Fixed Point Results for Hybrid Multi-Valued Mappings in Metric Spaces

Abstract

The purpose of the present paper is to investigate fixed point results for a class of hybrid multi-valued mappings defined on complete metric spaces. The contractive condition considered in the study combines the ordinary distance between two points with the distances of the points from their respective images. In this sense, the condition incorporates features of both Nadler-type and Kannan-type contractions. By using the Hausdorff metric and an iterative selection procedure, a fixed point existence theorem is established for multi-valued mappings taking values in the family of nonempty closed and bounded subsets of a complete metric space. Several immediate consequences are derived, including the classical multi-valued contraction case and a Kannan-type multi-valued fixed point result. The corresponding single-valued version is also obtained, for which uniqueness of the fixed point follows. Two illustrative examples are presented, supported by explanatory figures. In particular, a finite metric-space example shows that the proposed hybrid condition may hold even when the standard Nadler contraction condition is not satisfied.