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

Abstract

For eachx0∈[0,2π)andk∈N, we obtain some existence theorems of periodic solutions to the two-point boundary value problemu′′(x)+k2u(x-x0)+g(x,u(x-x0))=h(x)in(0,2π)withu(0)-u(2π)=u′(0)-u′(2π)=0wheng:(0,2π)×R→Ris a Caratheodory function which grows linearly inuasu→∞, andh∈L1(0,2π)may satisfy a generalized Landesman-Lazer condition(1+sign(β))∫02πh(x)v(x)dx<∫v(x)>0gβ+(x)vx1-βdx+∫v(x)<0gβ-(x)vx1-βdxfor allv∈N(L)\{0}. HereN(L)denotes the subspace ofL1(0,2π)spanned bysin⁡kxandcos⁡kx,-1<β≤0,gβ+(x)=lim  infu→∞(gx,uu/u1-β), andgβ-(x)=lim  infu→-∞(gx,uu/u1-β).