## 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 $$ for all $v\in N(L)\backslash \{\mathrm{0}\}$. Here $N(L)$ denotes the subspace of ${L}^{\mathrm{1}}(\mathrm{0,2}\pi )$ spanned by $\mathrm{sin}kx$ and $\mathrm{cos}kx$, $$, ${g}_{\beta}^{+}(x)={\mathrm{l}\mathrm{i}\mathrm{m}\u200a\u200a\mathrm{i}\mathrm{n}\mathrm{f}}_{u\to \mathrm{\infty}}(g\left(x,u\right)u/{\left|u\right|}^{\mathrm{1}-\beta})$, and ${g}_{\beta}^{-}(x)={\mathrm{l}\mathrm{i}\mathrm{m}\u200a\u200a\mathrm{i}\mathrm{n}\mathrm{f}}_{u\to -\mathrm{\infty}}(g\left(x,u\right)u/{\left|u\right|}^{\mathrm{1}-\beta})$.

## Citation

Nai-Sher Yeh. "On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay." Abstr. Appl. Anal. 2018 1 - 6, 2018. https://doi.org/10.1155/2018/5321314