On Ćirić maps with a generalized contractive iterate at a point and Fisher’s quasi-contractions in cone metric spaces

In this paper, we generalize and unify some results of Sehgal and Guseman, and Ćirić’s theorem for mappings with a generalized contractive iterate at a point to cone metric spaces, in which the cone does not need to be normal. As corollaries, we obtain recent results of Huang and Zhang, and Raja and Vaezpour. Furthermore, we introduce the definition of Fisher quasi-contractions on cone metric spaces and study their properties. Among other things, using new method of proof, we solve the open problem for the interval of contractive constant λλ of (Ćirić) quasi-contraction in non-normal cone metric spaces, and as sn immediate corollary, we recover the recent result of Rezapour and Hamlbarani.