Electron. J. Differential Equations, Vol. 2021 (2021), No. 32, pp. 1-12.

Improved oscillation criteria for first-order delay differential equations with variable delay

Julio G. Dix

This article concerns the oscillation of solutions to the delay differential equation $x'(t)+p(t)x(\tau(t))=0$. Conditions for oscillation have been stated as lower bounds for the limit superior and limit inferior of $\int_\tau^t p$. In this article we match the bound for the best case in [7], without using one of their hypotheses. Then assuming that hypothesis, we obtain a bound lower than the one in [12]. Then we apply our results to an equation with several delays. We employ iterated estimates of the solution.

Submitted January 29, 2021. Published April 24, 2021.
Math Subject Classifications: 34K11, 34C10.
Key Words: Oscillation of solutions; first-order delay differential equation; eventually positive solution.

An addendum was posted on October 13, 2022. It indicates that the proof of Lemma 2.2 is incorrect. See the last page of this article.

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Julio G. Dix
Department of Mathematics
Texas State University
601 University Drive
San Marcos, TX 78666, USA
email: jd01@txstate.edu

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