A Fixed Point Theorem for a New Class of Set-Valued Mappings in R-Complete (Not Necessarily Complete) Metric Spaces

Abstract

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.