Fixed point theorem for question of set-valued quasi-contraction

Abstract

In this work, we give a partial positive answer to the question concerning the set-valued quasi-contraction proposed by Amini-Harandi (Appl. Math. Lett. 24:1791--1794 2011). By a useful lemma,  we prove a fixed point theorem for the set-valued quasi-contraction, which extends the range of contraction constant in result of Amini-Harandi from $\left[0, \frac{1}{2}\right)$ to $\left[0, \frac{1}{\sqrt[3]{3}}\right)$. Also, we give a new simple proof for the result of quasi-contraction type proposed by Haghi et al. (Appl. Math. Lett. 25:843--846 2012).  Finally, a counterexample and a theorem concerning cyclic set-valued mapping are given, which improve some recent results.