A GENERAL FIXED POINT THEOREM

Abstract

In this paper we prove a theorem which ensures the existence of a unique fixed point and is applicable to contractive type mappings as well as mappings which do not satisfy any contractive type condition. Our theorem contains the well known fixed point theorems respectively due to Banach, Kannan, Chatterjea, \'{C}iri\'c and Suzuki as particular cases; and is independent of Caristi's fixed point theorem. Moreover, our theorem provides new solutions to Rhoades problem on discontinuity at the fixed point as it admits contractive mappings which are discontinuous at the fixed point. It is also shown that the weaker form of continuity employed by us is a necessary and sufficient condition for the existence of the fixed point.