Existence and stability analysis of a problem of the Caputo fractional derivative with mixed conditions

Naimi Abdellouahab; Keltum Bouhali; Loay Alkhalifa; Khaled Zennir; Naimi Abdellouahab; Keltum Bouhali; Loay Alkhalifa; Khaled Zennir

doi:10.3934/math.2025312

AIMS Mathematics

2025, Volume 10, Issue 3: 6805-6826. doi: 10.3934/math.2025312

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Existence and stability analysis of a problem of the Caputo fractional derivative with mixed conditions

1.
Department of Mathematics, Faculty of Science and Technology, Université de Ghardaia, Algeria
2.
Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia

Received: 31 January 2025 Revised: 12 March 2025 Accepted: 17 March 2025 Published: 26 March 2025
MSC : 26A33, 34A08, 34B15, 34K20

Recently, initial-boundary-value problems with the Caputo fractional derivative for ordinary differential equations have been intensively studied. This paper studies a nonlinear integro-differential equation of fractional order, containing a composition of fractional derivatives with different origins and mixed conditions. The equation under consideration acts as a model equation of motion in a fractal medium. First, we use three fixed-point theorems to prove the existence and uniqueness results. Then, the Ulam stability criterion of the solution is given. The main results will be illustrated by a proposed example.
- fractional derivative,
- existence,
- Ulam-stability,
- mixed conditions,
- iterative methods
Citation: Naimi Abdellouahab, Keltum Bouhali, Loay Alkhalifa, Khaled Zennir. Existence and stability analysis of a problem of the Caputo fractional derivative with mixed conditions[J]. AIMS Mathematics, 2025, 10(3): 6805-6826. doi: 10.3934/math.2025312

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Abstract

Recently, initial-boundary-value problems with the Caputo fractional derivative for ordinary differential equations have been intensively studied. This paper studies a nonlinear integro-differential equation of fractional order, containing a composition of fractional derivatives with different origins and mixed conditions. The equation under consideration acts as a model equation of motion in a fractal medium. First, we use three fixed-point theorems to prove the existence and uniqueness results. Then, the Ulam stability criterion of the solution is given. The main results will be illustrated by a proposed example.

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