Abstract

In this Der, we show the existence of solutions of functional equations fx∈sx∩tx and x=fx∈sx∩Tx under certain contraction and asymptotic regularity conditions, where f, S and T are single-valued and multl-valued mappings on a metric space, respectively. We also observe that MukherJee's fixed point theorem for a single-valued mapping commuting with a multl-valued mapping admits of a counterexample and suggest some modifications. While doing so, we also answer an open question raised in [I] and [2]. Moreover, our results extend and unify a multitude of fixed point theorems for multi-valued mappings.