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Erschienen in:
01.02.2015 | Original Research
Positive solutions for a class of fractional differential equation multi-point boundary value problems with changing sign nonlinearity
Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2015
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Abstract
In this paper, we consider the multi-point boundary value problem of nonlinear fractional differential equation where \(\lambda \) is a parameter, \(\alpha \ge 2\), \(n-1<\alpha \le n\), \(i\in N\), \(0\le i\le n-2, {\eta }_{j}\ge 0\ (j=1,2,\ldots ,m-2)\), \(0<\xi _1<\xi _2<\cdots <\xi _{m-2}<1, \Delta -(\alpha -1) \sum \nolimits _{j=1}^{m-2}{\eta }_{j}\xi _{j}^{\alpha -2}>0\),
\(D_{0{+}}^{\alpha }\)is the Riemann–Liouville’s fractional derivative, \(f\) may change sign and may be singular at \(t=0,1\). We give the corresponding Green’s function for the boundary value problem and its some properties. Moreover, we derive an interval of \(\lambda \) such that for any \(\lambda \) lying in this interval, the semipositone boundary value problem has positive solutions.
$$\begin{aligned} \left\{ \begin{array}{lcl} D_{0{+}}^{\alpha }u(t)+{\lambda }f(t,u(t))=0, \quad 0<t<1,\\ u(0)=u^{\prime }(0)=\cdots =u^{(n-2)}(0)=0,\\ u^{(i)}(1)=\sum \nolimits _{j=0}^{m-2}{\eta }_{j}u^{\prime }(\xi _{j}), \end{array} \right. \end{aligned}$$
$$\begin{aligned} \Delta = \left\{ \begin{array}{ll} {1}, &{} \quad {i = 0}; \\ {(\alpha - 1)(\alpha - 2) \cdots (\alpha - i)}, &{} \quad {i \ge 1}. \\ \end{array} \right. \end{aligned}$$