Conditional reciprocal continuity and common fixed points

The aim of the present paper is to obtain common fixed point theorems by employing the recently introduced notion of conditional reciprocal continuity. We demonstrate that conditional reciprocal continuity ensures the existence of fixed points under contractive conditions which otherwise do not ensure the existence of fixed points. Our results generalize and extend several well-known fixed point theorems in the setting of metric spaces. We also provide more answers to the open problem posed by B. E. Rhoades [Contractive Definitions and Continuity, Contemporary Math. 72 (1988), 233-245] regarding existence of a contractive condition which is strong enough to generate a fixed point, but which does not force the map to be continuous at the fixed point.