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The aim of this study is to investigate the existence and other properties of solution of nonlinear fractional integro–differential equations with constant coefficient. Also with the help of Pachpatte’s inequality, we prove the continuous dependence of the solutions.

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.