Some Theorems on Cauchy Problem for Nonlinear Fractional Differential Equations with Positive Constant Coefficient

In this paper, we consider a Cauchy problem for nonlinear fractional differential equation with constant coefficient $$\lambda >0$$ of the type: $${^c}{D}^{\alpha }x(t)=\lambda x(t)+f(t,x(t))$$ with $$x(0)=x_{0}.$$ The aim of this paper is to investigate the existence and interval of existence of solutions, uniqueness, continuous dependence of solutions on initial conditions, estimates on solutions and continuous dependence on parameters and functions involved in the equations. Finally, one illustrative example is given to demonstrate the theoretical results.