Common Fixed Points for Nonexpansive Type Mappings

Vyomesh Pant

The aim of the present paper is to obtain common fixed point theorems for three mappings satisfying nonexpansive type condition. For this purpose we use the notion of pointwise R-weak commutativity or R-weak commutativity of type $(A_{g})$ but without assuming the completeness of the space or continuity of the mappings involved. We further generalize the results obtained in first three theorems by replacing the condition of noncompatibility of maps with the property (E.A). 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 are, thus, able to provide one more answer to the problem posed by Rhoades [13] regarding the existence of contractive definition which is strong enough to generate fixed point but does not forces the maps to become continuous. In Theorem 6, we use the notion of conditionally commuting maps recently introduced by Pant and Pant [12] and prove a common fixed point theorem under minimal commutative condition.