Abstract
In this paper, we consider a class of third-order nonlinear delay dynamic equations. First, we establish a Kiguradze-type lemma and some useful estimates. Second, we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero. Third, we obtain new oscillation criteria by employing the Pötzsche chain rule. Then, using the generalized Riccati transformation technique and averaging method, we establish the Philos-type oscillation criteria. Surprisingly, the integral value of the Philos-type oscillation criteria, which guarantees that all unbounded solutions oscillate, is greater than θ4(t1, T). The results of Theorem 3.5 and Remark 3.6 are novel. Finally, we offer four examples to illustrate our results.
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This work was supported by the National Natural Science Foundation of China (12071491, 12001113).
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Huang, X., Deng, X. & Wang, Q. The asymptotic behavior and oscillation for a class of third-order nonlinear delay dynamic equations. Acta Math Sci 44, 925–946 (2024). https://doi.org/10.1007/s10473-024-0309-6
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DOI: https://doi.org/10.1007/s10473-024-0309-6
Key words
- nonlinear delay dynamic equations
- nonoscillation
- asymptotic behavior
- Philos-type oscillation criteria
- generalized Riccati transformation