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Qualitative Analysis Of Implicit Delay Mittag-Leffler-Type Fractional Differential Equations

Author

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  • SHAO-WEN YAO

    (School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, P. R. China)

  • YASMEEN SUGHRA

    (Department of Mathematics, COMSATS University of Islamabad, Sahiwal Campus, Punjab, Pakistan)

  • ASMA

    (Department of Mathematics, COMSATS University of Islamabad, Sahiwal Campus, Punjab, Pakistan)

  • MUSTAFA INC

    (Department of Mathematics, Science Faculty, Firat University, 23119 Elâziğ, Turkey4Department of Medical Research, China Medical University, 40402 Taichung, Taiwan)

  • KHURSHEED J. ANSARI

    (Department of Mathematics, College of Science, King Khalid University, 61413 Abha, Saudi Arabia)

Abstract

This research work is devoted to endeavor some results for a delay implicit impulsive type problem under Atangana–Baleanu fractional derivative. The concerned derivative utilizes a nonlocal and non-singular kernel. We build some hypotheses to prove our results. We use Banach and Krasnoselskii fixed point theorems to derive the required results. We consider the following problem involving nonlocal and non-singular fractional derivative with delay term: 𠒜ℬ𠒞𠒟𠜗𠔵(ζ) = Φ(ζ)𠔵(ζ) + Ω(ζ, 𠔵(ζ − τ),𠒜ℬ𠒞𠒟𠜗𠔵(ζ)),ζ ∈ Δ = [0,𠜃], 0 < 𠜗 ≤ 1,𠔵(0) = 𠔵0 +∫0𠜃(𠜃−ψ)𠜗−1 Γ(𠜗) 𠔣(𠔵(ψ))dψ,(1) here 0 < 𠜗,τ ≤ 1, represent the order of the derivative λ, Φ : Δ →ℜ is bounded linear operator and Ω : Δ ×ℜ→ℜ shows a nonlinear continuous function. Stability theory of Ulam–Hyers is used to established the stability results. We provide some examples to demonstrate our theoretical findings.

Suggested Citation

  • Shao-Wen Yao & Yasmeen Sughra & Asma & Mustafa Inc & Khursheed J. Ansari, 2022. "Qualitative Analysis Of Implicit Delay Mittag-Leffler-Type Fractional Differential Equations," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 30(08), pages 1-14, December.
    • Handle: RePEc:wsi:fracta:v:30:y:2022:i:08:n:s0218348x22402083
    DOI: 10.1142/S0218348X22402083
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