### 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.

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