Set Contractions and KKM Mappings in Banach Spaces

A Razani,N Karamikabir

doi:10.1155/2013/346094

Abstract

Some fixed point theorems for generalized set contraction maps and KKM type ones in Banach spaces are presented. Moreover, a new generalized set contraction is introduced. As an application, some coincidence theorems for KKM type set contractions are obtained.

Highlights

IntroductionPbd(E), Pcl(E), Pcv(E), Pcp(E), Pco(E), Pcl,bd(E), Pcp,cv(E), and Prcp(E) denote the classes of all bounded, closed, convex, compact, connected, closed-bounded, compact-convex, and relatively compact subsets of E, respectively [1]
Let E be a Banach space and Pp (E)= {A ⊂ E : A is a nonempty and has a property p} . (1) Pbd(E), Pcl(E), Pcv(E), Pcp(E), Pco(E), Pcl,bd(E), Pcp,cv(E), and Prcp(E) denote the classes of all bounded, closed, convex, compact, connected, closed-bounded, compact-convex, and relatively compact subsets of E, respectively [1]
Chen and Chang obtained some fixed point theorems for KKM type set contraction mappings in various spaces [9,10,11,12]

Summary

Introduction

Pbd(E), Pcl(E), Pcv(E), Pcp(E), Pco(E), Pcl,bd(E), Pcp,cv(E), and Prcp(E) denote the classes of all bounded, closed, convex, compact, connected, closed-bounded, compact-convex, and relatively compact subsets of E, respectively [1]. Let X be a nonempty, closed, convex, and bounded subset of a Banach space E and let T : X → Pcl,cv(X) be a closed and nonlinear D-set contraction. A multivalued mapping T : E → Pcl,bd(E) is called C-set contraction if there exists a continuous (c)comparison function φ such that μ(T(A)) ≤ φ(μ(A)) for all A ∈ Pcl,bd(E) with T(A) ∈ Pcl,bd(E). Let X be a nonempty compact and connected metric space and let F : X → Pcp(X) be a multivalued ε-contractive map, F has a fixed point. Chen and Chang obtained some fixed point theorems for KKM type set contraction mappings in various spaces [9,10,11,12].

Generalized Set Contractions

Asymptotic Generalized Set Contractions

Some Applications in KKM Theory