Generalized Krasnoselskii fixed point theorem involving auxiliary functions in bimetric spaces and application to two-point boundary value problem

Keywords: Bimetric space Fixed point Coincidence point Two-point boundary value problem a b s t r a c t In this paper, we introduce a generalized contraction of Krasnoselskii-type using auxiliary functions, and obtained some sufficient conditions for existence and uniqueness of fixed point for such mappings on bimetric spaces. We also establish a result on coincidence point of two mappings, and derive several corollaries of our main theorems. As application, we establish an existence result for a two-point boundary value problem of second order differential equation. Fixed point theory is of an intrinsic theoretical interest but it is also a useful tool for studying a wide class of practical problems. In particular, there is an exhaustive variety of results concerning fixed point theory in both Banach spaces and metric spaces which are subject to different types of contractive conditions (see [1–16]). In [13], Jachymski shows the equivalence between eight contractive definitions. The contraction of Krasnoselskii is one among them, therefore a generalization of Krasnoselskii result implies also the generalization of all others results. In this paper, we introduce a generalized contraction of Krasnoselskii-type and we prove fixed and common fixed point results for such contractive mappings in the setting of bimetric spaces. Several interesting corollaries are also obtained. The results presented in this paper generalize and extend several well-known results in the literature. As application, we establish an existence result for a second order differential equation. We recall Maia's fixed point theorem: Theorem 1.1 [15]. Let ðX; d; dÞ be a bimetric space. Assume that for T : X ! X, the following conditions are satisfied: