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Upper and lower limit oscillation conditions for first-order difference equations

Part of: Difference equations Difference and functional equations

Published online by Cambridge University Press:  07 July 2022

Özkan Öcalan Open the ORCID record for Özkan Öcalan [Opens in a new window]
Özkan Öcalan*
Affiliation:
Department of Mathematics, Faculty of Science, Akdeniz University, Antalya 07058, Turkey ([email protected])
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Abstract

In this work, we consider the first-order difference equation with general argument

\[ \varDelta x(n)+p(n)x\left( \tau (n)\right) =0,\quad n\geq 0, \]
where $(p(n))$ is a sequence of non-negative real numbers, $(\tau (n))$ is a sequence of integers such that $\tau (n)< n$ for$\ n\in \mathbf {N},\,$and$\ \lim _{n\rightarrow \infty }\tau (n)=\infty.$ Under the assumption that the deviating argument is not necessarily monotone, we obtain some new oscillation conditions and improve the all known results for the above equation in the literature, involving only upper and only lower limit conditions. Two examples illustrating the results are also given.

Keywords

delay difference equationgeneral argumentoscillation

MSC classification

Primary: 39A10: Difference equations, additive 39A21: Oscillation theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 3 , August 2022 , pp. 618 - 631
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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