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

Abstract

For each $x_{\mathrm{0}}\in [\mathrm{0,2}\pi )$ and $k\in \mathbf{N}$, we obtain some existence theorems of periodic solutions to the two-point boundary value problem $u^{\mathrm{\prime }\mathrm{\prime }}(x)+k^{\mathrm{2}}u(x-x_{\mathrm{0}})+g(x,u(x-x_{\mathrm{0}}))=h(x)$ in $(\mathrm{0},\mathrm{2}\pi )$ with $u(\mathrm{0})-u(\mathrm{2}\pi )=u^{\mathrm{\prime }}(\mathrm{0})-u^{\mathrm{\prime }}(\mathrm{2}\pi )=\mathrm{0}$ when $g:(\mathrm{0,2}\pi )\times \mathbf{R}\to \mathbf{R}$ is a Caratheodory function which grows linearly in $u$ as $\left|u\right|\to \mathrm{\infty }$, and $h\in L^{\mathrm{1}}(\mathrm{0,2}\pi )$ may satisfy a generalized Landesman-Lazer condition for all $v\in N(L)\backslash \{\mathrm{0}\}$. Here $N(L)$ denotes the subspace of $L^{\mathrm{1}}(\mathrm{0,2}\pi )$ spanned by $\sin kx$ and $\cos kx$, , $g_{\beta }^{+}(x)={\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}}_{u\to \mathrm{\infty }}(g\left(x,u\right)u/{\left|u\right|}^{\mathrm{1}-\beta })$, and $g_{\beta }^{-}(x)={\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}}_{u\to -\mathrm{\infty }}(g\left(x,u\right)u/{\left|u\right|}^{\mathrm{1}-\beta })$.