Orthogonal sets: Coincidence and fixed point theorems in incomplete metric spaces

In this paper, as motivated by a work of Daffer et al. [6], we state and prove some theorems for set valued mappings and by them we conclude the existence of coincidence points and fixed points of a general class of set valued mappings satisfying a new generalized contractive condition which extends some well-known results in the literature. For this reason, firstly, by using a recent work of Eshaghi et al [11], we define the notion of orthogonal sets and by the notion, we consider our results in strongly orthogonal complete metric spaces (not necessarily complete metric spaces). In addition, this article has a new and different view on the subject and consists of several non-trivial examples which signify the motivation of such investigations. Also, in the end of this paper, by using our examples, we show that our results are real generalization of the previous results in the literature.