Oscillatory and nonoscillatory behavior of nonlinear delay dynamic equations on time scales

Svetlin G. Georgiev; Youssef N. Raffoul; Sanket Tikare; Svetlin G. Georgiev; Youssef N. Raffoul; Sanket Tikare

doi:10.3934/era.2026157

Electronic Research Archive

2026, Volume 34, Issue 5: 3516-3529. doi: 10.3934/era.2026157

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Oscillatory and nonoscillatory behavior of nonlinear delay dynamic equations on time scales

1.
Department of Mathematics, Sorbonne University, Paris 75005, France
2.
Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, USA
3.
Department of Mathematics, Ramniranjan Jhunjhunwala College, Mumbai 400086, Maharashtra, India

Received: 07 January 2026 Revised: 12 March 2026 Accepted: 03 April 2026 Published: 23 April 2026

This paper is devoted to the qualitative analysis of first-order nonlinear delay dynamic equations on time scales. We establish sufficient conditions for oscillation and nonoscillation of solutions to dynamic equations of the form
$ x^{\Delta}(t) + p(t)\big(x(\tau(t))\big)^{\alpha} = 0, \quad t \ge t_0, $
where $ \alpha $ is a quotient of two odd integers and $ p $, $ \tau $ satisfy standard regularity and delay conditions on an arbitrary time scale $ \mathbb{T} $. Employing comparison principles, the time scales version of L'Hôpital's rule, and integral transformations, we derive new oscillation criteria that generalize several classical results from the differential and difference equation settings to the unified framework of time scales. Furthermore, we provide sufficient conditions ensuring the existence of eventually positive (nonoscillatory) solutions. Our results extend and complement the existing theory in the literature.
- chain rule,
- delay dynamic equations,
- eventually positive solutions,
- nonoscillations,
- oscillations,
- time scales
Citation: Svetlin G. Georgiev, Youssef N. Raffoul, Sanket Tikare. Oscillatory and nonoscillatory behavior of nonlinear delay dynamic equations on time scales[J]. Electronic Research Archive, 2026, 34(5): 3516-3529. doi: 10.3934/era.2026157

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Abstract

This paper is devoted to the qualitative analysis of first-order nonlinear delay dynamic equations on time scales. We establish sufficient conditions for oscillation and nonoscillation of solutions to dynamic equations of the form

$ x^{\Delta}(t) + p(t)\big(x(\tau(t))\big)^{\alpha} = 0, \quad t \ge t_0, $

where $ \alpha $ is a quotient of two odd integers and $ p $, $ \tau $ satisfy standard regularity and delay conditions on an arbitrary time scale $ \mathbb{T} $. Employing comparison principles, the time scales version of L'Hôpital's rule, and integral transformations, we derive new oscillation criteria that generalize several classical results from the differential and difference equation settings to the unified framework of time scales. Furthermore, we provide sufficient conditions ensuring the existence of eventually positive (nonoscillatory) solutions. Our results extend and complement the existing theory in the literature.

References

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