Multiple positive solutions for (n-1, 1)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations

Chengjun Yuan, Harbin University, Harbin, Heilongjiang, P. R. China

E. J. Qualitative Theory of Diff. Equ., No. 36. (2010), pp. 1-12.

Abstract: In this paper, we consider (n-1, 1)-type conjugate boundary value problem for the nonlinear fractional differential equation
\begin{gather*}\begin{array}{ll}
\mathbf{D}_{0+}^\alpha u(t)+\lambda f(t,u(t))=0,\quad 0<t<1, \lambda >0,\\
u^{(j)}(0)=0, 0\leq j\leq n-2,\\
u(1)=0,
\end{array}\end{gather*}
where $\lambda$ is a parameter, $\alpha\in(n-1, n]$ is a real number and $n\geq 3$, and $\mathbf{D}_{0+}^\alpha$ is the Riemann-Liouville's fractional derivative, and $f$ is continuous and semipositone. We give properties of Green's function of the boundary value problems, and derive an interval of $\lambda$ such that any $\lambda$ lying in this interval, the semipositone boundary value problem has multiple positive solutions.

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