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

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

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

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

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

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doi:10.1007/s10473-024-0309-6

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (3): 925-946.doi: 10.1007/s10473-024-0309-6

Xianyong Huang¹, Xunhuan Deng^2,*, Qiru Wang³

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

收稿日期:2022-07-28 修回日期:2023-11-26 出版日期:2024-06-25 发布日期:2024-05-21

Xianyong Huang¹, Xunhuan Deng^2,*, Qiru Wang³

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).

摘要： 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.

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

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

中图分类号:

34C10

Xianyong Huang, Xunhuan Deng, Qiru Wang. THE ASYMPTOTIC BEHAVIOR AND OSCILLATION FOR A CLASS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS[J]. 数学物理学报(英文版), 2024, 44(3): 925-946.

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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