Existence And Stability Domain Of Solutions For A Class Of Hadamard Type Fractional Differential Equations

Fractional order differential equation is an important branch of differential equation,which is suitable to describe the physics and force with memory and genetic phenomena.It has been widely used in physics,mechanics and profound application.We mainly deal with two problems: the first one is the existence and uniqueness of solutions for nonlinear Hadamard type fractional differential equation(without and with relaxation factor respectively);the second one is the existence of the local stable manifold for two-dimensional nonlinear Hadamard type fractional differential equations with relaxation factor.Firstly,we apply the classical Picard iterative method to derive the existence and uniqueness of solution for fully nonlinear problem without relaxation factor and give the extension theorem;meanwhile,we turn the existence and uniqueness of solution for nonlinear problem with relaxation factor into a fixed point problem in a suitable weight functions space.As a result,sufficient conditions are given to guarantee the existence and uniqueness results.Secondly,we derive another interesting local stable manifold theorem for our problem by adopting Lyapunov-Perron operator approach and establishing new estimation of Mittag-Leffler function associated with Hadamard fractional derivative.Finally,examples are given to illustrate our theoretical results.