Initial value problems of first order fractional differential equations via monotone iterative techniques

Upper and lower generalized normal solutions of initial value problem (IVP) for first order fractional differential equation (FDE) and the corresponding integral equation are studied in this paper. It will be proved that a continuous generalized normal upper (or lower) solution of the IVP satisfying a suitable condition is an upper (or lower) solution of the corresponding integral equation. The minimal and maximal generalized normal solutions of the IVP are studied via the corresponding integral equation where the monotone iterative techniques for compact maps in Banach spaces can be applied. Two types of functions are provided and proved to be upper and lower solutions of the integral equation and the minimal and maximal generalized normal solutions of the IVP for the first order FDE are obtained. These new results are applied to obtain the existence of the minimal and maximal positive generalized normal solutions for the IVPs of the first order FDEs with nonlinearities arising from logistic type population models.