In the present paper, firstly, we review the notion of R-complete metric spaces, where 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, Topology and its Applications 159 (2012) 49–56), we prove the existence of coincidence points and fixed points of a general class of multivalued mappings satisfying a new generalized contractive condition in 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