Hyers-Ulam-Rassias-Kummer stability of the fractional integro-differential equations

Zahra Eidinejad; Reza Saadati; Zahra Eidinejad; Reza Saadati

doi:10.3934/mbe.2022308

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 7: 6536-6550. doi: 10.3934/mbe.2022308

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Hyers-Ulam-Rassias-Kummer stability of the fractional integro-differential equations

Zahra Eidinejad ,
Reza Saadati ^,

School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 13114-16846, Iran

Academic Editor: Rosana Rodríguez-López

Received: 15 October 2021 Revised: 23 March 2022 Accepted: 08 April 2022 Published: 25 April 2022

In this paper, using the fractional integral with respect to the $ \Psi $ function and the $ \Psi $-Hilfer fractional derivative, we consider the Volterra fractional equations. Considering the Gauss Hypergeometric function as a control function, we introduce the concept of the Hyers-Ulam-Rassias-Kummer stability of this fractional equations and study existence, uniqueness, and an approximation for two classes of fractional Volterra integro-differential and fractional Volterra integral. We apply the Cădariu-Radu method derived from the Diaz-Margolis alternative fixed point theorem. After proving each of the main theorems, we provide an applied example of each of the results obtained.
Citation: Zahra Eidinejad, Reza Saadati. Hyers-Ulam-Rassias-Kummer stability of the fractional integro-differential equations[J]. Mathematical Biosciences and Engineering, 2022, 19(7): 6536-6550. doi: 10.3934/mbe.2022308

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Abstract

In this paper, using the fractional integral with respect to the $ \Psi $ function and the $ \Psi $-Hilfer fractional derivative, we consider the Volterra fractional equations. Considering the Gauss Hypergeometric function as a control function, we introduce the concept of the Hyers-Ulam-Rassias-Kummer stability of this fractional equations and study existence, uniqueness, and an approximation for two classes of fractional Volterra integro-differential and fractional Volterra integral. We apply the Cădariu-Radu method derived from the Diaz-Margolis alternative fixed point theorem. After proving each of the main theorems, we provide an applied example of each of the results obtained.

