Some results on set-valued contractions in abstract metric spaces

In this paper, the concept of contractive set-valued maps in the frame of abstract metric spaces is studied and the existence of fixed points for such maps is guaranteed under certain conditions. Consequently, several known fixed point results are either generalized or extended, including the corresponding recent fixed point results of Wardowski [D. Wardowski, Endpoints and fixed points of set-valued contractions in cone metric spaces, Nonlinear Anal. 71 (2009) 512–516] as well as Klim and Wardowski [D. Klim, D. Wardowski, Dynamic process and fixed points of set-valued nonlinear contractions in cone metric spaces, Nonlinear Anal. 71 (2009) 5170–5175]. Examples are given to show that our results are distinct from the existing ones.