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 θ4(t1,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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