On nonlinear $ (p, q) $-fractional hybrid pantograph equations: Existence, and uniqueness

Mouataz Billah Mesmouli; Yasir A. Madani; Ioan-Lucian Popa; Taher S. Hassan; Mouataz Billah Mesmouli; Yasir A. Madani; Ioan-Lucian Popa; Taher S. Hassan

doi:10.3934/math.2026475

AIMS Mathematics

2026, Volume 11, Issue 4: 11546-11558. doi: 10.3934/math.2026475

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On nonlinear $ (p, q) $-fractional hybrid pantograph equations: Existence, and uniqueness

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia; yas.madani@uoh.edu.sa, t.hassan@uoh.edu.sa
2.
Department of Computing, Mathematics and Electronics, 1 Decembrie 1918 University of Alba Iulia, 510009 Alba Iulia, Romania; lucian.popa@uab.ro
3.
Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania

Received: 01 January 2026 Revised: 19 March 2026 Accepted: 31 March 2026 Published: 27 April 2026
MSC : 26A33, 39A13, 47H10, 47H09, 45J05, 34K37, 39A23

We consider a nonlinear class of hybrid pantograph equations governed by Caputo-type $ (p, q) $-fractional derivatives and proportional delay arguments. By rewriting the problem in an equivalent integral form, we first established the existence of solutions through Dhage's hybrid fixed point theorem in a Banach algebra. Under additional Lipschitz conditions, the Banach contraction principle was employed to guarantee the existence and uniqueness of solutions. The results generalize and extend studies on fractional and hybrid pantograph equations within the $ (p, q) $-fractional setting.
- $ (p, q) $-fractional derivative,
- hybrid pantograph equation,
- fixed point methods,
- Banach contraction principle,
- Dhage's theorem,
- proportional delays
Citation: Mouataz Billah Mesmouli, Yasir A. Madani, Ioan-Lucian Popa, Taher S. Hassan. On nonlinear $ (p, q) $-fractional hybrid pantograph equations: Existence, and uniqueness[J]. AIMS Mathematics, 2026, 11(4): 11546-11558. doi: 10.3934/math.2026475

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Abstract

We consider a nonlinear class of hybrid pantograph equations governed by Caputo-type $ (p, q) $-fractional derivatives and proportional delay arguments. By rewriting the problem in an equivalent integral form, we first established the existence of solutions through Dhage's hybrid fixed point theorem in a Banach algebra. Under additional Lipschitz conditions, the Banach contraction principle was employed to guarantee the existence and uniqueness of solutions. The results generalize and extend studies on fractional and hybrid pantograph equations within the $ (p, q) $-fractional setting.

