On the existence of solutions and Hyers–Ulam stability for Caputo ϑ-fractional iterative integro-differential equations

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Abstract

This work investigated a Caputo ϑ-fractional iterative integro-differential equation (Caputo ϑ-FIIDE). First, the existence of solutions was established by means of Schauder’s fixed point theorem, and uniqueness was obtained under suitable sufficient conditions. The continuous dependence of solutions on the given data was then analyzed. Furthermore, the Ulam-Hyers stability (the U-H stability) and the Ulam-Hyers-Rassias stability (the U-H-R stability) of the proposed problem were examined. A new illustrative example was presented to demonstrate the applicability of the theoretical results and to validate the developed analysis. Thus, this paper presents novel theoretical results, supported by an illustrative example that highlights the applicability of the proposed approach and contributes to the existing literature. Since the stability analysis of fractional iterative integro-differential equations plays a significant role in both theory and applications, the results obtained here provide a useful framework for future research and stimulate further advances in this area.

Keywords

iterative integro-differential equation / fractional derivative / existence and uniqueness of solutions / Ulam-type stability

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References

[1] H. Singh, H. M. Srivastava, J. J. Nieto, Handbook of Fractional Calculus for Engineering and Science, 1st edition, CRC Press, Boca Raton, 2022. https://doi.org/10.1201/9781003263517

[2] W. Chen, H. G. Sun, X. Li, Fractional Derivative Modeling in Mechanics and Engineering, 1st edition, Springer, Singapore, 2022. https://doi.org/10.1007/978-981-16-8802-7

[3] K. B. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Software, 41 (2010), 9-12. https://doi.org/10.1016/j.advengsoft.2008.12.012

[4] R. Herrmann, Fractional Calculus: An Introduction for Physicists, 1st edition, World Scientific, Hackensack, 2011. https://doi.org/10.1142/8072

[5] R. Hilfer, Applications of Fractional Calculus in Physics, 1st edition, World Scientific, River Edge, 2000. https://doi.org/10.1142/3779

[6] H. G. Sun, A. Chang, Y. Zhang, W. Chen, A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications, Fract. Calc. Appl. Anal., 22 (2019), 27-59. https://doi.org/10.1515/fca-2019-0003

[7] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, 1st edition, Springer, Berlin, 2010. https://doi.org/10.1007/978-3-642-14574-2

[8] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

[9] Y. Yang, H. H. Zhang, Fractional Calculus With Its Applications In Engineering and Technology, 1st edition, Springer, Cham, 2019. https://doi.org/10.1007/978-3-031-79625-8

[10] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, 1st edition, Academic Press, San Diego, 1999. https://doi.org/10.1016/s0076-5392(99)x8001-5

[11] D. Luo, K. Shah, Z. Luo, On the novel Ulam-Hyers stability for a class of nonlinear Hilfer fractional differential equation with time-varying delay, Mediterr. J. Math., 16 (2019), 112. https://doi.org/10.1007/s00009-019-1387-x

[12] E. C. de Oliveira, J. V. da C. Sousa, Ulam-Hyers-Rassias stability for a class of fractional integro-differential equations, Results Math., 73 (2018), 111. https://doi.org/10.1007/s00025-018-0872-z

[13] J. R. Graef, O. Tunç, C. Tunç, Ulam-Hyers-Rassias stability of ψ-Hilfer Volterra integro-differential equations of fractional order containing multiple variable delays, Fractal Fract., 9 (2025), 304. https://doi.org/10.3390/fractalfract9050304

[14] C. Tunç, F. S. Alshammari, F. T. Akyildiz, On the existence of solutions and Ulam-type stability for a nonlinear ψ-Hilfer fractional-order delay integro-differential equation, Fractal Fract., 9 (2025), 409. https://doi.org/10.3390/fractalfract9070409

[15] C. Tunç, O. Tunç, Ulam-type stability results for fractional integro-delay differential and integral equations via the ψ-Hilfer operator, Fractal Fract., 10 (2026), 57. https://doi.org/10.3390/fractalfract10010057

