On classical and sequential conformable fractional boundary value problems: new results via alternative fixed point method

Saleh S. Almuthaybiri; Saleh S. Almuthaybiri

doi:10.3934/math.2025862

AIMS Mathematics

2025, Volume 10, Issue 8: 19280-19299. doi: 10.3934/math.2025862

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Research article

On classical and sequential conformable fractional boundary value problems: new results via alternative fixed point method

Saleh S. Almuthaybiri ^,

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

Received: 27 June 2025 Revised: 07 August 2025 Accepted: 20 August 2025 Published: 25 August 2025
MSC : 34A08, 35R11, 47H10

This paper investigates the existence and uniqueness of solutions for two types of conformable fractional boundary value problems: a classical CFBVP of order $ \alpha \in (\frac{3}{2}, 2] $, and a sequential conformable fractional boundary value problem (SCFBVP) of order $ \beta \in(\frac{1}{2}, 1] $. By establishing new integral bounds for the Green's functions associated with both problems, we extend the results obtained by Z. Laadjal et al. (Numerical Methods for Partial Differential Equations, 40 (2024), e22760) by applying Rus's fixed point theorems. Furthermore, we establish an existence and uniqueness theorem for the SCFBVP based on the Banach fixed point theorem, which complements their findings. Our results improve their work by relaxing key assumptions and broadening applicability. Finally, we present a detailed numerical comparison between the results herein and the existing results, highlighting the advantages of our approach, followed by concluding remarks.
Citation: Saleh S. Almuthaybiri. On classical and sequential conformable fractional boundary value problems: new results via alternative fixed point method[J]. AIMS Mathematics, 2025, 10(8): 19280-19299. doi: 10.3934/math.2025862

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Abstract

This paper investigates the existence and uniqueness of solutions for two types of conformable fractional boundary value problems: a classical CFBVP of order $ \alpha \in (\frac{3}{2}, 2] $, and a sequential conformable fractional boundary value problem (SCFBVP) of order $ \beta \in(\frac{1}{2}, 1] $. By establishing new integral bounds for the Green's functions associated with both problems, we extend the results obtained by Z. Laadjal et al. (Numerical Methods for Partial Differential Equations, 40 (2024), e22760) by applying Rus's fixed point theorems. Furthermore, we establish an existence and uniqueness theorem for the SCFBVP based on the Banach fixed point theorem, which complements their findings. Our results improve their work by relaxing key assumptions and broadening applicability. Finally, we present a detailed numerical comparison between the results herein and the existing results, highlighting the advantages of our approach, followed by concluding remarks.

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