Coupled Langevin system with generalized proportional fractional derivatives relative to a function and Riemann-Stieltjes integral boundary conditions

Lamya Almaghamsi; Lamya Almaghamsi

doi:10.3934/math.2026738

AIMS Mathematics

2026, Volume 11, Issue 6: 18148-18170. doi: 10.3934/math.2026738

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Coupled Langevin system with generalized proportional fractional derivatives relative to a function and Riemann-Stieltjes integral boundary conditions

Lamya Almaghamsi ^, ,

Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabia

Received: 30 April 2026 Revised: 06 June 2026 Accepted: 09 June 2026 Published: 18 June 2026
MSC : 26A33, 34A08, 34A12, 34B15

We investigate a coupled Langevin system driven by the generalized proportional fractional derivative with respect to a function, closed by Riemann-Stieltjes integral boundary data that intertwine the two unknowns at the right endpoint. Under the assumption that the sum of the two fractional orders within each component exceeds one, the problem is recast as an equivalent system of Volterra-type integral equations. Existence is then established via Krasnoselskii's fixed point theorem and, alternatively, the Leray-Schauder nonlinear alternative; uniqueness follows from the Banach contraction principle. Three numerical examples illustrate the results, with explicit verification of every hypothesis.
- generalized proportional fractional derivative,
- coupled Langevin system,
- Riemann-Stieltjes integral,
- fixed point theory,
- existence and uniqueness
Citation: Lamya Almaghamsi. Coupled Langevin system with generalized proportional fractional derivatives relative to a function and Riemann-Stieltjes integral boundary conditions[J]. AIMS Mathematics, 2026, 11(6): 18148-18170. doi: 10.3934/math.2026738

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Abstract

We investigate a coupled Langevin system driven by the generalized proportional fractional derivative with respect to a function, closed by Riemann-Stieltjes integral boundary data that intertwine the two unknowns at the right endpoint. Under the assumption that the sum of the two fractional orders within each component exceeds one, the problem is recast as an equivalent system of Volterra-type integral equations. Existence is then established via Krasnoselskii's fixed point theorem and, alternatively, the Leray-Schauder nonlinear alternative; uniqueness follows from the Banach contraction principle. Three numerical examples illustrate the results, with explicit verification of every hypothesis.

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