Acta mathematica scientia,Series B ›› 2024, Vol. 44 ›› Issue (3): 925-946.doi: 10.1007/s10473-024-0309-6

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

Xianyong Huang1, Xunhuan Deng2,*, Qiru Wang3   

  1. 1. Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China;
    2. Department of Mathematics, College of Medical Information Engineering, Guangdong Pharmaceutical University, Guangzhou 510006, China;
    3. School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
  • Received:2022-07-28 Revised:2023-11-26 Online:2024-06-25 Published:2024-05-21
  • Contact: *Xunhuan Deng, E-mail:deng_xunhuan@163.com
  • About author:Xianyong Huang, E-mail:huangxianyong@gdei.edu.cn;Qiru Wang, E-mail:mcswqr@mail.sysu.edu.cn
  • Supported by:
    National Natural Science Foundation of China (12071491, 12001113).

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
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