Monotone Iterative Technique For Nonlinear Fractional Differential Equations

The subject of fractional calculus was initially appeared in a letter from G.W.Leibniz to L’Hospital in 1695.11 years later of the appearance of integral calculus. So far there have been more than 300 years. For long years, the studies of the theory of fractional calculus were mainly constraint to the field of pure theoretical mathematics. Fractional calculus and fractional differential equations have gained considerable popularity and importance during the past three decades or so.The monotone iteration technique is an interesting and effective way to investi-gate the existence of solutions to nonlinear fractional differential equations. Generally speaking, the methods of proving existence of solutions consist of three steps, namely.(i) constructing a sequence of approximate solutions of the nonlinear fractional differential equations:(ii) showing the convergence of the constructed sequence;(iii) proving that the limit function is a solution.This paper has three parts, that is three sections. In section 1, the introduction part, we introduce the development of fractional calculus and fractional differential equations. In section 2, we investigate a class of nonlinear fractional differential e-quations with nonlocal integral boundary value conditions. Firstly, we obtain some lemmas needed, such as the Green’s function and its properties:by monotone iteration technique, the existence of the positive solutions to the problem is proved and two successively iterative sequences to approximate the solutions are constructed; finally. an example is given to numerically simulate our conclusion. In Section 3, we study a coupled system of nonlinear fractional differential equations. By using the mono-tone iterative technique and upper and lower solution method, we proved the existence of the extremal system of solutions to the problem, and the successively iterative se-quences to approximate the solutions are constructed. At last, an example is presented to illustrate the main results.