A GENERALIZED COMMON FIXED POINT THEOREM FOR TWO PAIRS OF WEAKLY COMPATIBLE SELF-MAPS

Abstract

A common fixed point theorem of Rahman et al. [3] has been generalized under a weaker inequality through the notion of weak compatibility by dropping continuity and restricting the completeness of the space.

Keywords

Compatible and compatible maps of type A, A- and S-Compatible maps, weakly compatible self-maps and common fixed point.

INTRODUCTION

II. MAIN RESULT

Remark 2. The completeness of the space X is restricted to that of its subspace, namely T(X)S(X) (cf. Condition (c) of Theorem 2). Condition (b) imples (d) in view of Remark 1. Further the inequality (5) implies (7) with the choice q a b 2c e f . It is important to note that the continuity of none of the four maps is needed to obtain a fixed point. Thus Theorem 2 is a generalization of Theorem 1.

References

1. [1] Gerald Jungck, Compatible mappings and common fixed points, Int. Jour. Math. & Math. Sci, 9 (1986), 771-779.
2. [2] Gerald Jungck and B. E. Rhoades, Fixed point for set-valued functions with out continuity, Indian J. pure appl. Math. 29 (3) (1998), 227-238.
3. [3] S. Rahman, Y. Rohen and M. P. Singh, Generalized common fixed point Theorems of A-compatibility and S- compatible mappings, Amer. J. Appl. Math. and Stat. 1(2), (2013), 27-29.