Abstract

Let f:[0,π]×ℝN→ℝN, (N≥1) satisfy Caratheodory conditions, e(x)∈L1([0,π];ℝN). This paper studies the system of nonlinear Neumann boundary value problems x″(t)+f(t,x(t))=e(t), 0<t<π, x′(0)=x′(π)=0. This problem is at resonance since the associated linear boundary value problem x″(t)=λx(t), 0<t<π, x′(0)=x′(π)=0, has λ=0 as an eigenvalue. Asymptotic conditions on the nonlinearity f(t,x(t)) are offered to give existence of solutions for the nonlinear systems. The methods apply to the corresponding system of Lienard-type periodic boundary value problems.