Theory of Hybrid Fractional Differential Equations with Complex Order

We develop the theory of hybrid fractional differential equations with the complex order thetain mathbb{C}, theta=m+ialpha, 0<mleq 1, alphain mathbb{R}, in Caputo sense. Using Dhage's type fixed point theorem for the product of abstract nonlinear operators in Banach algebra; one of the operators is mathfrak{D}- Lipschitzian and the other one is completely continuous, we prove the existence of mild solutions of initial value problems for hybrid fractional differential equations. Finally, an application to solve one-variable linear fractional Schr"odinger equation with complex order is given.