On fixed points of quasi-contraction type multifunctions

In 2009, Ilić and Rakoc˘ević proved that quasi-contraction maps on normal cone metric spaces have a unique fixed point (Ilić and Rakoc˘ević, 2009 [6]). Then, Kadelburg, Radenović and Rakoc˘ević generalized their results by considering an additional assumption (Kadelburg et al., 2009 [7]). Also, they proved that quasi-contraction maps on cone metric spaces have the property (P) whenever λ∈(0,12). Later, Haghi, Rezapour and Shahzad proved same results without the additional assumption and for λ∈(0,1)λ∈(0,1) by providing a new technical proof (Rezapour et al., 2010 [4]). In 2011, Wardowski published a paper (Wardowski, 2011 [8]) and tried to test fixed point results for multifunctions on normal cone metric spaces. Of course, he used a special view in his results. Recently, Amini-Harandi proved a result on the existence of fixed points of set-valued quasi-contraction maps in metric spaces by using the technique of Rezapour et al. (2010) [4]. But, like Kadelburg et al. (2009) [7], he could prove it only for λ∈(0,12) (Amini-Harandi (2011) [3]). In this work, we prove again the main result of Amini-Harandi (2011) [3] by using a simple method. Also, we introduce quasi-contraction type multifunctions and show that the main result of Amini-Harandi (2011) [3] holds for quasi-contraction type multifunctions.