×

Dynamics on \((P_{cp}(X), H_d)\) generated by a finite family of multi-valued operators on \((X,d)\). (English) Zbl 1011.47043

Let \((X,d)\) be a complete metric space, \(P(X)\) the set of all nonempty subsets of \(X\) and \(P_p(X)\) the subset of \(P(X)\) containing all \(Y\subset X\) having the property \(p\), where \(p\) can be \(cp=\) compact, \(cl=\) closed, \(b=\) bounded, etc. Let \(H_d\) be the Hausdorff-Pompeiu functional [see, e.g., G. Beer, “Topologies on Closed and Closed Convex Sets”, Kluwer, Dordrecht (1994; Zbl 0792.54008)]. If \(F_i:X\to P_p(X)\), \(i\in\{1,\dots,m\}\), are multi-valued operators, then one can consider the operator \(T_F:(P_p(X),H_d)\to(P_p(X),H_d)\), \(T_F(Y)=\bigcup_{i=1}^mF_i(Y)\). The authors partially answer the question whether some special contraction properties are inherited from operators of the family \((F_i)\) to the operator \(T_F\). As a consequence, they obtain some fixed-point results.

MSC:

47H10 Fixed-point theorems
47H06 Nonlinear accretive operators, dissipative operators, etc.

Citations:

Zbl 0792.54008
PDFBibTeX XMLCite