[16] J. V. da C. Sousa, F. G. Rodrigues, E. C. de Oliveira, Stability of the fractional Volterra integro-differential equation by means of ψ-Hilfer operator, Math. Methods Appl. Sci., 42 (2019), 3033-3043. https://doi.org/10.1002/mma.5563

[17] J. V. da C. Sousa, E. C. de Oliveira, Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Lett., 81 (2018), 50-56. https://doi.org/10.1016/j.aml.2018.01.016

[18] H. Vu, N. V. Hoa, Existence of solutions and Hyers-Ulam stability for K-fractional iterative differential equations, Math. Slovaca, 75 (2025), 913-926. https://doi.org/10.1515/ms-2025-0066

[19] H. Vu, J. M. Rassias, N. V. Hoa, Hyers-Ulam stability for boundary value problem of fractional differential equations with K-Caputo fractional derivative, Math. Methods Appl. Sci., 46 (2023), 438-460. https://doi.org/10.1002/mma.8520

[20] X. Xu, Z. Shen, S. Liu, Q. Dong, Existence and Hyers-Ulam stability for boundary value problems of two-term fractional differential equations with K-Caputo derivative, J. Appl. Anal. Comput., 15 (2025), 2786-2804. https://doi.org/10.11948/20240526

[21] S. Pinelas, C. Tunç, O. Tunç, M. Ş. Oguz, On the Ulam-type stability of higher-order iterative Volterra integro-delay differential equations, AIMS Math., 11 (2026), 1219-1238. https://doi.org/10.3934/math.2026052

[22] S. M. Ulam Problems in Modern Mathematics, 1st edition, John Wiley & Sons, New York, 1964.

[23] H. D. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A., 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222

[24] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc., 72 (1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1

[25] E. Eder, The functional-differential equation x′(t) = x(x(t)), J. Differ. Equations, 54 (1984), 390-400. https://doi.org/10.1016/0022-0396(84)90150-5

[26] M. Feckan, On a certain type of functional-differential equations, Math. Slovaca, 43 (1993), 39-43.

[27] A. Buica, Existence and continuous dependence of solutions of some functional-differential equations, Semin. Fixed Point Theory, 3 (1995), 1-14.

[28] V. Berinde, Existence and approximation of solutions of some first order iterative differential equations, Miskolc Math. Notes, 11 (2010), 13-26. https://doi.org/10.18514/MMN.2010.256

[29] R. W. Ibrahim, Existence of deviating fractional differential equation, Cubo, 14 (2012), 129-142. https://doi.org/10.4067/S0719-06462012000300009

[30] M. Zhou, Well-posedness of nonlinear fractional quadratic iterative differential equations, J. Anal., 31 (2023), 881-897. https://doi.org/10.1007/s41478-022-00484-0

[31] S. Abbas, M. Benchohra, A. Petruşel, Ulam stability for Hilfer type fractional differential inclusions via the weakly Picard operators theory, Fract. Calc. Appl. Anal., 20 (2017), 384-398. https://doi.org/10.1515/fca-2017-0020

[32] C. Benzarouala, C. Tunç, Hyers-Ulam-Rassias stability of fractional delay differential equations with Caputo derivative, Math. Methods Appl. Sci., 47 (2024), 13499-13509. https://doi.org/10.1002/mma.10202

[33] M. Rhaima, L. Mchiri, A. B. Makhlouf, H. Ahmed, Ulam type stability for mixed Hadamard and Riemann-Liouville fractional stochastic differential equations, Chaos, Solitons Fractals, 178 (2024), 114356. https://doi.org/10.1016/j.chaos.2023.114356

[34] A. Granas, J. Dugundji, Fixed Point Theory, 1st edition, Springer-Verlag, New York, 2003. https://doi.org/10.1007/978-0-387-21593-8

[35] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460-481. https://doi.org/10.1016/j.cnsns.2016.09.006

[36] E. R. Kaufmann, A fourth-order iterative boundary value problem with Lidstone boundary conditions, Differ. Equations Appl., 14 (2022), 305-312. https://doi.org/10.7153/dea-2022-14-21