Existence and data dependence results for neutral fractional order integro-differential equations

Veliappan Vijayaraj; Chokkalingam Ravichandran; Thongchai Botmart; Kottakkaran Sooppy Nisar; Kasthurisamy Jothimani; Veliappan Vijayaraj; Chokkalingam Ravichandran; Thongchai Botmart; Kottakkaran Sooppy Nisar; Kasthurisamy Jothimani

doi:10.3934/math.2023052

AIMS Mathematics

2023, Volume 8, Issue 1: 1055-1071. doi: 10.3934/math.2023052

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Research article

Existence and data dependence results for neutral fractional order integro-differential equations

1.
Department of Mathematics, Kongunadu Arts and Science College (Autonomous), Coimbatore 641029, Tamil Nadu, India
2.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
3.
Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser 11991, Saudi Arabia
4.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, Tamil Nadu, India

Received: 10 August 2022 Revised: 17 September 2022 Accepted: 20 September 2022 Published: 14 October 2022
MSC : 34A60, 33E12, 34G20

We assess the multi-derivative nonlinear neutral fractional order integro-differential equations with Atangana-Baleanu fractional derivative of the Riemann-Liouville sense. We discuss results about the existence and difference solution on some data, based on the Prabhakar fractional integral operator $ \varepsilon^{\alpha}_{\delta, \eta, \mathcal{V}; c+} $ with generalized Mittag-Leffler function. The results are obtained by using Krasnoselskii's fixed point theorem and the Gronwall-Bellman inequality.
- fractional calculus,
- fixed point techniques,
- Gronwall-Bellman inequality
Citation: Veliappan Vijayaraj, Chokkalingam Ravichandran, Thongchai Botmart, Kottakkaran Sooppy Nisar, Kasthurisamy Jothimani. Existence and data dependence results for neutral fractional order integro-differential equations[J]. AIMS Mathematics, 2023, 8(1): 1055-1071. doi: 10.3934/math.2023052

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Abstract

We assess the multi-derivative nonlinear neutral fractional order integro-differential equations with Atangana-Baleanu fractional derivative of the Riemann-Liouville sense. We discuss results about the existence and difference solution on some data, based on the Prabhakar fractional integral operator $ \varepsilon^{\alpha}_{\delta, \eta, \mathcal{V}; c+} $ with generalized Mittag-Leffler function. The results are obtained by using Krasnoselskii's fixed point theorem and the Gronwall-Bellman inequality.

References

[1]	T. Abdeljawad, Fractional operators with generalized Mittag-Leffler kernels and their iterated differintegrals, Chaos, 29 (2019), 023102. https://doi.org/10.1063/1.5085726 doi: 10.1063/1.5085726
[2]	T. Abdeljawad, D. Baleanu, On fractional derivatives with generalized Mittag-Leffler kernels, Adv. Differ. Equ., 2018 (2018), 468. https://doi.org/10.1186/s13662-018-1914-2 doi: 10.1186/s13662-018-1914-2
[3]	A. A. Hamoud, Existence and uniqueness of solutions for fractional neutral Volterra-Fredholm integro-diffrential equations, Adv. Theor. Nonlinear Anal. Appl., 4 (2020), 321–331. https://doi.org/10.31197/atnaa.799854 doi: 10.31197/atnaa.799854
[4]	A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.48550/arXiv.1602.03408 doi: 10.48550/arXiv.1602.03408
[5]	A. Atangana, S. I. Araz, Modeling and forecasting the spread of COVID-19 with stochastic and deterministic approaches: Africa and Europe, Adv. Differ. Equ., 2021 (2021), 57. https://doi.org/10.1186/s13662-021-03213-2 doi: 10.1186/s13662-021-03213-2
[6]	A. Atangana, S. I. Araz, Rhythmic behaviors of the human heart with piecewise derivative, Math. Biosci. Eng., 19 (2022), 3091–3109. https://doi.org/10.3934/mbe.2022143 doi: 10.3934/mbe.2022143
[7]	R. P. Agarwal, Y. Zhou, Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl., 59 (2010), 1095–1100. https://doi.org/10.1016/j.camwa.2009.05.010 doi: 10.1016/j.camwa.2009.05.010
[8]	N. H. Aljahdaly, R. Shah, R. P. Agarwal, T. Botmart, The analysis of the fractional-order system of third-order KdV equation within different operators, Alex. Eng. J., 61 (2022), 11825–11834. https://doi.org/10.1016/j.aej.2022.05.032 doi: 10.1016/j.aej.2022.05.032
[9]	M. Alshammari, N. Iqbal, W. W. Mohammed, T. Botmart, The solution of fractional-order system of KdV equations with exponential-decay kernel, Results Phys., 38 (2022), 105615. https://doi.org/10.1016/j.rinp.2022.105615 doi: 10.1016/j.rinp.2022.105615
[10]	E. Bonyah, R. Zarin Fatmawati, Mathematical modeling of cancer and hepatitis co-dynamics with non-local and non-singular kernel, Commun. Math. Biol. Neu., 2022 (2022). http://doi.org/10.28919/cmbn/5029 doi: 10.28919/cmbn/5029
[11]	A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, McGraw-Hill, New York, 1953.
[12]	A. Fernandez, T. Abdeljawad, D. Baleanu, Relations between fractional models with three-parameter Mittag-Leffler kernels, Adv. Differ. Equ., 2020 (2020), 186. https://doi.org/10.1186/s13662-020-02638-5 doi: 10.1186/s13662-020-02638-5
[13]	N. Iqbal, T. Botmart, W. W. Mohammed, A. Ali, Numerical investigation of fractional-order Kersten-Krasil'shchik coupled KdV-mKdV system with Atangana-Baleanu derivative, Adv. Cont. Dis. Model., 2022 (2022), 37. https://doi.org/10.1186/s13662-022-03709-5 doi: 10.1186/s13662-022-03709-5
[14]	F. Jarad, T. Abdeljawad, Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos Soliton. Fract., 117 (2018), 16–20. https://doi.org/10.1016/j.chaos.2018.10.006 doi: 10.1016/j.chaos.2018.10.006
[15]	F. Mainardi, R. Garrappa, On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics, J. Comput. Phys., 293 (2015), 70–80. https://doi.org/10.1016/j.jcp.2014.08.006 doi: 10.1016/j.jcp.2014.08.006
[16]	K. Kumar, R. Patel, V. Vijayakumar, A. Shukla, C. Ravichandran, A discussion on boundary controllability of nonlocal impulsive neutral integrodifferential evolution equations, Math. Meth. Appl. Sci., 45 (2022), 8193–8215. https://doi.org/10.1002/mma.8117 doi: 10.1002/mma.8117
[17]	A. A. Kilbas, M. Saigo, K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integr. Transf. Spec. F., 15 (2004), 31–49. https://doi.org/10.1080/10652460310001600717 doi: 10.1080/10652460310001600717
[18]	K. S. Nisar, K. Jothimani, K. Kaliraj, C. Ravichandran, An analysis of controllability results for nonlinear Hilfer neutral fractional derivatives with non-dense domain, Chaos Soliton. Fract., 146 (2021), 110915. https://doi.org/10.1016/j.chaos.2021.110915 doi: 10.1016/j.chaos.2021.110915
[19]	K. D. Kucche, S. T. Sutar, Analysis of nonlinear fractional diferential equations involving AB Caputo derivative, Chaos Soliton. Fract., 143 (2021), 110556. https://doi.org/10.1016/j.chaos.2020.110556 doi: 10.1016/j.chaos.2020.110556
