On sequential pantograph fractional differential equations with Atangana–Baleanu derivatives and a $ p $-Laplacian operator

Hammou Benmehidi; Tayeb Mahrouz; Zoubir Dahmani; Abdelkader Moumen; Hussien Albala; Hammou Benmehidi; Tayeb Mahrouz; Zoubir Dahmani; Abdelkader Moumen; Hussien Albala

doi:10.3934/math.20261374

AIMS Mathematics

2026, Volume 11, Issue 10: 35403-35428. doi: 10.3934/math.20261374

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On sequential pantograph fractional differential equations with Atangana–Baleanu derivatives and a $ p $-Laplacian operator

1.
Faculty of Nature and Life Sciences, Department of Biology, Ibn Khaldoun University, P.O. Box 78, Tiaret 14000, Algeria
2.
Laboratory of Pure and Applied Mathematics, University of Mostaganem, B.P. 227, Mostaganem 27000, Algeria
3.
Department of Computer Science, Laboratory of Research in Artificial Intelligence and Systems (LRAIS), Ibn Khaldoun University, P.O. Box 78, Tiaret 14000, Algeria
4.
LAMDA-RO Laboratory, Faculty of Science, Department of Mathematics, Saad Dahlab Blida 1 University, P.O. Box 270, Blida 09000, Algeria
5.
Department of Mathematics, College of Science, University of Ha'il, 55473 Ha'il, Saudi Arabia
6.
Department of Mathematics, Faculty of Sciences, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia

Received: 08 July 2026 Revised: 08 October 2026 Accepted: 14 October 2026 Published: 28 October 2026
MSC : 34A08, 34B15, 34D20, 47H10

This study investigated a class of sequential pantograph fractional differential equations that couple the Caputo–Hadamard and the Atangana–Baleanu–Caputo (ABC) fractional derivatives through a $ p $-Laplacian operator, together with hybrid nonlocal terms and proportional (pantograph) delays. An equivalent integral representation of the problem was first derived. Banach's contraction principle was then used to establish the existence and uniqueness of solutions under suitable assumptions, whereas Dhage's Krasnoselskii-type hybrid fixed-point theorem in Banach algebras yielded the existence of at least one solution under weaker conditions. We further analyzed the Ulam–Hyers stability of the proposed problem. Two illustrative examples were provided.
- Caputo–Hadamard derivative,
- Atangana–Baleanu derivative,
- $ p $-Laplacian operator,
- pantograph equation
Citation: Hammou Benmehidi, Tayeb Mahrouz, Zoubir Dahmani, Abdelkader Moumen, Hussien Albala. On sequential pantograph fractional differential equations with Atangana–Baleanu derivatives and a $ p $-Laplacian operator[J]. AIMS Mathematics, 2026, 11(10): 35403-35428. doi: 10.3934/math.20261374

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Abstract

This study investigated a class of sequential pantograph fractional differential equations that couple the Caputo–Hadamard and the Atangana–Baleanu–Caputo (ABC) fractional derivatives through a $ p $-Laplacian operator, together with hybrid nonlocal terms and proportional (pantograph) delays. An equivalent integral representation of the problem was first derived. Banach's contraction principle was then used to establish the existence and uniqueness of solutions under suitable assumptions, whereas Dhage's Krasnoselskii-type hybrid fixed-point theorem in Banach algebras yielded the existence of at least one solution under weaker conditions. We further analyzed the Ulam–Hyers stability of the proposed problem. Two illustrative examples were provided.

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