Existence, and Ulam's types stability of higher-order fractional Langevin equations on a star graph

Gang Chen; Jinbo Ni; Xinyu Fu; Gang Chen; Jinbo Ni; Xinyu Fu

doi:10.3934/math.2024581

AIMS Mathematics

2024, Volume 9, Issue 5: 11877-11909. doi: 10.3934/math.2024581

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Research article

Existence, and Ulam's types stability of higher-order fractional Langevin equations on a star graph

School of mathematics and big data, Anhui University of Science and Technology, Huainan 232001, China

Received: 25 January 2024 Revised: 07 March 2024 Accepted: 22 March 2024 Published: 26 March 2024
MSC : 34A08, 34A34, 34B15

A study was conducted on the existence of solutions for a class of nonlinear Caputo type higher-order fractional Langevin equations with mixed boundary conditions on a star graph with $ k+1 $ nodes and $ k $ edges. By applying a variable transformation, a system of fractional differential equations with mixed boundary conditions and different domains was converted into an equivalent system with identical boundary conditions and domains. Subsequently, the existence and uniqueness of solutions were verified using Krasnoselskii's fixed point theorem and Banach's contraction principle. In addition, the stability results of different types of solutions for the system were further discussed. Finally, two examples are illustrated to reinforce the main study outcomes.
- fractional Langevin equation,
- mixed boundary condition,
- existence and uniqueness,
- Ulam stability
Citation: Gang Chen, Jinbo Ni, Xinyu Fu. Existence, and Ulam's types stability of higher-order fractional Langevin equations on a star graph[J]. AIMS Mathematics, 2024, 9(5): 11877-11909. doi: 10.3934/math.2024581

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Abstract

A study was conducted on the existence of solutions for a class of nonlinear Caputo type higher-order fractional Langevin equations with mixed boundary conditions on a star graph with $ k+1 $ nodes and $ k $ edges. By applying a variable transformation, a system of fractional differential equations with mixed boundary conditions and different domains was converted into an equivalent system with identical boundary conditions and domains. Subsequently, the existence and uniqueness of solutions were verified using Krasnoselskii's fixed point theorem and Banach's contraction principle. In addition, the stability results of different types of solutions for the system were further discussed. Finally, two examples are illustrated to reinforce the main study outcomes.

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