[20]	K. D. Kucche, J. J. Trujillo, Theory of system of nonlinear fractional diferential equations, Prog. Fract. Differ. Appl., 3 (2017), 7–18. http://doi.org/10.18576/pfda/030102 doi: 10.18576/pfda/030102
[21]	K. D. Kucche, J. J. Nieto, V. Venktesh, Theory of nonlinear implicit fractional diferential equations, Differ. Equat. Dyn. Sys., 28 (2020), 1–17. http://doi.org/10.1007/s12591-016-0297-7 doi: 10.1007/s12591-016-0297-7
[22]	K. Logeswari, C. Ravichandran, K. S. Nisar, Mathematical model for spreading of COVID-19 virus with the Mittag-Leffler kernel, Numer. Meth. Part. D. E., 2020, 1–16. http://doi.org/10.1002/num.22652 doi: 10.1002/num.22652
[23]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons, New York, 1993.
[24]	M. S. Abdo, S. K. Panchal, Existence and continuous dependence for fractional neutral functional differential equations, J. Math. Model., 5 (2017), 153–170. https://dx.doi.org/10.22124/jmm.2017.2535 doi: 10.22124/jmm.2017.2535
[25]	A. S. Mohamed, R. A. Mahmoud, Picard, Adomian and perdictor-corrector methods for an initial value problem of arbitrary (fractional) prders differential equation, J. Egyptian Math. Soc., 24 (2016), 165–170. https://doi.org/10.1016/J.JOEMS.2015.01.001 doi: 10.1016/J.JOEMS.2015.01.001
[26]	I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
[27]	T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama. Math. J., 19 (1971), 7–15.
[28]	B. G. Pachpatte, Inequalities for differential and integral equations, Academic Press, San Diago, 1998.
[29]	C. Ravichandran, V. Sowbakiya, K. S. Nisar, Study on existence and data dependence results for fractional order differential equations, Chaos Soliton. Fract., 160 (2022), 112232. https://doi.org/10.1016/j.chaos.2022.112232 doi: 10.1016/j.chaos.2022.112232
[30]	C. Ravichandran, D. Baleanu, On the controllability of fractional functional integro-differential systems with an infinite delay in Banach spaces, Adv. Differ. Equ., 2013 (2013), 291. https://doi.org/10.1186/1687-1847-2013-291 doi: 10.1186/1687-1847-2013-291
[31]	S. K. Verma, R. K. Vats, A. Kumar, V. Vijayakumar, A. Shukla, A discussion on the existence and uniqueness analysis for the coupled two-term fractional differential equations, Turkish J. Math., 46 (2022), 516–532. https://doi.org/10.3906/mat-2107-30 doi: 10.3906/mat-2107-30
[32]	S. T. Sutar, K. D. Kucche, Existence and data dependence results for fractional differential equations involving Atangana-Baleanu derivative, Rend. Circ. Mat. Palerm., 71 (2022), 647–663. https://doi.org/10.1007/s12215-021-00622-w doi: 10.1007/s12215-021-00622-w
[33]	S. T. Sutar, K. D. Kucche, On nonlinear hybrid fractional diferential equations with AB-Caputo derivative, Chaos Soliton. Fract., 143 (2021), 110557. https://doi.org/10.1016/j.chaos.2020.110557 doi: 10.1016/j.chaos.2020.110557
[34]	J. V. C. Sousa, , K. D. Kucche, E. C. Oliveira, Stability of mild solutions of the fractional nonlinear abstract Cauchy problem, Electronic Research Archive, 30 (2022), 272–288. http://doi.org/10.3934/era.2022015 doi: 10.3934/era.2022015
[35]	X. B. Shu, Y. J. Shi, A study on the mild solution of impulsive fractional evolution equations, Appl. Math. Comput., 273 (2016), 465–476. https://doi.org/10.1016/j.amc.2015.10.020 doi: 10.1016/j.amc.2015.10.020
[36]	Y. Guo, M. Q. Chen, X. B. Shu, F. Xu, The existence and Hyers-Ulam stability of solution for almost periodical fractional stochastic differential equation with fBm, Stoch. Anal. Appl., 39 (2021), 643–666. https://doi.org/10.1080/07362994.2020.1824677 doi: 10.1080/07362994.2020.1824677
[37]	Y. Guo, X. B. Shu, Y. J. Li, F. Xu, The existence and Hyers-Ulam stability of solution for an impulsive Riemann-Liouville fractional neutral functional stochastic differential equation with infinite delay of order $1 < \beta < 2$, Boundary Value Probl., 2019 (2019), 59. https://doi.org/10.1186/s13661-019-1172-6 doi: 10.1186/s13661-019-1172-6

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