New fixed point theorems for 1-set-contractive operators and variational iterative method (Q2885012)

The article deals with some results about the calculation of the degree \(\deg (I - A,\Omega,p)\) of vector fields \(I - A\) with \(1\)-set-contractive operators \(A\) on the boundary of a bounded open set \(\Omega\) with respect to a point \(p\). More exactly, the authors formulate a series of conditions that guarantee the well-known Leray-Schauder condition \(Ax \neq tx\), \(x \in \partial\Omega\), \(t \geq 1\).NEWLINENEWLINEThe main among such conditions are the following ones: (1) there exist \(\alpha > 1\) and \(\beta,\gamma > 0\), \(n \in {\mathbb N}\), such that NEWLINE\[NEWLINE\|Ax + x\|^{n(\alpha+\beta)+\gamma} \leq \|Ax - x\|^{n\alpha+\gamma} \|x\|^{n\beta} + \|x\|^{n\alpha+\gamma} \|Ax\|^{n\beta}, \quad x \in \partial\Omega.NEWLINE\]NEWLINE and (2) there exist \(\alpha > 1\) and \(\beta,\gamma > 0\), \(n \in {\mathbb N}\), such that NEWLINE\[NEWLINE\|Ax + 2x\|^{n(\alpha+\beta)+\gamma} \leq \|Ax + x\|^{n\alpha+\gamma} \|x\|^{n\beta} + \|x\|^{n(\alpha+\beta)+\gamma}, \quad x \in \partial\Omega.NEWLINE\]NEWLINE It is given also the following generalization of Altman's condition: NEWLINE\[NEWLINE\|Ax - x\|^n \geq \underbrace{\begin{vmatrix} \|Ax + x\| & \|x\| & \ldots & \|x\| \\ \|x\| & \|Ax + x\| & \ldots & \|x\| \\ \ldots & \ldots & \ldots & \ldots \\ \|x\| & \|x\| & \ldots & \|Ax + x\|\end{vmatrix}}_{n \times n}, \quad x \in \partial\Omega.NEWLINE\]NEWLINE As examples of these results, the authors consider some simple nonlinear integral equations.