A probabilistic Meir-Keeler type fixed point theorem which characterizes metric completeness

A probabilistic version of the Meir-Keeler type fixed point theorem, which characterizes completeness of the metric space is established. In addition to it, a fixed point theorem for non-expansive mappings satisfying type condition in Menger probabilistic metric space (Menger PM-space) is proved. As a byproduct we find an affirmative answer to the open question on the existence of contractive mappings which admit discontinuity at the fixed point (see Rhoades, B. E., Contractive definitions and continuity, Contemporary Mathematics 72 (1988), 233–245, p. 242) in the setting of Menger PM-space.