Common fixed points under Lipschitz type condition
Bull. Korean Math. Soc. 2008 Vol. 45, No. 3, 467-475 Printed September 1, 2008
Abstract : The aim of the present paper is three fold. Firstly, we obtain common fixed point theorems for a pair of selfmaps satisfying nonexpansive or Lipschitz type condition by using the notion of pointwise $R$-weak commutativity but without assuming the completeness of the space or continuity of the mappings involved (Theorem 1, Theorem 2 and Theorem 3). Secondly, we generalize the results obtained in first three theorems for four mappings by replacing the condition of noncompatibility of maps with the property (E.A) and using the $R$-weak commutativity of type ($A_g$) (Theorem 4). Thirdly, in Theorem 5, we show that if the aspect of noncompatibility is taken in place of the property (E.A), the maps become discontinuous at their common fixed point. We, thus, provide one more answer to the problem posed by Rhoades  regarding the existence of contractive definition which is strong enough to generate fixed point but does not forces the maps to become continuous.
Keywords : Lipschitz type mapping pairs, nonexpansive conditions, noncompatible mappings, pointwise $R$-weak commutativity, $R$-weak commutativity of type $(A_g)$, contractive conditions, property (E.A)