Analyzing the existence, uniqueness, and stability of solutions to boundary value problems involving a generalized fractional derivative

This paper examines tempered fractional derivatives with respect to a kernel function, extending classical operators like Caputo and tempered derivatives. We investigate fractional differential equations (FDEs) that incorporate these generalized derivatives, focusing on the existence and uniqueness of solutions for boundary value problems. Using fixed-point theorems, we establish conditions for the existence and uniqueness of solutions. Additionally, we analyze the stability of these equations under different criteria. Our approach addresses inaccuracies in previous studies and contributes to the broader theory of fractional equations with generalized derivatives.