Lipschitz Type Analogue of Strict Contractive Conditions and Common Fixed Points

Ravindra K. Bisht

Abstract

In this paper, we obtain a common xed point theorem for a pair of self mappings satisfying a Lipschitz type analogue of strict contractive condition by using a relatively new notion of conditional reciprocal continuity wherein we never require conditions on the completeness of the space, noncompatibility or property (E.A.), continuity of any mapping and completeness (or closedness) of the range of any one of the involved mappings. Our results substantially improve the results of Pant [Discontinuity and fixed points, J. Math. Anal. Appl. 240(1999) 284--289], Pant and Pant [Common xed points under strict contractive conditions, J. Math. Anal. Appl. 248 (2000) 327--332], Imdad et al. [Coincidencefixed points in symmetric spaces under strict contractions, J. Math. Anal. Appl. 320 (2006) 352--360] and Jin-Xuan and Yang [Common fixed point theorems under strict contractive conditions in Menger spaces, Nonlinear Anal. 70 (2009) 184--193].