The asymptotic behavior and oscillation for a class of third-order nonlinear delay dynamic equations. (English) Zbl 07815377
Summary: 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 \(\theta_4(t_1, T)\). The results of Theorem 3.5 and Remark 3.6 are novel. Finally, we offer four examples to illustrate our results.
MSC:
| 34K42 | Functional-differential equations on time scales or measure chains |
| 34K11 | Oscillation theory of functional-differential equations |
| 34N05 | Dynamic equations on time scales or measure chains |
| 34K45 | Functional-differential equations with impulses |
| 34K33 | Averaging for functional-differential equations |
Keywords:
nonlinear delay dynamic equations; non-oscillation; asymptotic behavior; Philos-type oscillation criteria; generalized Riccati transformationReferences:
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