PERIODIC SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS

The periodic boundary value problems of a class of nonlinear differential equations are investigated.

Inspired by the work of Li [5], we give sufficient conditions for the existence and uniqueness of the 2π -periodic solution of (1.1) by using the initial value problem method and homeomorphism of R n to R n .

Initial value problem and eigenvalues problem.
Throughout this paper, we denote the interval [0, 2π] and M n denotes the set of all complex n × n matrices.We also assume the solutions of (1.1) exist on I for any initial value (x(0), x (0)) ∈ R 2n .
Let us consider the initial value problem Lemma 2.1 [3,4].Assume that g ∈ C(I × R n , R n ) and possesses continuous partial derivatives ∂g/∂u on I × R n .Let the solution u 0 (t) = u(t, t 0 ,u 0 ) of (2.1) exist for t ∈ I and let exists and is the solution of such that φ(t 0 ,t 0 ,u 0 ) is the unit matrix.
Lemma 2.2 [2].Suppose A ∈ M n .Then λ is an eigenvalue of the matrix A if and only if exp λ is an eigenvalue of the matrix exp A.
Lemma 2.4 [7].If A ≥ B ≥ 0, and A and B are two real symmetric n × n matrices, where Lemma 2.5 [6].Assume that Proof.From the following matrices equality we obtain the result of Lemma 2.6 immediately.

Main results.
Rewrite (1.1) as follows where b(t)y =: Consider the variation equation of (3.2) with respect to u ξ = ∂g(t, u) ∂u ξ, ( where Then Z(t) is a fundamental solution matrix of (3.3) and Z(0) = I 2n .Meanwhile, by Lemma 2.1 we know that is also a fundamental solution matrix of (3.3).Therefore Therefore, solving periodic solution of (1.1) is equivalent to finding the fixed points of h(v) or the zero points of H(v).From (3.8) and (3.9) (3.10) Proof.By Lemma 2.5, we need only to show that H (V ) is invertible and that there exists a constant M > 0 such that [H (v)] −1 ≤ M.