On fixed points of quasi-contraction type multifunctions

Abstract

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 , 1 2 ) . Later, Haghi, Rezapour and Shahzad proved same results without the additional assumption and for 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 , 1 2 ) (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.