On a probabilistic version of Meir-Keeler type fixed point theorem for a family of discontinuous operators

<p>A Meir-Keeler type fixed point theorem for a family of mappings is proved in Menger probabilistic metric space (Menger PM-space). We establish that completeness of the space is equivalent to fixed point property for a larger class of mappings that includes continuous as well as discontinuous mappings. In addition to it, a probabilistic fixed point theorem satisfying (ϵ - δ) type non-expansive mappings is established.</p>

A probabilistic Meir-Keeler type fixed point theorem whichcharacterizes metric completeness

A probabilistic version of the Meir-Keeler type fixed point theorem, which characterizes completeness of themetric space is established. In addition to it, a fixed point theorem for non-expansive mappings satisfying(−δ)type condition in Menger probabilistic metric space (Menger PM-space) is proved. As a byproduct we find anaffirmative answer to the open question on the existence of contractive mappings which admit discontinuity atthe fixed point (see Rhoades, B. E.,Contractive definitions and continuity, Contemporary Mathematics72(1988),233–245, p. 242) in the setting of Menger PM-space.

Multivalued operator with respect generalized distance on Menger probabilistic metric spaces

In this paper, we recall the concept of r-distance on a Menger probabilistic metric space. Further we prove a fixed point theorem for contractive type multi-valued operators in terms of a r-distance on a complete Menger probabilistic metric space.