Some Fixed Point Theorems for Multivalued Maps Satisfying an Implicit Relation on Metrically Convex Spaces

Abstract

Abstract. In this paper, we give some fixed point theorems for multivalued maps satis-fying an implicit relation on metrically convex spaces. Our results extend and generalizesome fixed point theorem in the literature. 1. IntroductionIn recent years several fixed point results have been obtained on metrically con-vex spaces. Assad and Kirk [6] gave a sufficient condition enunciating fixed pointof set-valued mappings enjoying specific boundary condition in metrically convexmetric spaces. A significant generalization of the fixed point theorem of Assad [5]and the theorem of Assad and Kirk [6] for multivalued contraction non-self map-pings is obtained by Itoh [17] in 1977. In the current years the work due to Assadand Kirk [6] has inspired extensive activities which includes Ahmad and Imdad [1],[2], Imdad et al. [13], Imdad and Ali [14], Itoh [17], Khan [19] and some others.Most recently, Dhage et al. [9] and Huang and Cho [12] proved some fixed pointtheorems for a sequence of set-valued mappings which generalize several results dueto Ahmad and Khan [3], Itoh [17], Khan [19] and others. The purpose of this pa-per is to prove some coincidence and common fixed point theorems for multivaluedmappings satisfying an implicit relation on metrically convex spaces. Our resultseither partially or completely generalize earlier results due to Ahmad and Imdad[1], [2], Ahmad and Khan [3], Ciri´c [7], Imdad and Khan [15], Itoh [17], Khan [19],´Khan et al. [20], Rhoades [26] and several others. See also the related Theorem 3.1of [14].