Existence of the Unique Nontrivial Solution for Mixed Fractional Differential Equations

Abstract

In this paper, we consider the differential equations with right-sided Caputo and left-sided Riemann-Liouville fractional derivatives. Furthermore, the expression of Green’s function is derived, and its properties are investigated. By the fixed-point theorem for both $\phi -\left(h,e\right)$ -concave operators and mixed monotone operators, we get the existence and uniqueness of the solution, respectively. As applications, some examples are provided to illustrate our main results.