Nonlinear second order system of Neumann boundary value problem at resonance

Let f:[0,;]×R^N⇒R^N, (N≥1) satisfying Carathéodory conditions, e(x)∈L¹([0,;];R^N). This paper studies the system of nonlinear Neumann boundary value problems x∈(t)+f(t,x(t)=e(t),0∈t∈;, with x'(0)=x'(;)=0. This problem is at resonance since the associated linear boundary value problem x∈(t)=λx(t), 0∈t∈;, with x'(0)=x'(;)=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.