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 0, 1 2 to 0, 1 3 √ 3. 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.