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Zbl 0643.34077
Futák, Ján
Asymptotic formulas for solutions of functional differential equations. (English)
[J] Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. 85, Math. 25, 181-192 (1986). ISSN 0231-9721

The initial problem $(1.1)\quad y'=A(t)y+f(t,\sigma (y;h(t))),$ (1.2) $y(0)=y\sb 0$, $y\sb 0$ is a constant vector, is considered, where the function $f: R\sb+\times R\sp n\to R\sp n$ satisfies the Carathéodory local conditions, the matrix $A: R\sb+\to R\sp{n\times n}$ is local integrable, $h: R\sb+\to R$ is a continuous function, with h(t)$\le t$ and $\sigma$ is an operator defined by $\sigma (u;t)=u(t)$ for $t\in R\sb+$, $=0$ for $t<0$. Sufficient conditions are given for that: a) all solutions of (1.1), (1.2) exist on $R\sb+$ under small initial conditions $\Vert y\sb 0\Vert$ and are of asymptotic representation $(1.4)\quad y(t)=X(t)[c+o(1)]$ as $t\to \infty$, where c is a constant vector; b) the family of solutions of the form (1.4) of (1.1) is stable (in some sense) with regard to small changes of both initial conditions and right- handside of (1.1); c) any solution of (1.1), (1.2) has the form (1.4) under arbitrary initial conditions $\Vert y\sb 0\Vert$.
[J.Futák]

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

Keywords: Carathéodory local conditions

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