Fixed point theorems for α-set-valued quasi-contractions in b-metric spaces

Recently, Samet et al. [B. Samet, C. Vetro and P. Vetro, Fixed point theorems for alpha-psi-contractive type mappings, Nonlinear Anal., 75 (2012), 2154-2165] introduced the notion of alpha-psi-contractive mappings and established some fixed point results in the setting of complete metric spaces. In this paper, we introduce the notions of alpha-set-valued contraction and alpha-set-valued quasi-contraction and we give some fixed point theorems for such classes of mappings in the setting of b-metric spaces and ordered b-metric spaces. The presented theorems extend, unify and generalize several well-known comparable results in the existing literature.