Asymptotic formulas for solutions of functional differential equations. (English) Zbl 0643.34077

The initial problem \((1.1)\quad y'=A(t)y+f(t,\sigma (y;h(t))),\) (1.2) \(y(0)=y_ 0\), \(y_ 0\) is a constant vector, is considered, where the function \(f: R_+\times R^ n\to R^ n\) satisfies the Carathéodory local conditions, the matrix \(A: R_+\to R^{n\times n}\) is local integrable, \(h: R_+\to R\) is a continuous function, with h(t)\(\leq t\) and \(\sigma\) is an operator defined by \(\sigma (u;t)=u(t)\) for \(t\in R_+\), \(=0\) for \(t<0\). Sufficient conditions are given for that: a) all solutions of (1.1), (1.2) exist on \(R_+\) under small initial conditions \(\| y_ 0\|\) 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 \(\| y_ 0\|\).
Reviewer: J.Futák


34K25 Asymptotic theory of functional-differential equations
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