Existence and multiplicity of periodic solutions for a second-order ODE at resonance with an Ahmad–Lazer–Paul condition

Abstract

In this work, we study the following resonant boundary value problem (1)−u′′(x)=g(x,u(x))+f(x);x∈(0,2π)u(0)=u(2π)u′(0)=u′(2π),where g:[0,2π]×R→R is continuous and bounded, and f∈L2(0,2π) is assumed to have mean-value zero. Using a variational approach and infinite-dimensional Morse theory, we prove existence and multiplicity of periodic solutions of the boundary problem (1), where the nonlinearity g satisfies an Ahmad–Lazer–Paul condition.