FIXED POINT THEOREMS FOR GENERALIZED NONEXPANSIVE SET-VALUED MAPPINGS IN CONE METRIC SPACES

Abstract

Abstract. In 2007, Huang and Zhang [1] introduced a cone metric spacewith a cone metric generalizing the usual metric space by replacing thereal numbers with Banach space ordered by the cone. They consideredsome xed point theorems for contractive mappings in cone metric spaces.Since then, the xed point theory for mappings in cone metric spaces hasbecome a subject of interest in [1-6] and references therein. In this paper,we consider some xed point theorems for generalized nonexpansive set-valued mappings under suitable conditions in sequentially compact conemetric spaces and complete cone metric spaces. 1. Introduction and PreliminariesIn 2007, Huang and Zhang [1] introduced a cone metric space with a conemetric generalizing the usual metric space by replacing the real numbers withBanach space ordered by the cone. They considered some xed point theoremsfor contractive mappings in cone metric spaces. Since then, the xed pointtheory for mappings in cone metric spaces has become a subject of interest in[1-6] and references therein. Especially, for single-valued mappings, Choudhuryand Metiya [6] considered some xed point theorems for weak contraction incone metric spaces in 2010. In 2011, Wardowski [2] introduced H-cone metricin the collection of subsets of a given cone metric space. And he consideredthe concept of set-valued contraction of Nadler-type and proved a xed pointtheorem for contractive set-valued mappings in H-cone metric spaces. Inspiredand encouraged by the previous works, in this paper we consider some xedpoint theorems for generalized nonexpansive set-valued mappings under suit-able conditions in sequentially compact cone metric spaces and complete conemetric spaces.Let Ebe a real Banach space and Pbe a subset of E.Pis called a cone if and only if