References

[1]	C. F. D. L. Godinho, I. V. Vancea, Fractional calculus in physics: A brief review of fundamental formalisms, Mathematics, 13 (2025), 3643. https://doi.org/10.3390/math13223643 doi: 10.3390/math13223643
[2]	A. V. Crişan, C. F. D. L. Godinho, C. M. Porto, I. V. Vancea, Conformable Lagrangian mechanics of actuated pendulum, Mathematics, 13 (2025), 1634. https://doi.org/10.3390/math13101634 doi: 10.3390/math13101634
[3]	C. M. Porto, C. F. D. L. Godinho, I. V. Vancea, Fractional Laplacian spinning particle in external electromagnetic field, Dynamics, 3 (2023), 855–870. https://doi.org/10.3390/dynamics3040046 doi: 10.3390/dynamics3040046
[4]	T. Li, D. A. Soba, A. Columbu, G. Viglialoro, Dissipative gradient nonlinearities prevent $\delta$-formations in local and nonlocal attraction–repulsion chemotaxis models, Stud. Appl. Math., 154 (2025), e70018. https://doi.org/10.1111/sapm.70018 doi: 10.1111/sapm.70018
[5]	Y. Zhou, Y. Zhang, Noether symmetries for fractional generalized Birkhoffian systems in terms of classical and combined Caputo derivatives, Acta Mech., 231 (2020), 3017–3029. https://doi.org/10.1007/s00707-020-02690-y doi: 10.1007/s00707-020-02690-y
[6]	R. P. Agarwal, Certain fractional $q$-integrals and $q$-derivatives, Proc. Cambridge Philos. Soc., 66 (1969), 365–370.
[7]	M. H. Annaby, Z. S. Mansour, $q$-fractional calculus and equations, Springer, 2056 (2012).
[8]	P. M. Rajković, S. D. Marinković, M. S. Stanković, On $q$-analogues of Caputo derivative and Mittag-Leffler function, Fract. Calc. Appl. Anal., 10 (2007), 359–373.
[9]	P. N. Sadjang, On the $(p, q)$-Gamma and $(p, q)$-Beta functions, arXiv Preprint, 2015.
[10]	P. N. Sadjang, On the fundamental theorem of $(p, q)$-calculus and some $(p, q)$-Taylor formulas, Results Math., 73 (2018), 39. https://doi.org/10.1007/s00025-018-0783-z doi: 10.1007/s00025-018-0783-z
[11]	J. Soontharanon, T. Sitthiwirattham, On fractional $(p, q)$-calculus, Adv. Differ. Equ., 2020 (2020), 35. https://doi.org/10.1186/s13662-020-2512-7 doi: 10.1186/s13662-020-2512-7
[12]	M. Tunç, E. Göv, $(p, q)$-integral inequalities, RGMIA Res. Rep. Coll., 19 (2016), 1–13.
[13]	M. B. Mesmouli, L. F. Iambor, A. A. Menaem, T. S. Hassan, Existence results and finite-time stability of a fractional $(p, q)$-integro-difference system, Mathematics, 12 (2024), 1399. https://doi.org/10.3390/math12091399 doi: 10.3390/math12091399
[14]	M. B. Mesmouli, L. F. Iambor, O. Tunç, T. S. Hassan, Existence of solutions and Ulam stability analysis of implicit $(p, q)$-fractional difference equations, Contemp. Math., 6 (2025), 7619–7635. https://doi.org/10.37256/cm.6620258140 doi: 10.37256/cm.6620258140
[15]	R. P. Agarwal, H. Al-Hutami, B. Ahmad, On solvability of fractional $(p, q)$-difference equations with $(p, q)$-difference anti-periodic boundary conditions, Mathematics, 10 (2022), 4419. https://doi.org/10.3390/math10234419 doi: 10.3390/math10234419
[16]	M. Zhou, Well-posedness for fractional $(p, q)$-difference equations: Initial value problem, J. Nonlinear Model. Anal., 5 (2023), 565–579. https://doi.org/10.12150/jnma.2023.565 doi: 10.12150/jnma.2023.565
[17]	G. A. 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
[18]	R. Ockendon, A. B. Taylor, The dynamics of a current collection system for an electric locomotive, Proc. A, 322 (1971), 447–468. https://doi.org/10.1098/rspa.1971.0078 doi: 10.1098/rspa.1971.0078
[19]	M. S. Abdo, T. Abdeljawad, K. D. Kucche, M. A. Alqudah, M. B. Jeelani, On nonlinear pantograph fractional differential equations with Atangana–Baleanu–Caputo derivative, Adv. Differ. Equ., 2021 (2021), 65. https://doi.org/10.1186/s13662-021-03229-8 doi: 10.1186/s13662-021-03229-8
[20]	I. Ahmad, J. J. Nieto, G. U. Rahman, K. Shah, Existence and stability for fractional-order pantograph equations with nonlocal conditions, Electron. J. Differ. Eq., 2020 (2020), 132. https://doi.org/10.58997/ejde.2020.132 doi: 10.58997/ejde.2020.132
[21]	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
[22]	E. T. Karimov, B. Lopez, K. Sadarangani, About the existence of solutions for a hybrid nonlinear generalized fractional pantograph equation, Fract. Differ. Calc., 6 (2016), 95–110. https://doi.org/10.7153/fdc-06-06 doi: 10.7153/fdc-06-06
[23]	M. Houas, Existence and stability results for hybrid fractional $q$-differential pantograph equations, Asia Math., 5 (2021), 20–35.
[24]	B. C. Dhage, G. T. Khurpe, A. Y Shete, J. N. Salunke, Existence and approximate solutions for nonlinear hybrid fractional integro-differential equations, Int. J. Anal. Appl., 11 (2016), 157–167.
[25]	B. C. Dhage, V. Lakshmikantham, Basic results on hybrid differential equations, Nonlinear Anal.-Hybri., 4 (2010), 414–424. https://doi.org/10.1016/j.nahs.2009.10.005 doi: 10.1016/j.nahs.2009.10.005

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