Existence and stability results of pantograph equation with three sequential fractional derivatives

Mohamed Houas; Kirti Kaushik; Anoop Kumar; Aziz Khan; Thabet Abdeljawad; Mohamed Houas; Kirti Kaushik; Anoop Kumar; Aziz Khan; Thabet Abdeljawad

doi:10.3934/math.2023262

AIMS Mathematics

2023, Volume 8, Issue 3: 5216-5232. doi: 10.3934/math.2023262

Previous Article Next Article

Research article Special Issues

Existence and stability results of pantograph equation with three sequential fractional derivatives

1.
Laboratory FIMA, UDBKM, Khemis Miliana University, Algeria
2.
Department of Mathematics and Statistics, Central University of Punjab, Bathinda, Punjab, India
3.
Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
4.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 18 October 2022 Revised: 23 November 2022 Accepted: 04 December 2022 Published: 14 December 2022
MSC : 34A08, 39A30, 37C25

The subject of this work is the existence and Mittag-Leffler-Ulam (MLU) stability of solutions for fractional pantograph equations with three sequential fractional derivatives. Sufficient conditions for the existence and uniqueness of solutions are constructed by utilizing well-known classical fixed point theorems such as the Banach contraction principle, and Leray-Schauder nonlinear alternative. The generalized singular Gronwall's inequality is used to show the MLU stability results. An illustrated example is provided to support the main findings.
- fractional derivative,
- fixed point,
- existence,
- fractional pantograph equation,
- Mittag-Leffler-Ulam stability
Citation: Mohamed Houas, Kirti Kaushik, Anoop Kumar, Aziz Khan, Thabet Abdeljawad. Existence and stability results of pantograph equation with three sequential fractional derivatives[J]. AIMS Mathematics, 2023, 8(3): 5216-5232. doi: 10.3934/math.2023262

Related Papers:

Abstract

The subject of this work is the existence and Mittag-Leffler-Ulam (MLU) stability of solutions for fractional pantograph equations with three sequential fractional derivatives. Sufficient conditions for the existence and uniqueness of solutions are constructed by utilizing well-known classical fixed point theorems such as the Banach contraction principle, and Leray-Schauder nonlinear alternative. The generalized singular Gronwall's inequality is used to show the MLU stability results. An illustrated example is provided to support the main findings.

