Positive solutions for three-point nonlinear fractional boundary value problems

A. Saadi, Bechar University, Bechar, Algeria
M. Benbachir, Bechar University, Bechar, Algeria

E. J. Qualitative Theory of Diff. Equ., No. 2. (2011), pp. 1-19.

Communicated by M. Benchohra.Received on 2010-01-24
Appeared on 2011-01-07

Abstract: 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.


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