Stability analysis of boundary value problems for Caputo proportional fractional derivative of a function with respect to another function via impulsive Langevin equation

Chutarat Treanbucha; Weerawat Sudsutad; Chutarat Treanbucha; Weerawat Sudsutad

doi:10.3934/math.2021391

AIMS Mathematics

2021, Volume 6, Issue 7: 6647-6686. doi: 10.3934/math.2021391

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Stability analysis of boundary value problems for Caputo proportional fractional derivative of a function with respect to another function via impulsive Langevin equation

Chutarat Treanbucha ¹,
Weerawat Sudsutad ^{1,2
,
,}

1.
Department of General Education, Faculty of Science and Health Technology, Navamindradhiraj University, Bangkok 10300, Thailand
2.
Department of Applied Statistics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand

Received: 06 February 2021 Accepted: 13 April 2021 Published: 19 April 2021
MSC : 26A33, 34A37, 34A08, 34D20

In this paper, we discuss existence and stability results for a new class of impulsive fractional boundary value problems with non-separated boundary conditions containing the Caputo proportional fractional derivative of a function with respect to another function. The uniqueness result is discussed via Banach's contraction mapping principle, and the existence of solutions is proved by using Schaefer's fixed point theorem. Furthermore, we utilize the theory of stability for presenting different kinds of Ulam's stability results of the proposed problem. Finally, an example is also constructed to demonstrate the application of the main results.
- existence and uniqueness,
- fractional Langevin equation,
- fixed point theorems,
- impulsive conditions,
- Ulam-Hyers stability
Citation: Chutarat Treanbucha, Weerawat Sudsutad. Stability analysis of boundary value problems for Caputo proportional fractional derivative of a function with respect to another function via impulsive Langevin equation[J]. AIMS Mathematics, 2021, 6(7): 6647-6686. doi: 10.3934/math.2021391

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Abstract

In this paper, we discuss existence and stability results for a new class of impulsive fractional boundary value problems with non-separated boundary conditions containing the Caputo proportional fractional derivative of a function with respect to another function. The uniqueness result is discussed via Banach's contraction mapping principle, and the existence of solutions is proved by using Schaefer's fixed point theorem. Furthermore, we utilize the theory of stability for presenting different kinds of Ulam's stability results of the proposed problem. Finally, an example is also constructed to demonstrate the application of the main results.

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