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International Conference on Differential & Difference Equations and Applications

ICDDEA 2017: Differential and Difference Equations with Applications pp 441–461Cite as

A Survey on the Oscillation of Delay Equations with A Monotone or Non-monotone Argument

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 230))

Abstract

Consider the first-order linear differential equation

$$\begin{aligned} x^{\prime }(t)+p(t)x(\tau (t))=0,\;\;\;t\ge t_{0}, \end{aligned}$$

where the functions \(p,\tau \in C([t_{0,}\infty ),\mathbb {R}^{+})\), (here \( \mathbb {R}^{+}=[0,\infty )),\tau (t)\le t\) for \(t\ge t_{0}\) and \( \lim _{t\rightarrow \infty }\tau (t)=\infty .\) A survey on the oscillation of all solutions to this equation is presented in the case of monotone and non-monotone argument and especially in the critical case where \( \liminf _{t\rightarrow \infty }p(t)=1/e\tau \) and also when the known oscillation conditions \(\underset{t\rightarrow \infty }{\lim \sup } \int \nolimits _{\tau (t)}^{t}p(s)ds>1\) and \(\liminf _{t\rightarrow \infty }\int _{\tau (t)}^{t}p(s)ds>\frac{1}{e}\) are not satisfied. Examples illustrating the results are given.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of South Africa, Pretoria, 0003, South Africa

    G. M. Moremedi & I. P. Stavroulakis

  2. Department of Mathematics, University of Ioannina, 451 10, Ioannina, Greece

    I. P. Stavroulakis

  3. Department of Mathematics, Ankara University, 06100, Ankara, Turkey

    I. P. Stavroulakis

Authors
  1. G. M. Moremedi
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  2. I. P. Stavroulakis
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Corresponding author

Correspondence to I. P. Stavroulakis .

Editor information

Editors and Affiliations

  1. Departamento de Ciências Exactas e Engenharia, Academia Militar, Amadora, Portugal

    Sandra Pinelas

  2. Facultad de Matemáticas, Universidad de Sevilla, Sevilla, Spain

    Tomás Caraballo

  3. Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, Hubei, China

    Peter Kloeden

  4. Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, Tennessee, USA

    John R. Graef

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Moremedi, G.M., Stavroulakis, I.P. (2018). A Survey on the Oscillation of Delay Equations with A Monotone or Non-monotone Argument. In: Pinelas, S., Caraballo, T., Kloeden, P., Graef, J. (eds) Differential and Difference Equations with Applications. ICDDEA 2017. Springer Proceedings in Mathematics & Statistics, vol 230. Springer, Cham. https://doi.org/10.1007/978-3-319-75647-9_36

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