A study of Caputo fractional differential equations of variable order via Darbo's fixed point theorem and Kuratowski measure of noncompactness

Mohammed Said Souid; Souhila Sabit; Zoubida Bouazza; Kanokwan Sitthithakerngkiet; Mohammed Said Souid; Souhila Sabit; Zoubida Bouazza; Kanokwan Sitthithakerngkiet

doi:10.3934/math.2025691

AIMS Mathematics

2025, Volume 10, Issue 7: 15410-15432. doi: 10.3934/math.2025691

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A study of Caputo fractional differential equations of variable order via Darbo's fixed point theorem and Kuratowski measure of noncompactness

1.
Department of Mathematics, Saveetha School of Engineering, SIMATS, Chennai, India
2.
Department of Economic Sciences, University of Tiaret, Algeria
3.
Departement of Mathematics, University of Tiaret, Algeria
4.
Department of Computer Science, University of Tiaret, Algeria
5.
Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok, 10800, Thailand
6.
Intelligent and Nonlinear Dynamic Innovations Research Center, Science and Technology Research Institute, King Mongkut's University of Technology North Bangkok, Bangkok, 10800, Thailand

Received: 19 April 2025 Revised: 18 June 2025 Accepted: 23 June 2025 Published: 03 July 2025
MSC : 26A33, 34K37

This paper investigated the existence and stability of solutions for boundary value problems involving Caputo fractional differential equations of variable order. Unlike constant-order models, variable-order equations allow the fractional order to change over time, enabling more flexible and accurate modeling of complex systems with evolving dynamics and memory. Using Darbo's fixed point theorem and the Kuratowski measure of noncompactness, we established new existence results for solutions within a Banach space of continuous functions. Our approach treated the variable order as piecewise constant, transforming the problem into a sequence of more manageable constant-order subproblems. Furthermore, we demonstrated Ulam-Hyers stability of the solutions, ensuring that small perturbations in the system did not lead to significant deviations in the results. To validate the theoretical findings, we provided a detailed example supported by numerical simulations. These results offered a solid foundation for future applications in science and engineering where system dynamics evolve over time.
- boundary value problem,
- fractional differential equations,
- variable order,
- measure of noncompactness,
- Darbo's fixed point,
- Ulam-Hyers
Citation: Mohammed Said Souid, Souhila Sabit, Zoubida Bouazza, Kanokwan Sitthithakerngkiet. A study of Caputo fractional differential equations of variable order via Darbo's fixed point theorem and Kuratowski measure of noncompactness[J]. AIMS Mathematics, 2025, 10(7): 15410-15432. doi: 10.3934/math.2025691

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Abstract

This paper investigated the existence and stability of solutions for boundary value problems involving Caputo fractional differential equations of variable order. Unlike constant-order models, variable-order equations allow the fractional order to change over time, enabling more flexible and accurate modeling of complex systems with evolving dynamics and memory. Using Darbo's fixed point theorem and the Kuratowski measure of noncompactness, we established new existence results for solutions within a Banach space of continuous functions. Our approach treated the variable order as piecewise constant, transforming the problem into a sequence of more manageable constant-order subproblems. Furthermore, we demonstrated Ulam-Hyers stability of the solutions, ensuring that small perturbations in the system did not lead to significant deviations in the results. To validate the theoretical findings, we provided a detailed example supported by numerical simulations. These results offered a solid foundation for future applications in science and engineering where system dynamics evolve over time.

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