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On the Study of Solutions for a Class of Third-Order Semilinear Nonhomogeneous Delay Differential Equations

Author

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  • Wenjin Li

    (School of Applied Mathematics, Jilin University of Finance and Economics, Changchun 130117, China)

  • Jiaxuan Sun

    (School of Applied Mathematics, Jilin University of Finance and Economics, Changchun 130117, China)

  • Yanni Pang

    (School of Mathematics, Jilin University, Changchun 130021, China)

Abstract

This paper mainly investigates a class of third-order semilinear delay differential equations with a nonhomogeneous term ( [ x ″ ( t ) ] α ) ′ + q ( t ) x α ( σ ( t ) ) + f ( t ) = 0 , t ≥ t 0 . Under the oscillation criteria, we propose a sufficient condition to ensure that all solutions for the equation exhibit oscillatory behavior when α is the quotient of two positive odd integers, supported by concrete examples to verify the accuracy of these conditions. Furthermore, for the case α = 1 , a sufficient condition is established to guarantee that the solutions either oscillate or asymptotically converge to zero. Moreover, under these criteria, we demonstrate that the global oscillatory behavior of solutions remains unaffected by time-delay functions, nonhomogeneous terms, or nonlinear perturbations when α = 1 . Finally, numerical simulations are provided to validate the effectiveness of the derived conclusions.

Suggested Citation

  • Wenjin Li & Jiaxuan Sun & Yanni Pang, 2025. "On the Study of Solutions for a Class of Third-Order Semilinear Nonhomogeneous Delay Differential Equations," Mathematics, MDPI, vol. 13(12), pages 1-17, June.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:12:p:1926-:d:1675698
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    References listed on IDEAS

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    1. Tomas Ruzgas & Irma Jankauskienė & Audrius Zajančkauskas & Mantas Lukauskas & Matas Bazilevičius & Rugilė Kaluževičiūtė & Jurgita Arnastauskaitė, 2024. "Solving Linear and Nonlinear Delayed Differential Equations Using the Lambert W Function for Economic and Biological Problems," Mathematics, MDPI, vol. 12(17), pages 1-15, September.
    2. Reem Alrebdi & Hind K. Al-Jeaid, 2023. "Accurate Solution for the Pantograph Delay Differential Equation via Laplace Transform," Mathematics, MDPI, vol. 11(9), pages 1-15, April.
    3. Li Gao & Quanxin Zhang & Shouhua Liu, 2014. "New Oscillatory Behavior of Third-Order Nonlinear Delay Dynamic Equations on Time Scales," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-11, March.
    4. Li Gao & Quanxin Zhang & Shouhua Liu, 2014. "New Oscillatory Behavior of Third‐Order Nonlinear Delay Dynamic Equations on Time Scales," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    5. Zhou, Weigang & Huang, Chengdai & Xiao, Min & Cao, Jinde, 2019. "Hybrid tactics for bifurcation control in a fractional-order delayed predator–prey model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 515(C), pages 183-191.
    6. Luca Guerrini & Adam Krawiec & Marek Szydlowski, 2020. "Bifurcations in economic growth model with distributed time delay transformed to ODE," Papers 2002.05016, arXiv.org.
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