Some discussion on the existence of common fixed points for a pair of maps

In this paper, the concepts of conditionally sequential absorbing and pseudo-reciprocal continuous maps are introduced in connection to giving a brief discussion on the role of various types of commutativity (e.g., weakly compatible, occasionally weakly compatible, subcompatible, pseudo-compatible, etc.) and continuity-type conditions (e.g., reciprocal, weak reciprocal, g-reciprocal, conditionally reciprocal, subsequential and sequential continuity of type and ) in the context of existence of common fixed points of a pair of maps. Here, the utility of newly introduced maps (i.e., conditionally sequential absorbing and pseudo-reciprocal continuous) in view of common fixed points for a pair of maps satisfying contractive as well as nonexpansive Lipschitz-type conditions is shown. MSC: 47H10, 54H25.