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Asymptotic estimation for functional differential equations with several delays. (English) Zbl 1048.34126

The asymptotic behaviour of all solutions of the functional-differential equation \[ y'(x)=\sum _{i=1}^ma_i(x)y(\tau _i(x))+b(x)y(x) \] with \(b(x)<0\) is investigated. Asymptotic bounds for the solutions are given in terms of solutions of the functional equation \[ \sum _{i=1}^m| a_i(x)| \omega (\tau _i(x))+b(x)\omega (x)=0. \] Conditions, under which \(\omega \) exhibits the form \(\omega (x)=\text{exp}(\alpha \psi (x))\) with a constant \(\alpha \) and with \(\psi (x)\) being a solution of Abel’s functional equation, are discussed. Many partial cases of above equation, known from literature, are considered as applications. In the proof, there is used the transformation approach [see, e.g., F. Neuman, Proc. R. Soc. Edinb., Sect. A 115, 349–357 (1990; Zbl 0714.34108)].

MSC:

34K25 Asymptotic theory of functional-differential equations
39B99 Functional equations and inequalities

Citations:

Zbl 0714.34108
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