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Hyers–Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems

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  • Zada, Akbar
  • Shah, Omar
  • Shah, Rahim

Abstract

In this paper, the concepts of Hyers–Ulam stability are generalized for non-autonomous linear differential systems. We prove that the k-periodic linear differential matrix system Z˙(t)=A(t)Z(t),t∈R is Hyers–Ulam stable if and only if the matrix family L=E(k,0) has no eigenvalues on the unit circle, i.e. we study the Hyers–Ulam stability in terms of dichotomy of the differential matrix system Z˙(t)=A(t)Z(t),t∈R. Furthermore, we relate Hyers–Ulam stability of the system Z˙(t)=A(t)Z(t),t∈R to the boundedness of solution of the following Cauchy problem: {Y˙(t)=A(t)Y(t)+ρ(t),t≥0Y(0)=x−x0,where A(t) is a square matrix for any t∈R,ρ(t) is a bounded function and x,x0∈Cm.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:271:y:2015:i:c:p:512-518
    DOI: 10.1016/j.amc.2015.09.040
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. L. Chitra & K. Alagesan & Vediyappan Govindan & Salman Saleem & A. Al-Zubaidi & C. Vimala, 2023. "Applications of the Tarig Transform and Hyers–Ulam Stability to Linear Differential Equations," Mathematics, MDPI, vol. 11(12), pages 1-25, June.
    4. Naveed Ahmad & Zeeshan Ali & Kamal Shah & Akbar Zada & Ghaus ur Rahman, 2018. "Analysis of Implicit Type Nonlinear Dynamical Problem of Impulsive Fractional Differential Equations," Complexity, Hindawi, vol. 2018, pages 1-15, February.
    5. Onitsuka, Masakazu, 2018. "Hyers–Ulam stability of first-order nonhomogeneous linear difference equations with a constant stepsize," Applied Mathematics and Computation, Elsevier, vol. 330(C), pages 143-151.
    6. Samina & Kamal Shah & Rahmat Ali Khan, 2020. "Stability theory to a coupled system of nonlinear fractional hybrid differential equations," Indian Journal of Pure and Applied Mathematics, Springer, vol. 51(2), pages 669-687, June.

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