Existence and uniqueness of the solution to Caputo-Hadamard differential equations with delay and nonlocal conditions

Nguyen Hoang Tuan; Vo Viet Tri; Nguyen Hoang Tuan; Vo Viet Tri

doi:10.3934/cam.2025029

Communications in Analysis and Mechanics

2025, Volume 17, Issue 3: 725-748. doi: 10.3934/cam.2025029

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Research article

Existence and uniqueness of the solution to Caputo-Hadamard differential equations with delay and nonlocal conditions

Nguyen Hoang Tuan ^1,2,
Vo Viet Tri ^{3
,
,}

1.
Department of Applied Mathematics, Faculty of Applied Science, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet, District 10, Ward 14, Ho Chi Minh City, Vietnam
2.
Vietnam National University Ho Chi Minh City, Linh Trung Ward, Thu Duc City, Ho Chi Minh City
3.
Division of Applied Mathematics, Thu Dau Mot University, 06 Tran Van On, Binh Duong Province, Vietnam

Received: 10 December 2024 Revised: 07 April 2025 Accepted: 17 July 2025 Published: 18 August 2025
35R11, 26A33, 35B44, 34A08, 26A33

In this paper, we established the existence conditions for initial value problems involving the Caputo-Hadamard derivative operator with delay and nonlocal conditions. The problems were analyzed for various source functions. In a special case, we obtained a unique solution. These results were derived by combining the concept of the measure of non-compactness with appropriate applications of fixed point theorems. Additionally, we provided illustrative examples to support the theoretical findings.
- Caputo-Hadamard derivative,
- fractional differential,
- fractional equation,
- measure,
- noncompactness
Citation: Nguyen Hoang Tuan, Vo Viet Tri. Existence and uniqueness of the solution to Caputo-Hadamard differential equations with delay and nonlocal conditions[J]. Communications in Analysis and Mechanics, 2025, 17(3): 725-748. doi: 10.3934/cam.2025029

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Abstract

In this paper, we established the existence conditions for initial value problems involving the Caputo-Hadamard derivative operator with delay and nonlocal conditions. The problems were analyzed for various source functions. In a special case, we obtained a unique solution. These results were derived by combining the concept of the measure of non-compactness with appropriate applications of fixed point theorems. Additionally, we provided illustrative examples to support the theoretical findings.

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