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Results on Implicit Fractional Pantograph Equations with Mittag‐Leffler Kernel and Nonlocal Condition

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  • Mohammed A. Almalahi
  • Satish K. Panchal
  • Fahd Jarad

Abstract

In this study, the main focus is on an investigation of the sufficient conditions of existence and uniqueness of solution for two‐classess of nonlinear implicit fractional pantograph equations with nonlocal conditions via Atangana–Baleanu–Riemann–Liouville (ABR) and Atangana–Baleanu–Caputo (ABC) fractional derivative with order σ ∈ (1,2]. We introduce the properties of solutions as well as stability results for the proposed problem without using the semigroup property. In the beginning, we convert the given problems into equivalent fractional integral equations. Then, by employing some fixed‐point theorems such as Krasnoselskii and Banach’s techniques, we examine the existence and uniqueness of solutions to proposed problems. Moreover, by using techniques of nonlinear functional analysis, we analyze Ulam–Hyers (UH) and generalized Ulam–Hyers (GUH) stability results. As an application, we provide some examples to illustrate the validity of our results.

Suggested Citation

  • Mohammed A. Almalahi & Satish K. Panchal & Fahd Jarad, 2022. "Results on Implicit Fractional Pantograph Equations with Mittag‐Leffler Kernel and Nonlocal Condition," Journal of Mathematics, John Wiley & Sons, vol. 2022(1).
    • Handle: RePEc:wly:jjmath:v:2022:y:2022:i:1:n:9693005
    DOI: 10.1155/2022/9693005
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    References listed on IDEAS

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    1. Atangana, Abdon & Koca, Ilknur, 2016. "Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 447-454.
    2. Klaus Deimling, 1985. "Nonlinear Functional Analysis," Springer Books, Springer, number 978-3-662-00547-7, January.
    3. Almalahi, Mohammed A. & Panchal, Satish K. & Jarad, Fahd, 2021. "Stability results of positive solutions for a system of ψ -Hilfer fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
    4. Atangana, Abdon, 2017. "Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 396-406.
    5. Idris Ahmed & Poom Kumam & Kamal Shah & Piyachat Borisut & Kanokwan Sitthithakerngkiet & Musa Ahmed Demba, 2020. "Stability Results for Implicit Fractional Pantograph Differential Equations via ϕ -Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition," Mathematics, MDPI, vol. 8(1), pages 1-21, January.
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    Cited by:

    1. Houari Bouzid & Abdelkader Benali & Abdelkrim Salim & Reny George & Sina Etemad, 2025. "On Solutions of the Nonlocal Generalized Coupled Langevin‐Type Pantograph Systems," Journal of Mathematics, John Wiley & Sons, vol. 2025(1).

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