Some theorems of random operator equations (Q1608197)

This note is devoted to the existence of random solutions of the equation \[ A(\omega ,x)=\mu x,\tag{1} \] where \(A:\Omega\times\overline{D}\to X\) is a 1-set-contractive random operator, \(X\) is a closed convex cone in a Banach space, \(D\subset X\) is open in \(X\), bounded and such that \(0\in D\), and \(\mu\geq 1\). The authors assume that for every \(\omega\in\Omega\) the operator \(A(\omega ,\cdot)\) satisfies some inequalities on the boundary of \(D\). The proofs are based on a result of the first author [Adv. Math. Beijing 27, No. 5, 464-468 (1998; Zbl 1054.47520)], and on some elementary inequalities. Reviewer's remark: The beginning of the proof of Theorem 3 is not clear: If the equation (1) has no random solution on \(\partial D\), then we cannot deduce that \(A(\omega ,x)\neq\mu x\) a.s. for every \(x\in\partial D\).