LAUSR

In this work, we give a partial positive answer to the question concerning the set-valued quasicontraction 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.