Existence And Uniqueness Results For Caputo Fractional Diferential Equations With Initial Value Conditions

This paper will concern with two kinds of initial value problems of Caupto frac-tional diferential equations.The paper is mainly divided into four chapters.The first section is the introduction of the whole paper. We introduce the back-ground of this problem, and make plans for the research of the problems.Next section of the paper mainly give some defintions and basic conclusions whichare used to dicuss the problems in the next part. In this paper, we investigate theCaputo fractional diferential equationsDn0y(x)=f(x, y(x)) with initial value conditions, where0<n <1and the function f(x, y)meets some integrable conditions. Existence and uniqueness results of solutions areobtained by the method of successive approximations under some weak conditions. Anexample is given to illustrate the results at last.The third section mainly investigates some properties of Riemann-Liourille integraloperators.Let n≥1,we investigate the existence and uniqueness of initial solutions forthe fractional diferential equationsDn0y(x)=f(x, y(x)) involving the Caputo derivativeand the initial conditions Dk (0)=y(k)0(k=0,1,2,··· m1).The last chapter is the summation of this paper§we point out some unsolvedproblems and direction for further research.