An application of a global inversion theorem to an existence and uniqueness theorem for a class of nonlinear systems of differential equations

In this paper, a class of systems of nonlinear differential equations at resonance is considered. With the use of a global inversion theorem which is an extended form of a non-variational version of a max-min principle, we prove that this class of equations possesses a unique 2 pi-periodic solution under a rather weaker condition, for existence and uniqueness, than those given in papers [J. Chen, W. Li. Periodic solution for 2kth boundary value problem with resonance, J. Math. Anal. Appl. 314 (2006) 661-671; F. Cong, Periodic Solutions for 2kth order ordinary differential equations with nonresonance, Nonlinear-Anal. 32 (1998) 787-793: F. Cong. Periodic solutions for second order differential equations, Appl. Math. Lett. 18 (2005) 957-961; W. Li, Periodic Solutions for 2kth order ordinary differential equations with resonance, J. Math. Anal. Appl. 259 (2001) 157-167: W. Li, H. Li, A min-max theorem and its applications to nonconservative systems, lilt. J. Math. Math. Sci. 17 (2003) 1101-1110: W. Li, Z. Shen, A constructive proof of existence and uniqueness 2 pi-periodic Solution to Duffing equation, Nonlinear Anal. 42 (2000) 1209-1220]. This result extends the results known so far. (c) 2008 Elsevier Ltd. All rights reserved.