New fixed point theorems for 1-set-contractive operators in Banach spaces (Q881624)

This article presents a list of sufficient conditions for the existence of fixed points of semi-closed \(1\)-set-contractive operators in real Banach spaces. All results presented here are evident corollaries of the following analogue for semi-closed \(1\)-set-contractive operators of the Leray-Schauder fixed point principle: {If \(A\) is a semi-closed \(1\)-set-contractive operator such that \[ Ax - x_0 \neq k(x - x_0) \;\text{for some} \;x_0 \in \Omega \;\text{and for all} \;x \in \partial\Omega, \;1 < k < \infty,\tag{*} \] then \(A\) has at least one fixed point in \(\overline{\Omega}\)}. In turn, this result is an evident corollary of the statement that \(\deg (I - A,\Omega) = 1\) provided that \(A\) is semi-closed \(1\)-set-contractive operator satisfying the condition \(Ax - x_0 \neq k(x - x_0)\) for some \(x_0 \in \Omega\) and for all \(x \in \partial\Omega\), \(1 \leq k < \infty\). The main part of the article is devoted to inequalities that imply (*); among others, the inequality \[ \| Ax - x\| ^\alpha \geq \| Ax\| ^{\alpha+\beta} - \| x\| ^\alpha, \quad x \in \partial\Omega, \] (\(\alpha > 1\), \(\beta \geq 0\)) is treated.