Fractional cauchy problem in sense of the complex hadamard operators

Abstract

The theory of fractional calculus has found interesting applications in the theory of analytic functions and geometric functions. We extend the definitions of the Hadamard fractional operators into the open unit disk. A class of nonlinear fractional differential equations (Cauchy-type problem) in the unit disk is studied using these fractional operators. The existence and uniqueness of the solution are established. Some properties of the integral operator are inflict such as boundedness in a space of analytic function and semigroup property. Moreover, we prove that the linear Cauchy problem (homogenous and nonhomogeneous) is solvable in a holomorphic space and its solution approximates to the Mittag-Leffler function. Examples are illustrated.