In this paper, firstly, we introduce the notion of R-complete metric spaces. This notion let us to consider fixed point theorem in 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 for set-valued mappings and prove a fixed point theorem in R-complete metric spaces which extends some well-known results in the literature. Finally, some examples are given to support our main theorem and also we find the existence of solution for a first-order ordinary differential equation.

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.

In this study, our goal is to apply a new fixed point method to prove the Hyers-Ulam-Rassias stability of a quadratic functional equation in normed spaces which are not necessarily Banach spaces. The results of the present paper improve and extend some previous results.

In this paper, using the conditions of Taleb-Hanebaly’s theorem in a modular space where the modular is s-convex and symmetric with respect to the ordinate axis, we prove a new generalized modular version of the Schauder and Petryshyn fixed point theorems for nonexpansive mappings in s-convex sets. Our results can be applied to a nonlinear integral equation in Musielak-Orlicz space $L^p$ where $0<p\le 1$ and $0<s\le p$ .

In this paper, we prove convergence theorems for viscosity approximation processes involving $*$ −nonexpansive multi-valued mappings in complete convex metric spaces. We also consider finite and infinite families of such mappings and prove convergence of the proposed iteration schemes to common fixed points of them. Our results improve and extend some corresponding results.