## Abstract and Applied Analysis

### Positive Solutions to Boundary Value Problems of Nonlinear Fractional Differential Equations

#### Abstract

We study the existence of positive solutions for the boundary value problem of nonlinear fractional differential equations ${D}_{{0}^{+}}^{\alpha}u(t)+\lambda f(u(t))=0$, $0\lt t\lt 1$, $u(0)=u(1)=u'(0)=0$, where $2\lt \alpha \le 3$ is a real number, ${D}_{{0}^{+}}^{\alpha}$ is the Riemann-Liouville fractional derivative, $\lambda$ is a positive parameter, and $f:(0,+\infty )\to (0,+\infty )$ is continuous. By the properties of the Green function and Guo-Krasnosel'skii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are considered, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. As an application, some examples are presented to illustrate the main results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2011, Number 1 (2011), Article ID 390543, 16 pages.

Dates
First available in Project Euclid: 12 August 2011

https://projecteuclid.org/euclid.aaa/1313171387

Digital Object Identifier
doi:10.1155/2011/390543

Mathematical Reviews number (MathSciNet)
MR2746016

Zentralblatt MATH identifier
1210.34009

#### Citation

Zhao, Yige; Sun, Shurong; Han, Zhenlai; Li, Qiuping. Positive Solutions to Boundary Value Problems of Nonlinear Fractional Differential Equations. Abstr. Appl. Anal. 2011 (2011), no. 1, Article ID 390543, 16 pages. doi:10.1155/2011/390543. https://projecteuclid.org/euclid.aaa/1313171387