Inequalities and solution of an operator equation
The article deals with fixed points of an operator \(A:\overline{D} \to E\), where \(D\) is a bounded open convex set in a real Banach space \(E\), \(0 \in D\). It is assumed that \(I - A\) is a closed operator and \(\alpha(AD) < \alpha(D)\), where \(\alpha\) denotes the noncompact measure (such operators are called by the authors \textit{semi-closed \(1\)-set-contractive}). It is proved that \(A\) has a fixed point in \(D\) if one of the following conditions holds: \[ \| Ax + x\| ^{n+1} + \| Ax - x\| ^{n+1} + \| x\| \leq 2[\| Ax\| ^{n+1} + \| x\| ^{n+1}] + e \ln \| x\| , \quad x \in \partial D, \] with \(n \in {\mathbb N}\), \(n > 1\); \[ 2\| Ax - x\| ^p - \| Ax + x\| ^p \geq \| Ax\| ^p - \| x\| ^p, \quad x \in \partial D, \] with \(p > 1\); \[ \| Ax - x_0\| \leq \| x - x_0\| , \quad x \in \partial D, \] with \(x_0 \in D\). All these statements are direct corollaries of the analogue of Schauder's fixed point principle for semi-closed \(1\)-set-contractive operators. Moreover, the definition of these operators (cited above) is incorrect.
scientific article; zbMATH DE number 5277570
scientific article