Stability results for fractional integral pantograph differential equations involving two Caputo operators

Abdelkader Moumen; Ramsha Shafqat; Zakia Hammouch; Azmat Ullah Khan Niazi; Mdi Begum Jeelani; Abdelkader Moumen; Ramsha Shafqat; Zakia Hammouch; Azmat Ullah Khan Niazi; Mdi Begum Jeelani

doi:10.3934/math.2023303

AIMS Mathematics

2023, Volume 8, Issue 3: 6009-6025. doi: 10.3934/math.2023303

Previous Article Next Article

Research article Special Issues

Stability results for fractional integral pantograph differential equations involving two Caputo operators

1.
Department of Mathematics, Faculty of Sciences, University of Hail, Hail 55425, Saudi Arabia
2.
Department of Mathematics and Statistics, The University of Lahore, Sargodha 40100, Pakistan
3.
Department of Sciences, Ecole normale superieure, Moulay Ismail University, Morocco
4.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh 13314, Saudi Arabia

Received: 26 September 2022 Revised: 09 November 2022 Accepted: 14 November 2022 Published: 29 December 2022
MSC : 34A07, 34A08, 60G22

In this paper, we investigate the existence-uniqueness, and Ulam Hyers stability (UHS) of solutions to a fractional-order pantograph differential equation (FOPDE) with two Caputo operators. Banach's fixed point (BFP) and Leray-alternative Schauder's are used to prove the existence- uniqueness of solutions. In addition, we discuss and demonstrate various types of Ulam-stability for our problem. Finally, an example is provided for clarity.
- fractional derivative,
- fixed point,
- existence,
- fractional pantograph equation,
- Ulam-Hyers stability
Citation: Abdelkader Moumen, Ramsha Shafqat, Zakia Hammouch, Azmat Ullah Khan Niazi, Mdi Begum Jeelani. Stability results for fractional integral pantograph differential equations involving two Caputo operators[J]. AIMS Mathematics, 2023, 8(3): 6009-6025. doi: 10.3934/math.2023303

Related Papers:

Abstract

In this paper, we investigate the existence-uniqueness, and Ulam Hyers stability (UHS) of solutions to a fractional-order pantograph differential equation (FOPDE) with two Caputo operators. Banach's fixed point (BFP) and Leray-alternative Schauder's are used to prove the existence- uniqueness of solutions. In addition, we discuss and demonstrate various types of Ulam-stability for our problem. Finally, an example is provided for clarity.

