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

Abstract

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 π -periodic solution under a rather weaker condition, for existence and uniqueness, than those given in papers [J. Chen, W. Li, Periodic solution for 2 k th boundary value problem with resonance, J. Math. Anal. Appl. 314 (2006) 661–671; F. Cong, Periodic solutions for 2 k th 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 2 k th 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, Int. J. Math. Math. Sci. 17 (2003) 1101–1110; W. Li, Z. Shen, A constructive proof of existence and uniqueness of 2 π -periodic solution to Duffing equation, Nonlinear Anal. 42 (2000) 1209–1220]. This result extends the results known so far.