## 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 $$x \in X$$.

### MSC:

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