Existence, uniqueness and stability analysis for a $\mathfrak{f}$-Caputo generalized proportional fractional boundary problem with distinct generalized integral conditions

This paper aims to examine the existence, uniqueness, and stability properties of a novel category of $\mathfrak{f}$ -Caputo generalized proportional differential equations characterized by two distinct fractional orders. We explore and discuss various properties of the generalized proportional operator. Using the Schaefer fixed point theorem and the Banach contraction principle, we establish results on existence and uniqueness, along with different types of stabilities, including Ulam-Hyers and generalized Ulam-Hyers stability. An example is provided to complete the numerical validation of the theoretical results.