Startseite Mathematik Existence of solutions and Hyers-Ulam stability for κ-fractional iterative differential equations
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Existence of solutions and Hyers-Ulam stability for κ-fractional iterative differential equations

  • Ho Vu und Ngo Van Hoa EMAIL logo
Veröffentlicht/Copyright: 9. August 2025
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Abstract

The aim of this paper is to establish the existence and uniqueness of solutions to the κ-fractional iterative differential equations (κ-FIDEs) using Schauder’s fixed point theorem. We also present the continuous dependence of the solution on the input data and a Hyers-Ulam stability analysis for this problem. Finally, some examples are provided to illustrate our main results.

Acknowledgement

The authors would like to sincerely thank the anonymous referees for their valuable comments and insightful suggestions, which have significantly enhanced the quality and clarity of this paper.

  1. (Communicated by Michal Fečkan)

References

[1] Ali, F.—Sharma, M.—Jain, R.: An application of fractional calculus in electrical engineering, Adv. Eng. Technol. Appl. 5 (2016), 11–15.10.18576/aeta/050204Suche in Google Scholar

[2] Almeida, R.: A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul. 44 (2017), 460–481.10.1016/j.cnsns.2016.09.006Suche in Google Scholar

[3] Almeida, R.—Malinowska, A. B.—Monteiro, M. T. T.: Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Methods Appl. Sci. 41 (2018), 336–352.10.1002/mma.4617Suche in Google Scholar

[4] An, T. V.—Vu, H.—Hoa, N. V.: Finite-time stability of fractional delay differential equations involving the generalized Caputo fractional derivative with non-instantaneous impulses, Math. Methods Appl. Sci. 45 (2022).10.1002/mma.8084Suche in Google Scholar

[5] Berinde, V.: Existence and approximation of solutions of some first order iterative differential equations, Miskolc Math. Notes 11 (2010), 13–26.10.18514/MMN.2010.256Suche in Google Scholar

[6] Buică, A.: Existence and continuous dependence of solutions of some functional-differential equations, Sem. Fixed Point Theory 3 (1995), 1–14.Suche in Google Scholar

[7] Chen, W.—Sun, H. G.—Li, X.: Fractional Derivative Modeling in Mechanics and Engineering, Springer, 2022.10.1007/978-981-16-8802-7Suche in Google Scholar

[8] Eder, E.: The functional differential equation xʹ(t) = x(x(t) J. Differ. Equ. 54 (1984), 390–400.10.1016/0022-0396(84)90150-5Suche in Google Scholar

[9] Farman, M.—Aslam, M.—Akgül, A.—Jarad, F.: On solutions of the stiff differential equations in chemistry kinetics with fractal-fractional derivatives, J. Comput. Nonlinear Dyn. 17 (2022), Art. ID 071007.10.1115/1.4054347Suche in Google Scholar

[10] Fečkan, M.: On a certain type of functional differential equations, Math. Slovaca 43 (1993), 39–43.Suche in Google Scholar

[11] Granas, A.—Dugundji, J.: Fixed Point Theory, Springer, 2003.10.1007/978-0-387-21593-8Suche in Google Scholar

[12] Hilfer, R.: Applications of Fractional Calculus in Physics, World Scientific, 2000.10.1142/9789812817747Suche in Google Scholar

[13] Hoa, N. V.—O’Regan, D.: A remark on ψ-Hilfer fractional differential equations with non-instantaneous impulses, Math. Methods Appl. Sci. 43 (2020), 3354–3368.10.1002/mma.6125Suche in Google Scholar

[14] Ibrahim, R. W.: Existence of deviating fractional differential equation, Cubo 14 (2012), 129–142.10.4067/S0719-06462012000300009Suche in Google Scholar

[15] Kaufmann, E. R.: A fourth-order iterative boundary value problem with Lidstone boundary conditions, Differ. Equ. Appl. 14 (2022), 305–312.10.7153/dea-2022-14-21Suche in Google Scholar

[16] Kharade, J. P.–Kucche, K. D.: On the impulsive implicit ψ-Hilfer fractional differential equations with delay, Math. Methods Appl. Sci. 43 (2020), 1938–1952.10.1002/mma.6017Suche in Google Scholar

[17] Kilbas, A. A.—Marichev, O. I.—Samko, S. G.: Fractional Integrals and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993.Suche in Google Scholar

[18] Kilbas, A. A.—Srivastava, H. M.—Trujillo, J. J.: Theory and Applications of Fractional Differential Equations, Elsevier, 2006.Suche in Google Scholar

[19] Li, C.—Wu, R.—Ma, R.: Existence of solutions for Caputo fractional iterative equations under several boundary value conditions, AIMS Math. 8 (2023), 317–339.10.3934/math.2023015Suche in Google Scholar

[20] Oldham, K. B.: Fractional differential equations in electrochemistry, Adv. Eng. Softw. 41 (2010), 9–12.10.1016/j.advengsoft.2008.12.012Suche in Google Scholar

[21] Petuhov, V. R.: On a boundary value problem, Trudy Aem. Teorii Diff. Uravn. Otklon. Arg 3 (1965), 252–255.Suche in Google Scholar

[22] Prasad, K. R.—Khuddush, M.—Leela, D.: Existence, uniqueness and Hyers–Ulam stability of a fractional order iterative two-point boundary value problems, Afr. Matematika 32 (2021), 1227–1237.10.1007/s13370-021-00895-5Suche in Google Scholar

[23] Sousa, J. V. C.—Oliveira, E. C.: On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul. 60 (2018), 72–91.10.1016/j.cnsns.2018.01.005Suche in Google Scholar

[24] Sousa, J. V. C.—Rodrigues, F. G.—Oliveira, E. C.: Stability of the fractional Volterra integro-differential equation by means of ψ-Hilfer operator, Math. Methods Appl. Sci. 42 (2019), 3033–3043.10.1002/mma.5563Suche in Google Scholar

[25] Sun, H. G.—Chang, A.—Zhang, Y.—Chen, W.: A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications, Fract. Calc. Appl. Anal. 22 (2019), 27–59.10.1515/fca-2019-0003Suche in Google Scholar

[26] Turab, A.—Sintunavarat, W.: A unique solution of the iterative boundary value problem for a second-order differential equation approached by fixed point results, Alex. Eng. J. 60 (2021), 5797–5802.10.1016/j.aej.2021.04.031Suche in Google Scholar

[27] Vu, H.—Rassias, J. M.—HOA, N. V.: Hyers–Ulam stability for boundary value problem of fractional differential equations with κ-Caputo fractional derivative, Math. Methods Appl. Sci. 46 (2023), 438–460.10.1002/mma.8520Suche in Google Scholar

[28] Zhou, M.: Well-posedness of nonlinear fractional quadratic iterative differential equations, J. Anal. 31 (2023), 881–897.10.1007/s41478-022-00484-0Suche in Google Scholar

[29] Zhou, M.—Zhang, L.: Initial value problem for a class of semi-linear fractional iterative differential equations, J. Appl. Anal. Comput. 14 (2024), 2733–2749.10.11948/20230353Suche in Google Scholar

Received: 2024-11-12
Accepted: 2025-03-28
Published Online: 2025-08-09
Published in Print: 2025-08-26

© 2025 Mathematical Institute Slovak Academy of Sciences

Heruntergeladen am 15.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2025-0066/pdf
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