LAUSR

In this note, firstly, we introduce the notion of the $\mathrm{R}$-complete metric spaces. This notion let us to consider fixed point theorem in $\mathrm{R}$-complete instead of complete metric spaces. Secondly, as motivated by the recent work of Amini-Harandi (Fixed Point Theory Appl. 2012, 2012:215), we explain a new generalized contractive condition of set-valued mappings and prove a fixed point theorem in $\mathrm{R}$-complete metric spaces which extends some well-known results in the literature. Finally, some examples are also given to support our main theorem.