Existence of solutions for nonlinear Caputo-Hadamard fractional differential equations via the method of upper and lower solutions

The purpose of this paper is devoted to consider the existence of solutions for a class of nonlinear Caputo-Hadamard fractional differential equations with integral terms ((CHFDE), for short). Firstly, by applying the semi-group property of Hadamard fractional integral operator, a necessary condition of solvability for (CHFDE) is established. Then, under the suitable conditions, we prove the solution set of (CHFDE) is nonempty by using the method of upper and lower solutions, and Arzel\`{a}-Ascoli theorem. Finally, we present several numerical examples to explicate the main results.