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Hille–Nehari theorems for dynamic equations with a time scale independent critical constant

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  • Karpuz, Başak

Abstract

In this paper, we give Hille–Nehari test for nonoscillation/oscillation of the dynamic equation(rxΔ)Δ(t)+p(t)x(t)=0fort∈[t0,∞)T,where t0∈T,supT=∞,r∈Crd([t0,∞)T,R+) and p∈Crd([t0,∞)T,R0+). We show that the critical constant for this dynamic equation is 14 as in the well-known cases T=R and T=Z. We also present illustrating examples showing that the critical constant 14 is sharp on arbitrary time scales. With two different techniques, we extend our results to the dynamic equation(rxΔ)Δ(t)+p(t)xσ(t)=0fort∈[t0,∞)Tby preserving the constant 14. The second technique is new even for the discrete case T=Z.

Suggested Citation

  • Karpuz, Başak, 2019. "Hille–Nehari theorems for dynamic equations with a time scale independent critical constant," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 336-351.
    • Handle: RePEc:eee:apmaco:v:346:y:2019:i:c:p:336-351
    DOI: 10.1016/j.amc.2018.09.055
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    References listed on IDEAS

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    1. Elena Braverman & Başak Karpuz, 2011. "Nonoscillation of Second-Order Dynamic Equations with Several Delays," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-34, April.
    2. Agarwal, Ravi P. & Bohner, Martin & Li, Tongxing, 2015. "Oscillatory behavior of second-order half-linear damped dynamic equations," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 408-418.
    3. Shao-Yan Zhang & Qi-Ru Wang, 2014. "Interval Oscillation Criteria for Second-Order Forced Functional Dynamic Equations on Time Scales," Discrete Dynamics in Nature and Society, Hindawi, vol. 2014, pages 1-10, March.
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    1. Taher S. Hassan & Clemente Cesarano & Rami Ahmad El-Nabulsi & Waranont Anukool, 2022. "Improved Hille-Type Oscillation Criteria for Second-Order Quasilinear Dynamic Equations," Mathematics, MDPI, vol. 10(19), pages 1-18, October.
    2. Grace, Said R. & Negi, Shekhar Singh & Abbas, Syed, 2022. "New oscillatory results for non-linear delay dynamic equations with super-linear neutral term," Applied Mathematics and Computation, Elsevier, vol. 412(C).
    3. Taher S. Hassan & Qingkai Kong & Rami Ahmad El-Nabulsi & Waranont Anukool, 2022. "New Hille Type and Ohriska Type Criteria for Nonlinear Third-Order Dynamic Equations," Mathematics, MDPI, vol. 10(21), pages 1-12, November.

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