Electron. J. Diff. Eqns., Vol. 2006(2006), No. 36, pp. 1-12.

Positive solutions for boundary-value problems of nonlinear fractional differential equations

Shuqin Zhang

Abstract:
In this paper, we consider the existence and multiplicity of positive solutions for the nonlinear fractional differential equation boundary-value problem
$$\displaylines{ \hbox{\bf D}_{0+}^\alpha u(t)=f(t,u(t)),\quad 0 less than t less than 1\cr u(0)+u'(0)=0,\quad u(1)+u'(1)=0 }$$
where $1 less than \alpha\leq 2$ is a real number, and $\hbox{\bf D}_{0+}^\alpha$ is the Caputo's fractional derivative, and $f:[0,1]\times[0,+\infty)\to [0,+\infty)$ is continuous. By means of a fixed-point theorem on cones, some existence and multiplicity results of positive solutions are obtained.

Submitted November 30, 2005. Published March 21, 2006.
Math Subject Classifications: 34B15.
Key Words: Caputo's fractional derivative; fractional differential equation; boundary-value problem; positive solution; fractional Green's function; fixed-point theorem

An addendum as attached on November 9, 2009. It corrects the definition of n in Lemmas 2.2 and 2.3. See the last page of this article.

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Shuqin Zhang
Department of Mathematics
University of Mining and Technology
Beijing, 100083 China
email: zhangshuqin@tsinghua.org.cn

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