Existence and uniqueness theorems for solutions to Caputo fractional differential equations with nonlocal and irregular boundary conditions

Saleh Fahad Aljurbua; Salman Alotaibi; Saleh Fahad Aljurbua; Salman Alotaibi

doi:10.3934/math.2026190

AIMS Mathematics

2026, Volume 11, Issue 2: 4681-4690. doi: 10.3934/math.2026190

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Existence and uniqueness theorems for solutions to Caputo fractional differential equations with nonlocal and irregular boundary conditions

Saleh Fahad Aljurbua ^{1
,
,},
Salman Alotaibi ²

1.
Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah, 51452, Saudi Arabia
2.
Ministry of Education, Saudi Arabia

Received: 03 January 2026 Revised: 06 February 2026 Accepted: 10 February 2026 Published: 26 February 2026
MSC : 26A33, 34A08, 34A12, 34A34, 34K37

In this work, we introduce a analytical framework for proving the existence and uniqueness of solutions to nonlinear fractional differential equations (NFDEs) of order $ \nu \in (1, 2] $, subject to irregular boundary conditions, in a Banach space. Several fundamental definitions, theorems, and auxiliary lemmas are presented for the analysis of the proposed problem. We establish sufficient conditions for existence and uniqueness for the proposed system by applying Krasnosel'skii's fixed-point theorem and the Banach contraction principle. Finally, an illustrative example is constructed to validate the main theoretical results and demonstrate the effectiveness and applicability of the proposed approach, showing how the generalized framework can be systematically applied to analyze fractional boundary value problems.
- differential equations,
- fractional derivatives,
- existence,
- contraction principle
Citation: Saleh Fahad Aljurbua, Salman Alotaibi. Existence and uniqueness theorems for solutions to Caputo fractional differential equations with nonlocal and irregular boundary conditions[J]. AIMS Mathematics, 2026, 11(2): 4681-4690. doi: 10.3934/math.2026190

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Abstract

In this work, we introduce a analytical framework for proving the existence and uniqueness of solutions to nonlinear fractional differential equations (NFDEs) of order $ \nu \in (1, 2] $, subject to irregular boundary conditions, in a Banach space. Several fundamental definitions, theorems, and auxiliary lemmas are presented for the analysis of the proposed problem. We establish sufficient conditions for existence and uniqueness for the proposed system by applying Krasnosel'skii's fixed-point theorem and the Banach contraction principle. Finally, an illustrative example is constructed to validate the main theoretical results and demonstrate the effectiveness and applicability of the proposed approach, showing how the generalized framework can be systematically applied to analyze fractional boundary value problems.

References

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