Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations

Jing Wu; Xinguang Zhang

doi:10.1155/2012/740760

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2012 Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations

Jing Wu, Xinguang Zhang

Abstr. Appl. Anal. 2012(SI06): 1-14 (2012). DOI: 10.1155/2012/740760

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.

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Jing Wu. Xinguang Zhang. "Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations." Abstr. Appl. Anal. 2012 (SI06) 1 - 14, 2012. https://doi.org/10.1155/2012/740760