Continuous and integrable solutions of a nonlinear Cauchy problem of fractional order with nonlocal conditions

In this article, we discuss the existence of at least one solution as well as uniqueness for a nonlinear fractional differential equation with weighted initial data and nonlocal conditions. The existence of at least one L1L1 and continuous solution will be proved under the Carathèodory conditions via a classical fixed point theorem of Schauder. An example is also given to illustrate the efficiency of the main theorems.