Abstract. In this paper the concept of set-valued cyclic Meir-Keeler contraction map is introduced. The existence of best proximity point for such maps on a metric space with the UC property is presented.

In this paper, we study the existence and uniqueness of best proximity points for cyclic Meir-Keeler contraction mappings in metric spaces with the property W-WUC. Also, the existence of best proximity points for set-valued cyclic Meir-Keeler contraction mappings in metric spaces with the property WUC are obtained

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By introducing a new concept called "set-valued asymptotic contraction of final type" in metric spaces, the existence and uniqueness of compact fixed sets for such mappings have been obtained. Furthermore, by adding additional conditions, we prove the existence and uniqueness of endpoints for these maps.