Periodic solutions to non-autonomous second-order systems

where x = (x1, . . . , xn), V (t, x) = (v1(t, x), . . . , vn(t, x)) ∈ C(R× Rn,Rn) is periodic of period T in the t variable, and μ is a constant. The existence of solutions of (1.1) has been studied by many researchers, see Mawhin andWillem [1] and the references therein. The variational method has beenmostly used to prove the existence of solutions of (1.1). Fixed point theorems such as Rothe’s theorem can also be used to prove the existence of solutions of (1.1). In this paper, we choose the fixed point theorem in cones [2,3] to establish the existence of a solution for (1.1). We believe that the fixed point theorem in cones can be further used to treat other cases of this problem. The fixed point theorem in cones has been employed to establish the existence of positive solution boundary value problems with some superlinear and sublinear assumptions at zero and infinity, see e. g. Erbe and the author [4], Torres [5], the author [6], Graef, Kong and the author [15]. Systems of differential equations can be treated similarly by constructing appropriate product spaces [7–10]. In a recent paper, O’Regan and the author [11] obtained the existence, multiplicity and nonexistence of positive periodic solutions of general second-order systems. Precup [12,13] gave a vector version of the fixed point theorem in cones and applications to systems of equations whose components have different sublinear or superlinear behaviors. In this paper,whilewe assume that all components are sublinear at infinity, existence results may be established by using the theorems in [12]. It would be interesting to address it in future research.