On End and Coupled Endpoints of $\theta$-$F$-Contractive Set-Valued ‎Mappings‎

Subject Areas : International Journal of Industrial Mathematics

B.  Mohammadi ¹ (Department of Mathematics, Marand Branch, Islamic Azad University, Marand, Iran.)
A.  Alizadeh ² (Department of Mathematics, Marand Branch, Islamic Azad University, Marand, Iran.)

Received: 2017-04-21 Accepted : 2017-04-21 Published : 2017-04-01

Keywords: Endpoint, Coupled fixed point, Coupled endpoint, Set-valued ‎mappings.‎, $theta$-$F$-Contractive,

Abstract :

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 $M \times M$, 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.‎