Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions
In this paper, we consider a four-point coupled boundary value problem for systems of the nonlinear semipositone fractional differential equation \begin{gather*}\left\{ \begin{array}{ll} \mathbf{D}_{0+}^\alpha u+\lambda f(t,u,v)=0,\quad 0<t<1, \lambda >0,\\ \mathbf{D}_{0+}^\alpha v+\lamb...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2012-02-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=1171 |
| Summary: | In this paper, we consider a four-point coupled boundary value problem for systems of the nonlinear semipositone fractional differential equation
\begin{gather*}\left\{ \begin{array}{ll}
\mathbf{D}_{0+}^\alpha u+\lambda f(t,u,v)=0,\quad 0<t<1, \lambda >0,\\
\mathbf{D}_{0+}^\alpha v+\lambda g(t,u,v)=0,\\
u^{(i)}(0)=v^{(i)}(0)=0, 0\leq i\leq n-2,\\
u(1)=av(\xi), v(1)=bu(\eta), \xi,\eta\in(0,1)
\end{array}\right.\end{gather*}
where $\lambda$ is a parameter, $a, b, \xi,\eta$ satisfy $\xi,\eta\in(0,1)$, $0<ab\xi\eta<1$, $\alpha \in(n-1, n]$ is a real number and $n\geq 3$, and $\mathbf{D}_{0+}^\alpha$ is the Riemann-Liouville's fractional derivative, and $f,g$ are continuous and semipositone. We derive an interval on $\lambda$ such that for any $\lambda$ lying in this interval, the semipositone boundary value problem has multiple positive solutions. |
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| ISSN: | 1417-3875 1417-3875 |