Volume 37 • Number 1 • 2014   • Twin Positive Solutions of Second-Order $m$-Point Boundary Value Problem with Sign Changing Nonlinearities Fuyi Xu and Xiaoyan Guan Abstract. In this paper, we study second-order $m$-point boundary value problem $$%\left\{\begin{array}{lll} \begin{cases} u''(t)+a(t)u'(t)+f(t,u)=0, & 0 \leq t \leq 1, \\ %\cr\noalign{\vskip 2mm} u'(0)=0, \quad u(1)=\sum _{i=1}^{k}a_iu(\xi_{i})-\sum _{i=k+1}^{m-2}a_{i} u(\xi_{i}), & %\end{array}\right. \end{cases}$$ where $a_{i}>0(i=1,2,\ldots, m-2), 0<\sum _{i=1}^{k}a_i-\sum_{i=k+1}^{m-2}a_{i}<1, 0<\xi_{1}<\xi_{2}<\cdots<\xi_{m-2}<1$, $a\in C([0,1],(-\infty, 0))$ and $f$ is allowed to change sign. We show that there exist two positive solutions by using Leggett-Williams fixed-point theorem. The conclusions in this paper essentially extend and improve some known results. 2010 Mathematics Subject Classification: 34B15, 34B25 Full text: PDF