Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations

Abstract

We discuss the initial value problem for the nonlinear fractional differential equation $L(D)u=f(t,u),t\in (\mathrm{0,1}],u(0)=0$, where $L(D)=D^{s_n}-a_{n-1}{D}^{s_{n-1}}-\cdots -a_1{D}^{s_1}$, , and , $j=\mathrm{1,2},\dots ,n-1$, $D^{s_j}$ is the standard Riemann-Liouville fractional derivative and $f:[\mathrm{0,1}]\times ℝ\to ℝ$ is a given continuous function. We extend the basic theory of differential equation, the method of upper and lower solutions, and monotone iterative technique to the initial value problem. Some existence and uniqueness results are established.