Unique Solution Analysis for Generalized Caputo-Type Fractional BVP via Banach Contraction

In this manuscript, we investigate the existence of a unique solution to a boundary value problem (BVP) involving generalized fractional derivatives of the Caputo type. Our approach is grounded in the Banach contraction mapping theorem, which provides a rigorous framework for proving the existence of a fixed point and, consequently, a solution to the BVP. We extend this methodology to explore analogous problems, offering further insights and interpretations of the results derived from the main theorem. This work not only contributes to the theoretical understanding of fractional differential equations but also demonstrates how these techniques can be applied to a broader class of problems in mathematical physics and engineering. Through detailed analysis and extrapolation, we aim to establish a deeper connection between fractional calculus and fixed-point theory, providing a foundation for future research in this area.