THE ASYMPTOTIC BEHAVIOR AND OSCILLATION FOR A CLASS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS

doi:10.1007/s10473-024-0309-6

Abstract

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 P ötzsche 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.

Key words: nonlinear delay dynamic equations, nonoscillation, asymptotic behavior, Philos-type oscillation criteria, generalized Riccati transformation

CLC Number:

34C10

Xianyong Huang, Xunhuan Deng, Qiru Wang. THE ASYMPTOTIC BEHAVIOR AND OSCILLATION FOR A CLASS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS[J].Acta mathematica scientia,Series B, 2024, 44(3): 925-946.

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