Asymptotic integration of higher order nonlinear delay differential equations via principal solutions

Abstract

We study the asymptotic integration problem for a general class of n-th order nonlinear delay differential equations of the form ${\mathit{L}}_{\mathit{n}}\mathit{x}\mathrm{(}\mathit{t}\mathrm{)}=\mathit{f}\mathrm{(}\mathit{t}\mathrm{,}\mathit{x}\mathrm{(}\mathit{g}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{)}\mathrm{)}$ , where ${\mathit{L}}_{\mathit{n}}\mathit{x}={\mathit{x}}^{\mathrm{(}\mathit{n}\mathrm{)}}+{\mathit{a}}_{\mathit{n}-1}\mathrm{(}\mathit{t}\mathrm{)}{\mathit{x}}^{\mathrm{(}\mathit{n}-1\mathrm{)}}+\cdots +{\mathit{a}}_0\mathrm{(}\mathit{t}\mathrm{)}\mathit{x}$ . It is shown that if $\{{\mathit{u}}_1\mathrm{,}{\mathit{u}}_2\mathrm{,}\dots \mathrm{,}{\mathit{u}}_{\mathit{n}}\}$ is a set of principal solutions for ${\mathit{L}}_{\mathit{n}}\mathit{x}=0$ , then the n-th order nonlinear delay differential equation above has solutions ${\mathit{x}}_{\mathit{k}}$ with the property that ${\mathit{x}}_{\mathit{k}}\mathrm{(}\mathit{t}\mathrm{)}={\mathit{b}}_1{\mathit{u}}_1\mathrm{(}\mathit{t}\mathrm{)}+\cdots +{\mathit{b}}_{\mathit{k}}{\mathit{u}}_{\mathit{k}}\mathrm{(}\mathit{t}\mathrm{)}+\mathit{o}\mathrm{(}{\mathit{u}}_{\mathit{k}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{)}$ , $\mathit{t}\to \infty$ for $\mathit{k}=1\mathrm{,}2\mathrm{,}\dots \mathrm{,}\mathit{n}$ .