The Research Of Several Problems For Fractional Diferential Equations

Fractional diferential equations can describe many phenomena in various fields of en-gineering and scientific disciplines such as financial mathematics, control theory, physics,chemistry, biology and so on. And fractional diferential equations have obtained compre-hensive applications on these areas. It makes the study of fractional diferential equationstheory become more significance and importance. In this paper, we mainly research severalmore popular questions of fractional diferential equations. Such as the positive solution offractional diferential equations, fractional order impulsive diferential equations, the control-lability of fractional order diferential equations and so on. Fixed point theorems, upper andlower solution method, monotone iterative technique, and quasilinear method are methodswhich are used to study the problem.The thesis is arranged as follows:In chapter1, we introduce the research background, the present research situations athome and abroad and the main results of this paper.In chapter2, we introduce the definitions and properties of fractional calculus, and somefixed point theorems which are used in this paper.In chapter3, we study the existence and uniqueness of positive solutions for a coupledsystem of nonlinear fractional diferential equations by fixed point theorems.In chapter4, we study nonlocal boundary value problems for higher order nonlinear frac-tional impulsive diferential equations. In virtue of Schauder fixed point theorem and Banachfixed point theorem, we show the existence and uniqueness of solutions of the equations.In chapter5, we study the existence and uniqueness of fractional order impulsive dif-ferential equations. We obtain the conclusion by using upper and lower solution method,monotone iterative technique and comparison principle.In chapter6, we research the controllability of Riemann Liouville fractional order controlsystems. Firstly, we prove the controllability of Riemann-Liouville fractional order linearcontrol systems, then under appropriate assumptions, we attain the controllability of nonlinearRiemann Liouville fractional order control systems.In chapter7, we give a conclusion of our present research and introduce some furtherstudy ideas in the future.