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

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.