Coincidence and Fixed Points for Multivalued Mappings in Incomplete Metric Spaces with Applications

Abstract

In the present paper, firstly, we review the notion of
$\mathrm{R}$-complete metric spaces, where $\mathrm{R}$ is a binary
relation (not necessarily a partial order). This notion lets us to
consider some fixed point theorems for multivalued mappings in
incomplete metric spaces. Secondly, as motivated by the recent work
of Wei-Shih Du (On coincidence point and fixed point theorems for
nonlinear multivalued maps, {\it Topology and its Applications} 159
(2012) 49--56), we prove existence of coincidence points and fixed
points of a general class of multivalued mappings satisfying a new
generalized contractive condition in $\mathrm{R}$-complete metric
spaces which extends some well-known results in the literature. In
addition, this article consists of several non-trivial examples
which signify the motivation of such investigations. Finally, we
give an application to the nonlinear fractional boundary value
equations.