Single-valued and multi-valued Caristi type operators. (English) Zbl 1003.47041

The main result of the paper is the following. Let \((X,d)\) be a metric space, \( P_{\text{cl}} (X)\) the space of all closed subsets of \(X\). Let \(F:X\to P_{\text{cl}}(X)\) satisfy \(H(F(x), F(y)) \leq ad(x, y) + bD(x, F(x))+cd(y,F(y))\) for each \(x,y\in X\), where \(H\) denotes the Hausdorff distance, and \(a, b,c \geq 0\) with \(a + b + c < 1\). Then there exists a selection \(f\) of \(F\) satisfying the Caristi type condition \(d(x, fx) + \varphi(fx) \leq \varphi(x)\) for each \(x \in X\).


47H10 Fixed-point theorems
47H04 Set-valued operators