QUALITATIVE ANALYSIS OF IMPLICIT DELAY MITTAG-LEFFLER-TYPE FRACTIONAL DIFFERENTIAL EQUATIONS

Shao-Wen Yao (), Yasmeen Sughra (), Asma (), Mustafa Inc and Khursheed J. Ansari ()
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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

FRACTALS (fractals), 2022, vol. 30, issue 08, 1-14

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.

Keywords: Non-Singular Kernel; Krasnoselskii Fixed Point Theorem; Existence and Uniqueness; Stability Results (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0218348X22402083

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