Existence of Solutions to Fractional Differential Equations with Mixed Caputo-Riemann Derivative

Abstract

The study of fractional differential equations is gaining increasing significance due to their wide-ranging applications across various fields. Different methods, including fixed-point theory, variational approaches, and the lower and upper solutions method, are employed to analyze the existence and uniqueness of solutions to fractional differential equations. This paper investigates the existence and uniqueness of solutions to a class of nonlinear fractional differential equations involving mixed Caputo-Riemann fractional derivatives with integral initial conditions, set within a Banach space. Sufficient conditions are provided for the existence and uniqueness of solutions based on the problem's parameters. The results are derived by constructing the Green's function for the initial value problem. Schauder's fixed-point theorem is used to prove existence, while Banach's contraction mapping principle ensures uniqueness. Finally, an example is given to demonstrate the practical application of the results.