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Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of Grönwall-Bellman-Bihari’s type

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  • Zada, Akbar
  • Ali, Wajid
  • Park, Choonkil

Abstract

In this paper, we prove the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability of a class of higher order nonlinear delay differential equations with multiple bounded variable delays on a compact interval. Result of existence and uniqueness of solution is obtained by fixed point approach. Meanwhile, for the first time, integral inequality of Grönwall-Bellman-Bihari’s type with delay is applied to prove the stability theorem which made our results more generalized and interesting.

Suggested Citation

  • Zada, Akbar & Ali, Wajid & Park, Choonkil, 2019. "Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of Grönwall-Bellman-Bihari’s type," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 60-65.
    • Handle: RePEc:eee:apmaco:v:350:y:2019:i:c:p:60-65
    DOI: 10.1016/j.amc.2019.01.014
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    References listed on IDEAS

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    1. Zada, Akbar & Shah, Omar & Shah, Rahim, 2015. "Hyers–Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 512-518.
    2. Yongjin Li & Yan Shen, 2009. "Hyers-Ulam Stability of Nonhomogeneous Linear Differential Equations of Second Order," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2009, pages 1-7, October.
    3. Soon-Mo Jung & Janusz Brzdęk, 2010. "Hyers-Ulam Stability of the Delay Equation," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-10, December.
    4. Jinghao Huang & Yongjin Li, 2016. "Hyers–Ulam stability of delay differential equations of first order," Mathematische Nachrichten, Wiley Blackwell, vol. 289(1), pages 60-66, January.
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    Cited by:

    1. Rafia Majeed & Binlin Zhang & Mehboob Alam, 2023. "Fractional Langevin Coupled System with Stieltjes Integral Conditions," Mathematics, MDPI, vol. 11(10), pages 1-14, May.
    2. Akbar Zada & Shaheen Fatima & Zeeshan Ali & Jiafa Xu & Yujun Cui, 2019. "Stability Results for a Coupled System of Impulsive Fractional Differential Equations," Mathematics, MDPI, vol. 7(10), pages 1-29, October.
    3. Shah, Syed Omar & Zada, Akbar, 2019. "Existence, uniqueness and stability of solution to mixed integral dynamic systems with instantaneous and noninstantaneous impulses on time scales," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 202-213.
    4. Usman Riaz & Akbar Zada & Zeeshan Ali & Ioan-Lucian Popa & Shahram Rezapour & Sina Etemad, 2021. "On a Riemann–Liouville Type Implicit Coupled System via Generalized Boundary Conditions," Mathematics, MDPI, vol. 9(11), pages 1-22, May.
    5. Shuyi Wang & Fanwei Meng, 2021. "Ulam Stability of n -th Order Delay Integro-Differential Equations," Mathematics, MDPI, vol. 9(23), pages 1-17, November.
    6. Hamza Khalil & Akbar Zada & Mohamed Rhaima & Ioan-Lucian Popa, 2024. "Analysis of Neutral Implicit Stochastic Hilfer Fractional Differential Equation Involving Lévy Noise with Retarded and Advanced Arguments," Mathematics, MDPI, vol. 12(21), pages 1-15, October.
    7. Binlin Zhang & Rafia Majeed & Mehboob Alam, 2022. "On Fractional Langevin Equations with Stieltjes Integral Conditions," Mathematics, MDPI, vol. 10(20), pages 1-16, October.
    8. Alam, Mehboob & Shah, Dildar, 2021. "Hyers–Ulam stability of coupled implicit fractional integro-differential equations with Riemann–Liouville derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    9. Singh, Ajeet & Shukla, Anurag & Vijayakumar, V. & Udhayakumar, R., 2021. "Asymptotic stability of fractional order (1,2] stochastic delay differential equations in Banach spaces," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).

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