On fixed points of multi-valued maps in metric spaces and differential inclusions

A generalization of the Nadler fixed point theorem for multi-valued maps acting in metric spaces is proposed. The obtained result allows to study the existence of fixed points for multi-valued maps that have as images any arbitrary sets of the corresponding metric space and are not necessarily contracting, or even continuous, with respect to the Hausdorff metric. The mentioned result can be used for investigating differential and functional-differential equations with discontinuities and inclusions generated by multi-valued maps with arbitrary images. In the second part of the paper, as an application, conditions of existence and continuation of solutions to the Cauchy problem for a differential inclusion with noncompact in ${\mathbb R}^n$ right-hand side are derived.