On solvability of BVP for a coupled Hadamard fractional systems involving fractional derivative impulses

Hui Huang; Kaihong Zhao; Xiuduo Liu; Hui Huang; Kaihong Zhao; Xiuduo Liu

doi:10.3934/math.20221055

AIMS Mathematics

2022, Volume 7, Issue 10: 19221-19236. doi: 10.3934/math.20221055

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Research article

On solvability of BVP for a coupled Hadamard fractional systems involving fractional derivative impulses

Hui Huang ^{1
,
,},
Kaihong Zhao ^{2
,
,},
Xiuduo Liu ³

1.
School of Science, Shaoyang University, Shaoyang 422000, Hunan, China
2.
Department of Mathematics, School of Electronics & Information Engineering, Taizhou University, Taizhou 318000, Zhejiang, China
3.
Puyang Petrochemical Vocational and Technical College, Puyang 457001, Henan, China

Received: 08 June 2022 Revised: 16 August 2022 Accepted: 19 August 2022 Published: 30 August 2022
MSC : 34B10, 34B15, 34B37

Hadamard fractional calculus is one of the most important fractional calculus theories. Compared with a single Hadamard fractional order equation, Hadamard fractional differential equations have a more complex structure and a wide range of applications. It is difficult and challenging to study the dynamic behavior of Hadamard fractional differential equations. This manuscript mainly deals with the boundary value problem (BVP) of a nonlinear coupled Hadamard fractional system involving fractional derivative impulses. By applying nonlinear alternative of Leray-Schauder, we find some new conditions for the existence of solutions to this nonlinear coupled Hadamard fractional system. Our findings reveal that the impulsive function and its impulsive point have a great influence on the existence of the solution. As an application, we discuss an interesting example to verify the correctness and validity of our results.
- coupled Hadamard fractional system,
- boundary value problem,
- solvability,
- fractional derivative impulse,
- nonlinear alternative of Leray-Schauder
Citation: Hui Huang, Kaihong Zhao, Xiuduo Liu. On solvability of BVP for a coupled Hadamard fractional systems involving fractional derivative impulses[J]. AIMS Mathematics, 2022, 7(10): 19221-19236. doi: 10.3934/math.20221055

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Abstract

Hadamard fractional calculus is one of the most important fractional calculus theories. Compared with a single Hadamard fractional order equation, Hadamard fractional differential equations have a more complex structure and a wide range of applications. It is difficult and challenging to study the dynamic behavior of Hadamard fractional differential equations. This manuscript mainly deals with the boundary value problem (BVP) of a nonlinear coupled Hadamard fractional system involving fractional derivative impulses. By applying nonlinear alternative of Leray-Schauder, we find some new conditions for the existence of solutions to this nonlinear coupled Hadamard fractional system. Our findings reveal that the impulsive function and its impulsive point have a great influence on the existence of the solution. As an application, we discuss an interesting example to verify the correctness and validity of our results.

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