×

On a linear differential equation with a proportional delay. (English) Zbl 1128.34051

The paper deals with the non-autonomous linear delay differential equation
\[ \dot x(t)=c(t) \left( x(t)-p x(\lambda t)\right),\quad 0<\lambda< 1,\quad p\neq 0,\quad t>0, \]
where \(p\) and \(\lambda\) are real scalars and \(c\) is a continuous and non-oscillatory function defined on \((0,\infty)\). The equation is referred to as pantograph equation, since in a simplified version it models the collection of current by the pantograph of an electric locomotive. The asymptotic properties of the solutions are in focus. The following condition on the growth of \(c\) is imposed: \[ \limsup_{t\to\infty}{\lambda\,c(\lambda t)\over c(t)}<1\,. \] The main result of the paper says that if \(c\in C^1((0,\infty))\) fulfills this condition and is eventually positive, then there exist real constants \(L\) and \(\rho\), where \(\rho>0\), and a continuous periodic function \(g\) of period \(\log \lambda^{-1}\) such that
\[ x(t)=Lx^*(t)+t^k g(\log t)+ O(t^{\kappa_r-\rho})\,. \]
Here \(\kappa_r\) is the real part of the possible complex \(\kappa\) such that \(\lambda^k=1/p\), and \(x^*\) is the solution of the considered equation, such that \[ x^*(t)\sim \exp\left( \int_{\bar t}^t c(s)\,ds\right) \quad\text{ as}\quad t\to\infty \] (the existence of such a sulution \(x^*\) is proved in the paper). Though it is natural to distinguish the cases of the eventually positive and the eventually negative \(c\), it is shown that a resembling asymptotic formula is valid also in the case of \(c\) eventually negative. Finally, using a transformation approach these results are generalized to equations with a general form of the delay.

MSC:

34K25 Asymptotic theory of functional-differential equations
34K06 Linear functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] and , Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays, in: Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, University North Carolina atWilmington, USA, May 24–27, 2002, Discrete and Continuous Dynamical Systems Supplements, Special (AIMS, Springfield, MO 65801–2604, USA, 2003), pp. 100–107. · Zbl 1071.34080
[2] Carr, Proc. Roy. Soc. Edinburgh Sect. A 74 pp 165– (1974/75) · Zbl 0344.34059 · doi:10.1017/S0308210500016632
[3] Carr, Proc. Roy. Soc. Edinburgh Sect. A 75 pp 5– (1975/76) · Zbl 0355.34054 · doi:10.1017/S0080454100012516
[4] Čermák, Aequationes Math. 64 pp 89– (2002)
[5] Čermák, J. Comput. Appl. Math. 143 pp 81– (2002)
[6] Diblík, J. Math. Anal. Appl. 217 pp 200– (1998)
[7] Györi, Dynam. Systems Appl. 5 pp 277– (1996)
[8] Heard, J. Differential Equations 18 pp 1– (1975) · Zbl 0318.34069
[9] Iserles, European J. Appl. Math. 4 pp 1– (1992)
[10] Iserles, J. Math. Anal. Appl. 207 pp 73– (1997)
[11] Kato, Bull. Amer. Math. Soc. (N. S.) 77 pp 891– (1971)
[12] Krisztin, J. Math. Anal. Appl. 145 pp 17– (1990)
[13] Lim, J. Math. Anal. Appl. 55 pp 794– (1976)
[14] Makay, Electron. J. Qual. Theory Differ. Equ. 2 pp 1– (1998) · Zbl 0891.34047 · doi:10.14232/ejqtde.1998.1.2
[15] Neuman, Czechoslovak Math. J. 31 pp 87– (1981)
[16] Neuman, Czechoslovak Math. J. 32 pp 488– (1982)
[17] Ockendon, Proc. Roy. Soc. London Ser. A 322 pp 447– (1971)
[18] Zdun, Aequationes Math. 38 pp 163– (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.