References

[1]	A. A. Alderremy, K. M. Saad, P. Agarwal, S. Aly, S. Jain, Certain new models of the multi space-fractional Gardner equation, Phys. A: Stat. Mech. Appl., 545 (2020), 123806. https://doi.org/10.1016/j.physa.2019.123806 doi: 10.1016/j.physa.2019.123806
[2]	R. Caponetto, G. Dongola, L. Fortuna, I. Petráš, Fractional order systems: modeling and control applications, Vol. 72, Singapore: World Scientific, 2010. https://doi.org/10.1142/7709
[3]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Vol. 204, Elsevier, 2006.
[4]	R. Almeida, N. R. O. Bastos, M. T. T. Monteiro, Modeling some real phenomena by fractional differential equations, Math. Methods Appl. Sci., 39 (2016), 4846–4855. https://doi.org/10.1002/mma.3818 doi: 10.1002/mma.3818
[5]	J. Wang, Y. Yang, W. Wei, Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces, Opusc. Math., 30 (2010), 361–381. https://doi.org/10.7494/OpMath.2010.30.3.361 doi: 10.7494/OpMath.2010.30.3.361
[6]	C. M. Pappalardo, M. C. De Simone, D. Guida, Multibody modeling and nonlinear control of the pantograph / catenary system, Arch. Appl. Mech., 89 (2019), 1589–1626. https://doi.org/10.1007/s00419-019-01530-3 doi: 10.1007/s00419-019-01530-3
[7]	D. Li, C. Zhang, Long time numerical behaviors of fractional pantograph equations, Math. Comput. Simul., 172 (2020), 244–257. https://doi.org/10.1016/j.matcom.2019.12.004 doi: 10.1016/j.matcom.2019.12.004
[8]	G. Derfel, A. Iserles, The pantograph equation in the complex plane, J. Math. Anal. Appl., 213 (1997), 117–132. https://doi.org/10.1006/jmaa.1997.5483 doi: 10.1006/jmaa.1997.5483
[9]	R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl., 59 (2010), 1586–1593. https://doi.org/10.1016/j.camwa.2009.08.039 doi: 10.1016/j.camwa.2009.08.039
[10]	J. R. Ockendon, A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. Royal Soc. London A, Math. Phys. Sci., 322 (1971), 447–468. https://doi.org/10.1098/rspa.1971.0078 doi: 10.1098/rspa.1971.0078
[11]	I. Ahmed, P. Kumam, T. Abdeljawad, F. Jarad, P. Borisut, M. A. Demba, et al., Existence and uniqueness results for $\varphi$-Caputo implicit fractional pantograph differential equation with generalized anti-periodic boundary condition, Adv. Differ. Equ., 2020 (2020), 555. https://doi.org/10.1186/s13662-020-03008-x doi: 10.1186/s13662-020-03008-x
[12]	M. A. Darwish, K. Sadarangani, Existence of solutions for hybrid fractional pantograph equations, Appl. Anal. Discr. Math., (2015), 150–167. https://doi.org/10.2298/AADM150126002D doi: 10.2298/AADM150126002D
[13]	M. Houas, Existence and stability of fractional pantograph differential equations with Caputo-Hadamard type derivative, Turkish J. Ineq., 4 (2020), 29–38.
[14]	K. Shah, D. Vivek, K. Kanagarajan, Dynamics and stability of $\psi$-fractional pantograph equations with boundary conditions, Bol. Soc. Parana. Mat., 39 (2021), 43–55. https://doi.org/10.5269/bspm.41154 doi: 10.5269/bspm.41154
[15]	I. Ahmad, J. J. Nieto, K. Shah, Existence and stability for fractional order pantograph equations with nonlocal conditions, Electron. J. Differ. Equ., 2020 (2020), 1–16.
[16]	I. Ahmed, P. Kumam, T. Abdeljawad, F. Jarad, P. Borisut, M. A. Demba, et al., Existence and uniqueness results for $\varphi$-Caputo implicit fractional pantograph differential equation with generalized anti-periodic boundary condition, Adv. Differ. Equ., 2020 (2020), 555. https://doi.org/10.1186/s13662-020-03008-x doi: 10.1186/s13662-020-03008-x
[17]	D. Vivek, K. Kanagarajan, S. Sivasundaram, On the behavior of solutions of Hilfer Hadamard type fractional neutral pantograph equations with boundary conditions, Commun. Appl. Anal., 22 (2018), 211–232.
[18]	K. Balachandran, S. Kiruthika, 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
[19]	M. S. Hashemi, E. Ashpazzadeh, M. Moharrami, M. Lakestani, Fractional order Alpert multiwavelets for discretizing delay fractional differential equation of pantograph type, Appl. Numer. Math., 170 (2021), 1–13. https://doi.org/10.1016/j.apnum.2021.07.015 doi: 10.1016/j.apnum.2021.07.015
