## Abstract and Applied Analysis

### Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations

#### Abstract

We study the eigenvalue interval for the existence of positive solutions to a semipositone higher order fractional differential equation $-{{\mathrm{\scr D}}_{\mathbf{t}}}^{\mu }x(t)$ = $\lambda f(t,x(t),{{\mathrm{\scr D}}_{\mathbf{t}}}^{{\mu }_{1}}x(t),{{\mathrm{\scr D}}_{\mathbf{t}}}^{{\mu }_{2}}x(t),\dots ,{{\mathrm{\scr D}}_{\mathbf{t}}}^{{\mu }_{n-1}}x(t))\dots {{\mathrm{\scr D}}_{\mathbf{t}}}^{{\mu }_{i}}x(0)$ = $0,1\le i\le n-1,{{\mathrm{\scr D}}_{\mathbf{t}}}^{{\mu }_{n-1}+1}x(0)=0,{{\mathrm{\scr D}}_{\mathbf{t}}}^{{\mu }_{n-1}}x(1)={\sum }_{j=1}^{m-2}{a}_{j}{{\mathrm{\scr D}}_{\mathbf{t}}}^{{\mu }_{n-1}}x({\xi }_{j}),$  where $n-1<\mu \le n$,  $n\ge 3$, $0<{\mu }_{1}<{\mu }_{2}<\cdots <{\mu }_{n-2}<{\mu }_{n-1}$, $n-3<{\mu }_{n-1}<\mu -2$, ${a}_{j}\in \Bbb R,0<{\xi }_{1}<{\xi }_{2}<\cdots <{\xi }_{m-2}<1$ satisfying $0<{\sum }_{j=1}^{m-2}{a}_{j}{\xi }_{j}^{\mu -{\mu }_{n-1}-1}<1$, ${{\mathrm{\scr D}}_{\mathbf{t}}}^{\mu }$ is the standard Riemann-Liouville derivative, $f\in C((0,1)×{\Bbb R}^{n},(-\infty ,+\infty ))$, and $f$ is allowed to be changing-sign. By using reducing order method, the eigenvalue interval of existence for positive solutions is obtained.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 740760, 14 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365168876

Digital Object Identifier
doi:10.1155/2012/740760

Mathematical Reviews number (MathSciNet)
MR3004914

Zentralblatt MATH identifier
1260.34015

#### Citation

Wu, Jing; Zhang, Xinguang. Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 740760, 14 pages. doi:10.1155/2012/740760. https://projecteuclid.org/euclid.aaa/1365168876