On a generalization of Lazer-Leach conditions for a system of second order ODE's (Q1029919)

The authors consider the periodic boundary value problem associated to a nonlinear system of second order equations of the form \[ u''+m^2u+g(u)=p(t), \] where \(m\neq 0\) is an integer, \(p\in L^2([0,2\pi],{\mathbb R}^n)\) and \(g\) is continuous and bounded. In the case \(n=1\) it was proved by \textit{A. C. Lazer} and \textit{D. E. Leach} [Ann. Mat. Pura Appl., IV. Ser. 82, 49--68 (1969; Zbl 0194.12003)] that if \(g\) has limits at infinity and \(\sqrt{\alpha_p^2+\beta_p^2} < (2/\pi)|g(+\infty)-g(-\infty)| (\alpha_p,\beta_p\) being the \(m\)-th Fourier coefficients of \(p\)) then there exists at least one solution. In the present paper, the authors generalize this result to the case \(n>1.\) Using an abstract result, appropriate conditions of Lazer-Leach type are given which ensure the solvability of the given problem. These conditions are expressed in terms of uniform radial limits of \(g\) and the \(m\)-th Fourier coefficients of some suitable extensions of \(g\) to the infinite sphere. The proofs are performed in the framework of coincidence degree theory; more precisely, it is applied Mawhin's continuation theorem.