Journal of Applied Mathematics and Simulation 
Volume 2 (1989), Issue 3, Pages 169-184
doi:10.1155/S1048953389000134

Nonlinear second order system of Neumann boundary value problems at resonance

Chaitan P. Gupta1,2

 1Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439-4801, USA
2Department of Mathematical Science, Northern Illinois University, DeKalb, IL 60115, USA

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.