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Ulam Stability for Boundary Value Problems of Differential Equations—Main Misunderstandings and How to Avoid Them

Author

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  • Ravi P. Agarwal

    (Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA
    Department of Mathematics and Systems Engineering, Florida Institute of Technology, Melbourne, FL 32901, USA)

  • Snezhana Hristova

    (Faculty of Mathematics and Informatics, Plovdiv University, Tzar Asen 24, 4000 Plovdiv, Bulgaria)

  • Donal O’Regan

    (School of Mathematical and Statistical Sciences, University of Galway, H91 TK33 Galway, Ireland)

Abstract

Ulam type stability is an important property studied for different types of differential equations. When this type of stability is applied to boundary value problems, there are some misunderstandings in the literature. In connection with this, initially, we give a brief overview of the basic ideas of the application of Ulam type stability to initial value problems. We provide several examples with simulations to illustrate the main points in the application. Then, we focus on some misunderstandings in the application of Ulam stability to boundary value problems. We suggest a new way to avoid these misunderstandings and how to keep the main idea of Ulam type stability when it is applied to boundary value problems of differential equations. We present one possible way to connect both the solutions of the given problem and the solutions of the corresponding inequality. In addition, we provide several examples with simulations to illustrate the ideas for boundary value problems and we also show the necessity of the new way of applying the Ulam type stability. To illustrate the theoretical application of the suggested idea to Ulam type stability, we consider a linear boundary value problem for nonlinear impulsive fractional differential equations with the Caputo fractional derivative with respect to another function and piecewise-constant variable order. We define the Ulam–Hyers stability and obtain sufficient conditions on a finite interval. As partial cases, integral presentations of the solutions of boundary value problems for various types of fractional differential equations are obtained and their Ulam type stability is studied.

Suggested Citation

  • Ravi P. Agarwal & Snezhana Hristova & Donal O’Regan, 2024. "Ulam Stability for Boundary Value Problems of Differential Equations—Main Misunderstandings and How to Avoid Them," Mathematics, MDPI, vol. 12(11), pages 1-22, May.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:11:p:1626-:d:1399735
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    References listed on IDEAS

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    1. ur Rahman, Ghaus & Agarwal, Ravi P. & Ahmad, Dildar, 2022. "Existence and stability analysis of nth order multi term fractional delay differential equation," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    2. Abdelkrim Salim & Mouffak Benchohra & Jamal Eddine Lazreg & Gaston N’Guérékata, 2021. "Boundary Value Problem for Nonlinear Implicit Generalized Hilfer-Type Fractional Differential Equations with Impulses," Abstract and Applied Analysis, Hindawi, vol. 2021, pages 1-17, April.
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