COMMON FIXED POINT POINT THEOREM FOR A PAIR OF WEAKLY COMPATIBLE MAPPINGS ON BANACH SPACES

Abstract

Abstract. In this paper, we prove a common xed point theorem fornoncompatible, discontinuous mappings on a Banach space. Our maintheorem extends, improves, generalizes some known results on Banachspaces. 1. Introduction and PreliminariesHusain and Sehgal [1] proved common xed point theorems for a familyof mappings. Khan and Imdad [5] extended result of Husain and Sehgal [1].Imdad, Khan and Sessa [2] extended above results and proved common xedpoints for three mappings de ned on a closed subset of a uniformly convexBanach space.Rashwan [9] extended result of Imdad, Khan and Sessa [2] by employingfour compatible mappings of type (A) instead of weakly commuting mappingsand by using one continuous mapping as opposed to two.Sharma and Bamboria [12] improved results of Rashwan [9] by removingthe condition of continuity and replacing the compatibility of mappings of type(A) by weak compatibility. Sharma and Tilwankar [13] proved a common xedpoint theorem for four mappings under the condition of weak compatiblemappings by using the new property (E.A).For the study of discontinuous and noncompatible mappings in xed pointtheory we refer to Sharma and Deshpande [14], [15] and Sharma, Deshpandeand Tiwari [16]. Several observations motivated us to prove common xed pointtheorem for noncompatible, discontinuous mappings on a Banach space. Ourmain theorem extends, improves, generalizes some known results on Banachspaces.Now, we begin with some known de nitions: