In this article, we explore the question of existence and finite time stability for fuzzy Hilfer-Katugampola fractional delay differential equations. By using the generalized Gronwall inequality and Schauder’s fixed point theorem, we establish existence of the solution, and the finite time stability for the presented problems. Finally, the effectiveness of the theoretical result is shown through verification and simulations for an example.
BaleanuD.Fractional calculus: models and numerical methods, World Scientific, Singapore, 2012.
2.
HilferR.Applications of fractional calculus in physics, applications of fractional calculus in physics, World Scientific, Singapore, 2000.
3.
KilbasA.A., SrivastavaH.M. and TrujilloJ.J., Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
4.
PodlubnyI.Fractional differential equations, Academic Press, San Diego, 1999.
5.
AllahviranlooT.Fuzzy fractional differential operators and equations, Springer, New York, 2021.
6.
ChakravertyS., TapaswiniS., BeheraD.Fuzzy arbitrary order system: fuzzy fractional differential equations and applications, John Wiley &Sons, 2016.
7.
BedeB. and StefaniniL., Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems230 (2013), 119–141.
8.
GomesL.T. and BarrosL.C., A note on the generalized difference and the generalized differentiability, Fuzzy Sets and Systems280 (2015), 142–145.
9.
StefaniniL. and BedeB., Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis: Theory, Methods & Applications71 (2009), 1311–1328.
10.
AgarwalR.P., LakshmikanthamV. and NietoJ.J., On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis: Theory, Methods & Applications72 (2010), 2859–2862.
11.
AlikhaniR. and BahramiF., Global solutions for nonlinear fuzzy fractional integral and integro-differential equations, Communications in Nonlinear Science and Numerical Simulation18 (2013), 2007–2017.
12.
SalahshourS. and AbbasbandyS., A comment on “Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations”, Communications in Nonlinear Science and Numerical Simulation19 (2014), 1256–1258.
13.
FardO.S. and SalehiM., A survey on fuzzy fractional variational problems, Journal of Computational & Applied Mathematics271 (2014), 71–82.
14.
MazandaraniM. and KamyadA.V., Modified fractional Euler method for solving fuzzy fractional initial value problem, Communications in Nonlinear Science and Numerical Simulation18 (2013), 12–21.
15.
NajariyanM. and ZhaoY., Fuzzy fractional quadratic regulator problem under granular fuzzy fractional derivatives, IEEE Transactions on Fuzzy Systems26 (2017), 2273–2288.
16.
SalahshourS., AllahviranlooT. and AbbasbandyS., Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Communications in Nonlinear Science and Numerical Simulation17 (2012), 1372–1381.
17.
HoaN.V., LupulescuV. and O’ReganD., A note on initial value problems for fractional fuzzy differential equations, Fuzzy Sets and Systems347 (2018), 54–69.
18.
PrakashP., NietoJ.J., SenthilvelavanS. and SudhaG., Priya, Fuzzy fractional initial value problem, Journal of Intelligent & Fuzzy Systems28 (2015), 2691–2704.
19.
HoaN.V., VuH. and DucT.M., Fuzzy fractional differential equations under Caputo-Katugampola fractional derivative approach, Fuzzy Sets and Systems375 (2019), 70–99.
20.
ChenX.R., GuH.B. and WangX.Z., Existence and uniqueness for fuzzy differential equation with Hilfer-Katugampola fractional derivative, Advances in Difference Equations2020 (2020), 241.
21.
PhuN.D., LupulescuV. and HoaN.V., Neutral fuzzy fractional functional differential equations, Fuzzy Sets and Systems419 (2021), 1–34.
22.
VuH., GhanbariB. and HoaN.V., Fuzzy fractional differential equations with the generalized Atangana-Baleanu fractional derivative, Fuzzy Sets and Systems429 (2022), 1–27.
23.
HoaN.V. and VuH., A survey on the initial value problems of fuzzy implicit fractional differential equations, Fuzzy Sets and Systems400 (2020), 90–133.
24.
LongH.V., SonN.T.K., TamH.T.T. and YaoJ.C., Ulam stability for fractional partial integro-differential equation with uncertainty, Acta Mathematica Vietna42 (2017), 675–700.
25.
HoaN.V., On the stability for implicit uncertain fractional integral equations with fuzzy concept, Iranian Journal of Fuzzy Systems18(1) (2021), 185–201.
26.
VuH., AnT.V. and HoaN.V., Ulam-Hyers stability of uncertain functional differential equation in fuzzy setting with Caputo-Hadamard fractional derivative concept, Iranian Journal of Fuzzy Systems38 (2020), 2245–2259.
27.
VuH., RassiasJ.M. and HoaN.V., Ulam-Hyers-Rassias stability for fuzzy fractional integral equations, Iranian Journal of Fuzzy Systems17(2) (2020), 17–17.
28.
WangX., LuoD.F. and ZhuQ.X., Ulam-Hyers stability of caputo type fuzzy fractional differential equations with time-delays, Chaos, Solitons and Fractals156 (2022), 111822.
29.
CaoX.K. and WangJ.R., Finite-time stability of a class of oscillating systems with two delays, Mathematical Methods in the Applied Sciences41(13) (2018), 4943–4954.
30.
HienL.V., An explicit criterion for finite-time stability of linearnonautonomous systems with delays, Applied Mathematics Letters30 (2014), 12–18.
31.
OblozaM., Connections between Hyers and Lyapunov Stability of theOrdinary Differential Equations, Rocznik Nauk-Dydakt, PraceMat14 (1997), 141–146.
32.
OliveiraD.S. and de OliveiraE.C., Hilfer-Katugampola fractional derivatives, Computational and Applied Mathematics37(3) (2018), 3672–3690.
33.
GomesL.T., BarrosL.C., BedeB.Fuzzy differential equations in various approaches, Springer, NewYork, 2015.
34.
de BarrosL.C., BassaneziR.C.LodwickW.A.The extension principle of Zadeh and fuzzy numbers, in: A first course in fuzzy logic, fuzzy dynamical systems, and biomathematics, Springer, Berlin, Heidelberg, 2017.
35.
LiM.M. and WangJ.R., Finite time stability and relative controllability of Riemann-Liouville fractional delay differential equations, Mathematical Methods in the Applied Sciences42(18) (2019), 6607–6623.
36.
SousaJ.V.C., OliveiraE.C. Gronwall inequality and the Cauchy-type problem by means of Ψ-Hilfer operator, arXiv preprint arXiv, 2017.
