## Abstract and Applied Analysis

### 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 $L(D)u=f(t,u),\mathrm{ }t\in (0,1],\mathrm{ }u(0)=0$, where $L(D)={D}^{{s}_{n}}-{a}_{n-1}{D}^{{s}_{n-1}}-\cdots -{a}_{1}{D}^{{s}_{1}}$, $0<{s}_{1}<{s}_{2}<\cdots <{s}_{n}<1$, and ${a}_{j}<0$, $j=1,2,\dots ,n-1$, ${D}^{{s}_{j}}$ is the standard Riemann-Liouville fractional derivative and $f:[0,1]{\times}\Bbb R\to \Bbb R$ 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 615230, 14 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365168370

Digital Object Identifier
doi:10.1155/2012/615230

Mathematical Reviews number (MathSciNet)
MR2947733

Zentralblatt MATH identifier
1247.35189

#### Citation

Li, Qiuping; Sun, Shurong; Zhao, Ping; Han, Zhenlai. Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 615230, 14 pages. doi:10.1155/2012/615230. https://projecteuclid.org/euclid.aaa/1365168370

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