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Asymptotic behaviour of functional-differential equations with proportional time delays. (English) Zbl 0856.34078

The problem \[ y'(t)= Ay(t)+ \sum^\infty_{i= 1} B_i y(q_i t)+ \sum^\infty_{i= 1} C_i y'(p_i t),\quad t> 0, \] \(y(0)= y_0\), \(p_i, q_i\in (0, 1)\), is considered. References are indicated for applications, motivations and special cases previously studied. It is proved that every solution is \(o(e^{\alpha t})\) as \(t\to \infty\) for any fixed constant \(\alpha> \max\{\alpha(A), 0\}\), where \(\alpha(\cdot)\) denotes the maximal real part of the eigenvalues of its argument. Detailed analysis is performed for the cases \(\alpha(A)> 0\), \(\alpha(A)= 0\), \(\alpha(A)< 0\).

MSC:

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K40 Neutral functional-differential equations
34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
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[1] DOI: 10.1216/jiea/1181075805 · Zbl 0816.45005
[2] Carr, Proc. Royal Soc. 75A pp 5– (1975)
[3] Morris, Ordinary Differential Equations pp 513– (1972)
[4] Carr, Proc. Royal Soc. 74A pp 165– (1974)
[5] Kato, Delay and Functional Differential Equations and Their Applications (1972)
[6] Frederickson, null pp 247– (1971)
[7] DOI: 10.1090/S0002-9904-1971-12805-7 · Zbl 0236.34064
[8] de Bruijn, Nederl. Acad. Wetensh. Proc. 56 pp 449– (1953)
[9] Derfel, Operator Theory 46 pp 319– (1990)
[10] DOI: 10.1093/imamat/8.3.271 · Zbl 0251.34045
[11] DOI: 10.1007/BF01682121 · Zbl 0518.34058
[12] DOI: 10.1098/rspa.1971.0078
[13] Liu, Numer. Math. (1995)
[14] Iserles, On functional-differential equations with proportional delays (1993)
[15] DOI: 10.2307/2154725 · Zbl 0804.34065
[16] Katznelson, An Introduction to Harmonic Analysis (1976)
[17] Guckenheimer, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (1983) · Zbl 0515.34001
[18] Romanenko, Asymptotic Behaviour of Solutions of Differential Difference Equations pp 5– (1978)
[19] Iserles, Euro. J. Appl. Math. 4 pp 1– (1992)
[20] DOI: 10.1137/0521089 · Zbl 0719.34134
[21] Feldstein, Canadian Math. Bull. 33 pp 428– (1990) · Zbl 0723.34067
[22] Derfel, J. Math. Anal. Appl.
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