Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives

Tunhua Wu; Xinguang Zhang; Yinan Lu

doi:10.1155/2012/797398

2012 Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives

Tunhua Wu, Xinguang Zhang, Yinan Lu

Abstr. Appl. Anal. 2012(SI11): 1-16 (2012). DOI: 10.1155/2012/797398

Abstract

We study the singular fractional-order boundary-value problem with a sign-changing nonlinear term $-{\mathrm{𝒟}}^{\alpha }x(t)=p(t)f(t,x(t),{\mathrm{𝒟}}^{{\mu }_{1}}x(t),{\mathrm{𝒟}}^{{\mu }_{2}}x(t),\dots ,{\mathrm{𝒟}}^{{\mu }_{n-1}}x(t)),0<t<1,{\mathrm{𝒟}}^{{\mu }_{i}}x(0)=0,1\le i\le n-1,{\mathrm{𝒟}}^{{\mu }_{n-1}+1}x(0)=0$ , ${\mathrm{𝒟}}^{{\mu }_{n-1}}x(1)={\sum }_{j=1}^{p-2}{a}_{j}{\mathrm{𝒟}}^{{\mu }_{n-1}}x({\xi }_{j})$ , where $n-1<\alpha \le n$ , $n\in \mathbb{N}$ and $n\ge 3$ with $0<{\mu }_{1}<{\mu }_{2}<\cdots <{\mu }_{n-2}<{\mu }_{n-1}$ and $n-3<{\mu }_{n-1}<\alpha -2$ , ${a}_{j}\in \mathbb{R},0<{\xi }_{1}<{\xi }_{2}<\cdots <{\xi }_{p-2}<1$ satisfying $0<{\sum }_{j=1}^{p-2}{a}_{j}{\xi }_{j}^{\alpha -{\mu }_{n-1}-1}<1$ , ${\mathrm{𝒟}}^{\alpha }$ is the standard Riemann-Liouville derivative, $f:[0,1]×{\mathbb{R}}^{n}\to \mathbb{R}$ is a sign-changing continuous function and may be unbounded from below with respect to ${x}_{i}$ , and $p:(0,1)\to [0,\infty )$ is continuous. Some new results on the existence of nontrivial solutions for the above problem are obtained by computing the topological degree of a completely continuous field.

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Tunhua Wu. Xinguang Zhang. Yinan Lu. "Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives." Abstr. Appl. Anal. 2012 (SI11) 1 - 16, 2012. https://doi.org/10.1155/2012/797398