On End and Coupled Endpoints of θ-F-Contractive Set-Valued Mappings

In this paper, we introduce a new concept in set-valued mappings which we have called condition (UHS). Then, adding this condition to a new type of contractive set-valued mappings, recently has been introduced by Amini-Harandi [Fixed and coupled fixed points of a new type contractive set-valued mapping in complete metric spaces, Fixed point theory and applications, 215 (2012)], we prove that this mapping have a unique end point. Then, we state and prove a result about existence of coupled fixed point of this type of contractive set-valued mappings defined on M ×M, where M is a complete metric space (Recently, Amini-Harandi proved existence of coupled fixed point only for self mappings). Finally, we introduce one another new concept, which we have called condition (UHS)?. Then, adding this condition we state and prove existence of coupled endpoint for such contractive set-valued mappings. Some examples are given to illustrate the results.