Existence of Solutions for Fractional Order Differential Equations: Extended Results

This study investigates the existence of solutions for nonlinear fractional differential equations of order $ q \in (1, 2]$. We establish new existence results for the boundary conditions $\xi(\kappa) = \alpha\neq 0$ and $\xi(\omega)= \beta \neq 0$ by incorporating an intermediate point, extending existing methodologies. Our results rely on fixed point theorems and the contraction principle, which provide a robust framework for analyzing these equations. We also provide several illustrative examples to demonstrate our results, showcasing their relevance in theoretical and applied contexts.