[20]	J. Alzabut, A. G. M. Selvam, R. A. El-Nabulsi, V. Dhakshinamoorthy, M. E. Samei, Asymptotic stability of nonlinear discrete fractional pantograph equations with non-local initial conditions, Symmetry, 13 (2021), 473. https://doi.org/10.3390/sym13030473 doi: 10.3390/sym13030473
[21]	A. Abbas, R. Shafqat, M. B. Jeelani, N. H. Alharthi, Significance of chemical reaction and Lorentz force on third-grade fluid flow and heat transfer with Darcy-Forchheimer law over an inclined exponentially stretching sheet embedded in a porous medium, Symmetry, 14 (2022), 779. https://doi.org/10.3390/sym14040779 doi: 10.3390/sym14040779
[22]	A. Abbas, R. Shafqat, M. B. Jeelani, N. H. Alharthi, Convective heat and mass transfer in third-grade fluid with Darcy-Forchheimer relation in the presence of thermal-diffusion and diffusion-thermo effects over an exponentially inclined stretching sheet surrounded by a porous medium: a CFD study, Processes, 10 (2022), 776. https://doi.org/10.3390/pr10040776 doi: 10.3390/pr10040776
[23]	A. U. K. Niazi, J. He, R. Shafqat, B. Ahmed, Existence, uniqueness, and $Eq$-Ulam-type stability of fuzzy fractional differential equation, Fractal Fract., 5 (2021), 66. https://doi.org/10.3390/fractalfract5030066 doi: 10.3390/fractalfract5030066
[24]	N. Iqbal, A. U. K. Niazi, R. Shafqat, S. Zaland, Existence and uniqueness of mild solution for fractional-order controlled fuzzy evolution equation, J. Funct. Spaces, 2021 (2021), 5795065. https://doi.org/10.1155/2021/5795065 doi: 10.1155/2021/5795065
[25]	R. Shafqat, A. U. K. Niazi, M. B. Jeelani, N. H. Alharthi, Existence and uniqueness of mild solution where $\alpha \in (1, 2)$ for fuzzy fractional evolution equations with uncertainty, Fractal Fract., 6 (2022), 65. https://doi.org/10.3390/fractalfract6020065 doi: 10.3390/fractalfract6020065
[26]	R. Shafqat, A. U. K. Niazi, M. Yavuz, M. B. Jeelani, K. Saleem, Mild solution for the time-fractional Navier-Stokes equation incorporating MHD effects, Fractal Fract., 6 (2022), 580. https://doi.org/10.3390/fractalfract6100580 doi: 10.3390/fractalfract6100580
[27]	A. S. Alnahdi, R. Shafqat, A. U. K. Niazi, M. B. Jeelani, Pattern formation induced by fuzzy fractional-order model of COVID-19, Axioms, 2022 (2022), 313. https://doi.org/10.3390/axioms11070313 doi: 10.3390/axioms11070313
[28]	A. Khan, R. Shafqat, A. U. K. Niazi, Existence results of fuzzy delay impulsive fractional differential equation by fixed point theory approach, J. Funct. Spaces, 2022 (2022), 4123949. https://doi.org/10.1155/2022/4123949 doi: 10.1155/2022/4123949
[29]	H. Boulares, A. Benchaabane, N. Pakkaranang, R. Shafqat, B. Panyanak, Qualitative properties of positive solutions of a kind for fractional pantograph problems using technique fixed point theory, Fractal Fract., 6 (2022), 593. https://doi.org/10.3390/fractalfract6100593 doi: 10.3390/fractalfract6100593
[30]	K. Abuasbeh, R. Shafqat, A. U. K. Niazi, M. Awadalla, Local and global existence and uniqueness of solution for time-fractional fuzzy Navier-Stokes equations, Fractal Fract., 6 (2022), 330. https://doi.org/10.3390/fractalfract6060330 doi: 10.3390/fractalfract6060330
[31]	K. Abuasbeh, R. Shafqat, Fractional Brownian motion for a system of fuzzy fractional stochastic differential equation, J. Math., 2022 (2022), 3559035. https://doi.org/10.1155/2022/3559035 doi: 10.1155/2022/3559035
[32]	K. Abuasbeh, R. Shafqat, A. U. K. Niazi, M. Awadalla, Oscillatory behavior of solution for fractional order fuzzy neutral predator-prey system, AIMS Math., 7 (2022), 20383–20400. https://doi.org/10.3934/math.20221117 doi: 10.3934/math.20221117
[33]	M. Houas, Existence and Ulam stability of fractional pantograph differential equations with two Caputo-Hadamard derivatives, Acta Univ. Apulensis, 63 (2020), 35–49.
[34]	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
[35]	I. Podlubnv, Fractional differential equations, 1 Ed., San Diego, CA: Academic press, 1999.
[36]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Vol. 204, Elsevier, 2006.
[37]	A. Granas, J. Dugundji, Fixed point theory, New York: Springer, 2003. https://doi.org/10.1007/978-0-387-21593-8

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)