References

[1]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, in North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, (2006), 1–523. https://doi.org/10.1016/s0304-0208(06)x8001-5
[2]	K. B. Oldham, J. Spanier, Chapter 3: fractional derivatives and integrals: definitions and equivalences, Math Sci. Eng., 111 (1974), 45–66. https://doi.org/10.1016/s0076-5392(09)60225-3 doi: 10.1016/s0076-5392(09)60225-3
[3]	I. Podlubny, Chapter 8-numerical solution of fractional differential equations, Math. Sci. Eng., 198 (1999), 223–242. https://doi.org/10.1016/s0076-5392(99)80027-7 doi: 10.1016/s0076-5392(99)80027-7
[4]	J. V. da C. Sousa, E. C. de Oliveira, On the $\Psi$-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72–91. https://doi.org/10.1016/j.cnsns.2018.01.005 doi: 10.1016/j.cnsns.2018.01.005
[5]	H. Amann, Chapter III: linear differential equations, in Ordinary Differential Equations, Walter de Gruyter & Co., Berlin, (1990), 136–197. https://doi.org/10.1515/9783110853698.136
[6]	C. Chicone, Ordinary differential equations with applications, in Texts in Applied Mathematics, Springer, New York, (2006), 145–322. https://doi.org/10.1007/0-387-35794-7
[7]	M. Abramowitz, Handbook of mathematical functions with formulas, graphs, and mathematical tables, J. Appl. Mech., 32 (1965), 239–239. https://doi.org/10.1115/1.3625776 doi: 10.1115/1.3625776
[8]	J. V. da C.Sousa, E. C. de Oliveira, Two new fractional derivatives of variable order with non-singular kernel and fractional differential equation, Comput. Appl. Math., 37 (2018), 5375–5394. https://doi.org/10.1007/s40314-018-0639-x doi: 10.1007/s40314-018-0639-x
[9]	Z. Eidinejad, R. Saadati, M. De La Sen, Picard method for existence, uniqueness, and Gauss Hypergeomatric stability of the fractional-order differential equations, Math. Probl. Eng. (2021), 2–3. https://doi.org/10.1155/2021/7074694
[10]	M. S. Abdo, S. K. Panchal, H. A. Wahash, Ulam-Hyers-Mittag-Leffler stability for a $\psi$-Hilfer problem with fractional order and infinite delay, Results Appl. Math., 7 (2020), 100115. https://doi.org/10.1016/j.rinam.2020.100115 doi: 10.1016/j.rinam.2020.100115
[11]	K. Ravikumar, K. Ramkumar, D. Chalishajar, Existence and stability results for second-order neutral stochastic differential equations with random impulses and poisson jumps, Eur. J. Math. Anal., 1 (2021), 1–8. https://doi.org/10.28924/ada/ma.1.1 doi: 10.28924/ada/ma.1.1
[12]	D. Chalishajar, K. Ramkumar, A. Anguraj, K. Ravikumar, M. A. Diop, Controllability of neutral impulsive stochastic functional integrodifferential equations driven by a fractional Brownian motion with infinite delay via resolvent operator, J. Nonlinear Sci. Appl., 15 (2022), 172–185. http://dx.doi.org/10.22436/jnsa.015.03.01 doi: 10.22436/jnsa.015.03.01
[13]	K. Ramkumar, K. Ravikumar, D. Chalishajar, A. Anguraj, Asymptotic behavior of attracting and quasi-invariant sets of impulsive stochastic partial integrodifferential equations with delays and Poisson jumps, J. Nonlinear Sci. Appl., 14 (2021), 339–350. http://dx.doi.org/10.22436/jnsa.014.05.04 doi: 10.22436/jnsa.014.05.04
[14]	M. S. Abdo, S. K. Panchal, Fractional integro-differential equations involving $\psi$-Hilfer fractional derivative, Adv. Appl. Math. Mech., 11 (2019), 338–359. http://dx.doi.org/10.4208/aamm.OA-2018-0143 doi: 10.4208/aamm.OA-2018-0143
[15]	M. S. Abdo, S. K. Panchal, K. Satish, H. S. Hussien, Fractional integro-differential equations with nonlocal conditions and $\psi$-Hilfer fractional derivative, Math. Model. Anal., 24 (2019), 564–584. http://dx.doi.org/10.3846/mma.2019.034 doi: 10.3846/mma.2019.034
[16]	H. A. Wahash, M. S. Abdo, S. K. Panchal, Fractional integrodifferential equations with nonlocal conditions and generalized Hilfer fractional derivative, Ufa Math. J., 11 (2019), 151–171. http://dx.doi.org/10.13108/2019-11-4-151 doi: 10.13108/2019-11-4-151
[17]	J. V. d. C. Sousa, F. G. Rodrigues, E. C. de Oliveira, Stability of the fractional Volterra integro-differential equation by means of $\psi$-Hilfer operator, Math. Methods Appl. Sci., 42 (2019), 3033–3043. https://doi.org/10.1002/mma.5563 doi: 10.1002/mma.5563
[18]	Y. Zhou, Existence and uniqueness of solutions for a system of fractional differential equations, Fract. Calc. Appl. Anal., 12 (2009), 195–204.
[19]	Y. Zhou, J. Wang, L. Zhang, Basic theory of fractional differential equations, Second edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2017), 1–380. https://doi.org/10.1142/10238
[20]	J. Huang, Y. Li, Hyers-Ulam stability of delay differential equations of first order, Math. Nachr., 289 (2016), 60–66. http://dx.doi.org/10.1002/mana.201400298 doi: 10.1002/mana.201400298
[21]	E. Graily, S. M. Vaezpour, R. Saadati, Y. J. Cho, Generalization of fixed point theorems in ordered metric spaces concerning generalized distance, Fixed Point Theory Appl., 30 2011, 8. http://dx.doi.org/10.1186/1687-1812-2011-30
[22]	S. Shakeri, L. J. B. Ciric, Common fixed point theorem in partially ordered $L$-fuzzy metric spaces, Fixed Point Theory Appl., (2010), 13. doi: 10.1155/2010/125082
[23]	L. Ciric, M. Abbas, B. Damjanovic, Common fuzzy fixed point theorems in ordered metric spaces, Math. Comput. Modell., 53 (2011), 1737–1741. https://doi.org/10.1016/j.mcm.2010.12.050 doi: 10.1016/j.mcm.2010.12.050
[24]	Y. J. Cho, Lattictic non-Archimedean random stability of ACQ functional equation, Adv. Differ. Equations, 31 (2011), 12. https://doi.org/10.1186/1687-1847-2011-31 doi: 10.1186/1687-1847-2011-31
[25]	D. Mihet, R. Saadati, S. M. Vaezpour, The stability of an additive functional equation in Menger probabilistic $\phi$-normed spaces, Math. Slovaca., 61 (2011), 817–826. http://dx.doi.org/10.2478/s12175-011-0049-7 doi: 10.2478/s12175-011-0049-7
[26]	Y. J. Cho, C. Park, T. M. Rassias, R. Saadati, Stability of Functional Equations in Banach Algebras, Springer, (2015). http://dx.doi.org/10.1007/978-3-319-18708-2
[27]	J. V. da C. Sousa, E. C. de Oliveira, Mittag-Leffler functions and the truncated ${\mathcal {V}} $-fractional derivative, Mediterr. J. Math., 14 (2017), 61–26. https://doi.org/10.1007/s00009-017-1046-z doi: 10.1007/s00009-017-1046-z
[28]	V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal., 69 (2008), 3337–3343. https://doi.org/10.1016/j.na.2007.09.025 doi: 10.1016/j.na.2007.09.025
[29]	N. D. Phuong, N. A. Tuan, D. Kumar, N. H. Tuan, Initial value problem for fractional Volterra integrodifferential pseudo-parabolic equations, Math. Model. Nat. Phenom., 16 (2021), 14. http://dx.doi.org/10.1051/mmnp/2021015 doi: 10.1051/mmnp/2021015
[30]	A. Salim, M. Benchohra, E. Karapinar, J. E. Lazreg, Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations, Adv. Differ. Equations, 601 (2020), 21. http://doi.org/10.1186/s13662-020-03063-4 doi: 10.1186/s13662-020-03063-4
[31]	Z. Wang, D. Cen, Y. Mo, Sharp error estimate of a compact $L$1-ADI scheme for the two-dimensional time-fractional integro-differential equation with singular kernels, Appl. Numer. Math., 159 (2021), 190–203. http://doi.org/10.1016/j.apnum.2020.09.006 doi: 10.1016/j.apnum.2020.09.006
[32]	M. Janfada, G. Sadeghi, Stability of the Volterra integrodifferential equation, Folia Math., 18 (2013), 11–20.
[33]	J. B. Diaz, B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305–309. https://doi.org/10.1090/S0002-9904-1968-11933-0 doi: 10.1090/S0002-9904-1968-11933-0

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