Solutions of boundary value problems at resonance with periodic and antiperiodic boundary conditions

Abstract

We study the existence of solutions of the second-order boundary value problem at resonance $u^{\prime \prime }=f\left(t,u,u^{\prime }\right)$ satisfying the boundary conditions $u\left(0\right)+u\left(1\right)=0$, $u^{\prime }\left(0\right)-u^{\prime }\left(1\right)=0$, or $u\left(0\right)-u\left(1\right)=0$, $u^{\prime }\left(0\right)+u^{\prime }\left(1\right)=0$. We employ a shift method, making a substitution for the nonlinear term in the differential equation so that these problems are no longer at resonance. Existence of solutions of equivalent boundary value problems is obtained, and these solutions give the existence of solutions of the original boundary value problems.