Positive solutions for three-point nonlinear fractional boundary value problems

• Published in: Electronic journal of qualitative theory of differential equations Vol. 2011; no. 2; pp. 1 - 19
• Main Authors: Saadi, Abdelkader, Benbachir, Maamar
• Format: Journal Article
• Language: English
• Published: University of Szeged 2011
• ISSN: 1417-3875; 1417-3875
• DOI: 10.14232/ejqtde.2011.1.2

In this paper, we give sufficient conditions for the existence or the nonexistence of positive solutions of the nonlinear fractional boundary value problem \begin{gather*} D_{0^{+}}^{\alpha}u+a(t)f(u(t))=0, 0<t<1, 2<\alpha<3,\\ u(0)=u^{\prime}(0)=0, u^{\prime}(1)-\mu u^{\prime}(\eta)=\lambda, \end{gather*} where $D_{0^{+}}^{\alpha}$ is the standard Riemann-Liouville fractional differential operator of order $\alpha$, $\eta\in\left(0,1\right)$, $\mu\in\left[0,\dfrac{1}{\eta^{\alpha-2}}\right)$ are two arbitrary constants and $\lambda\in\left[ 0,\infty\right) $ is a parameter. The proof uses the Guo-Krasnosel'skii fixed point theorem and Schauder's fixed point theorem.