Existence of Unique Solutions to Fractional Differential Equations with Integral Boundary Conditions

Abstract

This paper is concerned with the existence of unique solutions to nonlinear fractional differential equations in the settings of Riemann-Liouville fractional derivatives with integral boundary conditions. The fractional-order boundary value problem proposed in this paper is first transformed into an equivalent Volterra-Fredholm integral equation involving Riemann-Liouville fractional integrals, and then we will study this integral equation with the aid of the fixed-point technique. By constructing a suitable Banach space of continuous functions, we establish the existence-uniqueness theorem via the Banach contraction principle, imposing the Lipschitz-type condition on the nonlinear term. Furthermore, an example is constructed to illustrate our theoretical findings. The Riemann-Liouville fractional differential equation presented herein is a more general form compared with some previously studied equations.