UNIFYING A MULTITUDE OF COMMON FIXED POINT THEOREMS EMPLOYING AN IMPLICIT RELATION

Abstract

A general common fixed point theorem for two pairs of weakly compatible mappings using an implicit function is proved without any continuity requirement which generalizes the result due to Popa (20, Theorem 3). In process, several previously known results due to Fisher, Kannan, Jeong and Rhoades, Imdad and Ali, Imdad and Khan, Khan, Shahzad and others are derived as special cases. Some related results and illustrative examples are also discussed. As an application of our main result, we prove an existence theorem for the solution of simultaneous Hammerstein type integral equations. As established in Jungck (12), a common fixed point theorem in metric spaces generally involves conditions on contraction, commutativity and continuity of the involved mappings besides a suitable containment of range of one mapping into the range of other. To prove a new metrical common fixed point theorem one is always required to improve one or more of these conditions. Of all these four conditions, the means of improving contraction condition to prove new results on fixed and common fixed point is much discussed which continue to attract the attention of the researchers of this domain. But in this paper we use an implicit function due to Popa (20). The tradition of improving commutativity condition in common fixed point theorems was initiated by Sessa (21). Inspired by the definition of weak com- mutativity of Sessa (21), researchers of this domain, introduced several def- initions of weak commutativity such as: Compatible mappings, Compatible mappings of type (A), Compatible mappings of type (B), Compatible map- pings of type (P), Compatible mappings of type (C), Biased maps, R-weakly commuting mappings and some others whose lucid comparison and illustration