A Note on Periodic Solutions of Second Order Nonautonomous Singular Coupled Systems

We establish the existence of periodic solutions of the second order nonautonomous singular coupled systems for a.e. , for a.e. . The proof relies on Schauder's fixed point theorem. 1. Introduction Some classical tools have been used in the literature to study the positive solutions for two-point boundary value problems of a coupled system of differential equations. These classical tools include some fixed point theorems in cones for completely continuous operators and Leray-Schauder fixed point theorem; for examples, see [1–3] and literatures therein. Recently, Schauder's fixed point theorem has been used to study the existence of positive solutions of periodic boundary value problems in several papers; see, for example, Torres [4], Chu et al. [5, 6], Cao and Jiang [7], and the references contained therein. However, there are few works on periodic solutions of second-order nonautonomous singular coupled systems. In these papers above, there are the major assumption that their associated Green's functions are positive. Since Green's functions are positive, in the paper, we continue to study the existence of periodic solutions to second-order nonautonomous singular coupled systems in the following form: with , Here we write if is an -caratheodory function, that is, the map is continuous for a.e. and the map is measurable for all and for every there exists such that for all and a.e. ; here “for a.e." means “for almost every". This paper is mainly motivated by the recent papers [4–6, 8, 9], in which the periodic singular problems have been studied. Some results in [4–6, 9] prove that in some situations weak singularities may help create periodic solutions. In [6], the authors consider the periodic solutions of second-order nonautonomous singular dynamical systems, in which the scalar periodic singular problems have been studied by Leray-Schauder alternative principle, a well-known fixed point theorem in cones, and Schauder's fixed point theorem, respectively. The remaining part of the paper is organized as follows. In Section 2, some preliminary results will be given. In Sections 3–5, by employing a basic application of Schauder's fixed point theorem, we state and prove the existence results for (1.1) under the nonnegative of the Green's function associated with (2.1)-(2.2). Our view point sheds some new light on problems with weak force potentials and proves that in some situations weak singularities may stimulate the existence of periodic solutions, just as pointed out in [9] for the scalar case. To illustrate our results, for example, we can select the