On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay

Abstract

For each x0∈[0,2π) and k∈N, we obtain some existence theorems of periodic solutions to the two-point boundary value problem u′′(x)+k2u(x-x0)+g(x,u(x-x0))=h(x) in (0,2π) with u(0)-u(2π)=u′(0)-u′(2π)=0 when g:(0,2π)×R→R is a Caratheodory function which grows linearly in u as u→∞, and h∈L1(0,2π) may satisfy a generalized Landesman-Lazer condition (1+sign(β))∫02πh(x)v(x)dx0gβ+(x)vx1-βdx+∫v(x)<0gβ-(x)vx1-βdx for all v∈N(L)\{0}. Here N(L) denotes the subspace of L1(0,2π) spanned by sin⁡kx and cos⁡kx, -1<β≤0, gβ+(x)=lim  infu→∞(gx,uu/u1-β), and gβ-(x)=lim  infu→-∞(gx,uu/u1-β)