A boundary value problem of φ-fractional differential equations with complex variable functions

In recent years, boundary value problems for fractional differential equations in the real number domain have been widely studied, but little attention has been paid to the fractional boundary value problem for differential equations with complex functions. In this paper, we define a new fractional derivative with respect to a regular function, namely phi-Srivastava-Owa fractional derivative, and give the related properties. The existence of solutions of phi-Srivastava-Owa fractional differential equations with boundary conditions is studied by using the Krasnoselskii fixed point theorem, and its uniqueness is proved by using Banach' s contraction theorem. In the given examples, we find that the special cases of the obtained results are equivalent to the existing theorems.