The Existence and Uniqueness of a Class of Fractional Differential Equations

Abstract

By using inequalities, fixed point theorems, and lower and upper solution method, the existence and uniqueness of a class of fractional initial value problems, $D_{\mathrm{0}+}^{q}x(t)=f(t,x(t),D_{\mathrm{0}+}^{q-\mathrm{1}}x(t)),t\in (\mathrm{0},T),$ $x(\mathrm{0})=\mathrm{0},D_{\mathrm{0}+}^{q-\mathrm{1}}x(\mathrm{0})=x_{\mathrm{0}},\hspace{0.17em}$ are discussed, where $f\in C([\mathrm{0},T]\times {\mathbb{R}}^{\mathrm{2}},\mathbb{R}),D_{\mathrm{0}+}^{q}x(t)$ is the standard Riemann-Liouville fractional derivative, . Some mistakes in the literature are pointed out and some new inequalities and existence and uniqueness results are obtained.