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Zbl 1063.34072
Pituk, Mihály
Special solutions of functional differential equations. (English)
[J] Stud. Univ. Žilina, Math. Ser. 17, No. 1, 115-122 (2003). ISSN 1336-149X

The author considers the delay differential equation $$ \frac{dx(t)}{dt}=F(t,x_{t}), $$ where $F$ is a mapping from $\bbfR\times C$ into $\bbfR^{n},$ with $C=C([-r,0],\bbfR^{n}).$ \par It is proved that if $F$ satisfies $$\vert F(t,\varphi )-F(t,\psi )\vert \leq L(t)\Vert \varphi -\psi \Vert ,\quad t\in \bbfR,\ \varphi ,\psi \in C,$$ $$\vert F(t,0)\vert \leq AL(t)\exp \left( e\int_{t}^{0}L(s)ds\right) ,\quad t\leq 0,$$ $$\text{ and }K =\sup_{t\in \bbfR}\int_{t-r}^{t}L(s)ds<\frac{1}{e},$$ then there exists a unique special solution $x(t)$ with $ x_{t_{0}}=\varphi $ and $x(t_{0})=x_{0}$ satisfying the growth condition$$ \sup_{t\leq 0}\vert x(t)\vert \exp \left( -e\int_{t}^{0}L(s)ds\right) <\infty . $$ Furthermore, he gives a sufficient condition under which an arbitrary solution is asymptotically equivalent to exactly one special solution.
[Takeshi Taniguchi (Kurume)]

MSC 2000:: *34K25 Asymptotic theory of functional-differential equations

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