On solutions of some fractional $m$-point boundary value problems at resonance

Zhanbing Bai, Shandong University of Science and Technology, Qingdao, P. R. China

E. J. Qualitative Theory of Diff. Equ., No. 37. (2010), pp. 1-15.

Abstract: In this paper, the following fractional order ordinary differential equation boundary value problem:
\begin{gather*}
D_{0+}^\alpha u(t) =f(t,u(t),D_{0+}^{\alpha-1}u(t))+e(t), 0<t<1,\\
I_{0+}^{2-\alpha}u(t)\mid_{t=0}=0, D_{0+}^{\alpha-1}u(1)=\sum_{i=1}^{m-2}\beta_i D_{0+}^{\alpha-1}u(\eta_i),
\end{gather*}
is considered, where $1< \alpha \leq 2,$ is a real number, $D_{0+}^\alpha$ and $I_{0+}^{\alpha}$ are the standard Riemann-Liouville differentiation and integration, and $f:[0,1]\times R^2 \to R$ is continuous and $e \in L^1[0,1]$, and $\eta_i \in (0, 1), \beta_i \in R, i=1,2, \cdots, m-2$, are given constants such that $\sum_{i=1}^{m-2}\beta_i=1$. By using the coincidence degree theory, some existence results of solutions are established.

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