In 2012, Samet, Vetro and Vetro introduced α-ψ-contractive mappings and gave some results on a fixed point of the mappings (Samet et al. in Nonlinear Anal. 75:2154-2165, 2012). In fact, their technique generalized some ordered fixed point results (see (Alikhani et al. in Filomat, 2012, to appear) and (Samet et al. in Nonlinear Anal. 75:2154-2165, 2012)). By using the main idea of (Samet et al. in Nonlinear Anal. 75:2154-2165, 2012), we give some new results for α-ψ-Ciric generalized multifunctions and some related self-maps. Also, we give an affirmative answer to a recent open problem which was raised by Haghi, Rezapour and Shahzad in 2012.

Recently Samet, Vetro and Vetro introduced the notion of α-ψ-contractive type mappings and established some fixed point theorems in complete metric spaces. In this paper, we introduce the notion of α ∗ - ψ -contractive multifunctions and give a fixed point result for these multifunctions. We also obtain a fixed point result for self-maps in complete metric spaces satisfying a contractive condition.