References

[1]	M. Ahmad, J. Jiang, A. Zada, Z. Ali, Z. Fu, J. Xu, Hyers-Ulam-Mittag-Leffler stability for a system of fractional neutral differential equations, Discrete Dyn. Nat. Soc., 2020 (2020), 2786041. https://doi.org/10.1155/2020/2786041 doi: 10.1155/2020/2786041
[2]	I. Ahmad, J. J. Nieto, G. U. Rahman, K. Shah, Existence and stability for fractional order pantograph equations with nonlocal conditions, Electron. J. Differ. Equ., 132 (2020), 1–16.
[3]	G. Ali, K. Shah, G. ur Rahman, Investigating a class of pantograph differential equations under multi-points boundary conditions with fractional order, Int. J. Appl. Comput. Math., 7 (2021), 2. https://doi.org/10.1007/s40819-020-00932-0 doi: 10.1007/s40819-020-00932-0
[4]	K. Balachandran, S. Kiruthika, J. J. Trujillo, Existence of solutions of nonlinear fractional pantograph equations, Acta Math. Sci., 33 (2013), 712–720. https://doi.org/10.1016/S0252-9602(13)60032-6 doi: 10.1016/S0252-9602(13)60032-6
[5]	A. Granas, J. Dugundji, Fixed point theory, New York: Springer-Verlag, 2003. https://doi.org/10.1007/978-0-387-21593-8
[6]	Y. Gouari, Z. Dahmani, I. Jebri, Application of fractional calculus on a new differential problem of Duffing type, Adv. Math.: Sci. J., 9 (2020), 10989–11002. https://doi.org/10.37418/amsj.9.12.82 doi: 10.37418/amsj.9.12.82
[7]	M. Houas, Existence and stability of fractional pantograph differential equations with Caputo-Hadamard type derivative, Turkish J. Ineqal., 4 (2020), 29–38.
[8]	M. Houas, M, Bezziou, Existence of solutions for neutral Caputo-type fractional integro-differential equations with nonlocal boundary conditions, Commun. Optim. Theory, 2021 (2021), 10.
[9]	M. Houas, Z. Dahmani, On existence of solutions for fractional differential equations with nonlocal multi-point boundary conditions, Lobachevskii J. Math., 37 (2016), 120–127. https://doi.org/10.1134/S1995080216020050 doi: 10.1134/S1995080216020050
[10]	A. Iserles, Exact and discretized stability of the pantograph equation, Appl. Numer. Math., 24 (1997), 295–308. https://doi.org/10.1016/S0168-9274(97)00027-5 doi: 10.1016/S0168-9274(97)00027-5
[11]	A. Iserles, On the generalized pantograph functional-differential equation, Eur. J. Appl. Math., 4 (1993), 1–38. https://doi.org/10.1017/S0956792500000966 doi: 10.1017/S0956792500000966
[12]	E. T. Karimov, B. Lopez, K. Sadarangani, About the existence of solutions for a hybrid nonlinear generalized fractional pantograph equation, Fract. Differ. Calculus, 6 (2016), 95–110. https://doi.org/10.7153/fdc-06-06 doi: 10.7153/fdc-06-06
[13]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Vol. 204, Elsevier Science B.V. Amsterdam, 2006.
[14]	V. Lakshmikantham, A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear. Anal., 69 (2008), 2677–2682. https://doi.org/10.1016/j.na.2007.08.042 doi: 10.1016/j.na.2007.08.042
[15]	S. Y. Lin, Generalized Gronwall inequalities and their applications to fractional differential equations, J. Inequal. Appl., 2013 (2013), 549. https://doi.org/10.1186/1029-242X-2013-549 doi: 10.1186/1029-242X-2013-549
[16]	S. K. Ntouyas, A. Alsaedi, B. Ahmad, Existence theorems for mixed Riemann-Liouville and Caputo fractional differential equations and inclusions with nonlocal fractional integro-differential boundary conditions, Fractal Fract., 3 (2019), 21. https://doi.org/10.3390/fractalfract3020021 doi: 10.3390/fractalfract3020021
[17]	S. K. Ntouyas, D. Vivek, Existence and uniqueness results for sequential Hilfer fractional differential equations with multi-point boundary conditions, Acta Math. Univ. Comen., 90 (2021), 171–185.
[18]	L. Podlubny, Fractional differential equations, New York: Academic Press, 1999.
[19]	K. Shah, D. Vivek, K. Kanagarajan, Dynamics and stability of $\psi -$fractional pantograph equations with boundary conditions, Bol. Soc. Paran. Mat., 39 (2021), 43–55. https://doi.org/10.5269/bspm.41154 doi: 10.5269/bspm.41154
[20]	M. Sezer, S. Yalçinbaş, N. Şahin, Approximate solution of multi-pantograph equation with variable coefficients, J. Comput. Appl. Math., 214 (2008), 406–416. https://doi.org/10.1016/j.cam.2007.03.024 doi: 10.1016/j.cam.2007.03.024
[21]	I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., 26 (2010), 103–107.
[22]	D. Vivek, K. Kanagarajan, S. Sivasundaram, Dynamics and stability of pantograph equations via Hilfer fractional derivative, Nonlinear Stud., 23 (2016), 685–698.
[23]	D. Vivek, E. M. Elsayed, K. Kanagarajan, Existence and Ulam stability results for a class of boundary value problem of neutral pantograph equations with complex order, SeMA J., 77 (2021), 243–256. https://doi.org/10.1007/s40324-020-00214-1 doi: 10.1007/s40324-020-00214-1
[24]	J. Wang, Y. Zhang, Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations, Optimization, 63 (2014), 1181–1190. https://doi.org/10.1080/02331934.2014.906597 doi: 10.1080/02331934.2014.906597
[25]	Z. H. Yu, Variational iteration method for solving the multi-pantograph delay equation, Phys. Lett. A, 372 (2008), 6475–6479. https://doi.org/10.1016/j.physleta.2008.09.013 doi: 10.1016/j.physleta.2008.09.013
[26]	M. R. Fatehi, M. Samavat, M. A. Vali, F. Khaleghi, State analysis and optimal control of linear time-invariant scaled systems using the Chebyshev wavelets, Contemp. Eng. Sci., 5 (2012), 91–105.
[27]	F. Ghomanjani, M. H. Farahi, A. V. Kamyad, Numerical solution of some linear optimal control systems with pantograph delays, IMA J. Math. Control Inf., 32 (2015), 225–243. https://doi.org/10.1093/imamci/dnt037 doi: 10.1093/imamci/dnt037
[28]	S. M. Hoseini, Optimal control of linear pantograph-type delay systems via composite Legendre method, J. Franklin Inst., 357 (2020), 5402–5427. https://doi.org/10.1016/j.jfranklin.2020.02.051 doi: 10.1016/j.jfranklin.2020.02.051
[29]	Z. Gong, C. Liu, K. L. Teo, X. Yi, Optimal control of nonlinear fractional systems with multiple pantograph-delays, Appl. Math. Comput., 425 (2022), 127094. https://doi.org/10.1016/j.amc.2022.127094 doi: 10.1016/j.amc.2022.127094
[30]	J. F. Gómez-Aguilar, Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel, Phys. A: Stat. Mech. Appl., 465 (2017), 562–572. https://doi.org/10.1016/j.physa.2016.08.072 doi: 10.1016/j.physa.2016.08.072
[31]	J. F. Gómez-Aguilar, A. Atangana, Time-fractional variable-order telegraph equation involving operators with Mittag-Leffler kernel, J. Electromagnet. Waves Appl., 33 (2019), 165–177. https://doi.org/10.1080/09205071.2018.1531791 doi: 10.1080/09205071.2018.1531791
[32]	P. Pandey, J. F. Gómez-Aguilar, On solution of a class of nonlinear variable order fractional reaction-diffusion equation with Mittag-Leffler kernel, Numer. Methods Partial Differ. Equ., 37 (2021), 998–1011. https://doi.org/10.1002/num.22563 doi: 10.1002/num.22563
[33]	J. F. Gómez-Aguilar, H. Yépez-Martínez, J. Torres-Jiménez, T. Córdova-Fraga, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel, Adv. Differ. Equ., 2017 (2017), 68. https://doi.org/10.1186/s13662-017-1120-7 doi: 10.1186/s13662-017-1120-7

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)