENEWLINE\end{itemize}}NEWLINENEWLINEAssumption 2.4 For all \(\{x^{k} \} \subset \Omega, \| x^{k}\| \to \infty\) (as \(k \to \infty\)), there exists a \(\theta^{0}\in \Omega\) such thatNEWLINENEWLINENEWLINE\[NEWLINE{\lim_{k \to \infty}(\theta^{0}-x^{k})^{T} \left[ \sum^{m}_{i=1} \eta_{i}(x^{k}, y^{k}_{i})+ \sum^{l}_{j=1} \zeta_{j}(x^{k}, z^{k}_{j})\right]<0}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\displaystyle{\lim_{k \to \infty}(\theta^{0}-x^{k})^{T}(x^{k}-F(x^{k}))<0}.NEWLINE\]NEWLINENEWLINENEWLINEThe authors prove the following main result:NEWLINENEWLINETheorem 2.1 Suppose that \(\Omega\) is defined as above, the hair mappings \(\eta_{i}(x,y_{i}):\mathbb{R}^{n+1} \to \mathbb{R}^{n}\), \(i=1,2, \dots{}, m\) of \(\Omega_{i}\) and \(\zeta_{j}(x,z_{j}):\mathbb{R}^{n+1} \to \mathbb{R}^{n}\), \(j=1,2, \dots{}, l\) of \(\hat{\Omega}_{j}\) are twice continuously differentiable. If assumptions 2.1--2.4 hold for any twice continuous differentiable mapping \(F: \Omega \to \Omega\), thenNEWLINENEWLINE{\parindent=8mmNEWLINE\begin{itemize}\item[(1)] \(F\) has a fixed point in \(\Omega\);NEWLINENEWLINE\item[(2)] for almost all \((x^{(0)}, y^{(0)}, z^{(0)}) \in \Omega^{(0)}\times \mathbb{R}^{m}_{++}\times \mathbb{R}^{l}\), the modified homotopy equation determines a smooth curve \(\Gamma_{w^{(0)}}\subset \Omega^{0}\times \mathbb{R}^{m}_{++}\times \mathbb{R}^{l}\times (0,1]\) starting from \((w^{(0)},1)\).NEWLINENEWLINE\end{itemize}}NEWLINENEWLINEWhen \(t \to 0\), the limit set \(T \subset \Omega \times \mathbb{R}^{m}_{+} \times \mathbb{R}^{l} \times 0\) of \(\Gamma_{w^{(0)}}\) is nonempty, and the \(x\)-component of any point in \(T\) is a fixed point of \(F(\Omega)\) in \(\Omega\).