Existence of positive solutions for second order $m$-point boundary value problems
Tom 79 / 2002
Streszczenie
Let $\alpha, \beta, \gamma, \delta \geq 0$ and $\varrho:=\gamma\beta+\alpha\gamma+\alpha\delta>0$. Let $\psi(t)=\beta+\alpha t$, $\phi(t)=\gamma+\delta -\gamma t$, $t\in [0,1]$. We study the existence of positive solutions for the $m$-point boundary value problem $$\cases{ u' ' + h(t) f(u)=0, \quad\ 0< t< 1 ,\cr \alpha u(0)- \beta u'(0)=\sum^{m-2}_{i=1}a_i u(\xi_i),\cr \gamma u(1)+\delta u'(1)=\sum^{m-2}_{i=1}b_i u(\xi_i),\cr } $$ where $\xi_i\in (0,1)$, $a_i, b_i\in (0,\infty)$ (for $i\in \{1,\ldots, m-2\}$) are given constants satisfying $\varrho -\sum^{m-2}_{i=1} a_i \phi(\xi_i)>0$, $\varrho -\sum^{m-2}_{i=1} b_i \psi(\xi_i)>0$ and $${\mit\Delta}:= \left| \matrix{ -\sum^{m-2}_{i=1}a_i\psi(\xi_i) &\varrho -\sum^{m-2}_{i=1}a_i\phi(\xi_i)\cr \varrho -\sum^{m-2}_{i=1}b_i\psi(\xi_i) &-\sum^{m-2}_{i=1}b_i\phi(\xi_i)\cr} \right|< 0. $$ We show the existence of positive solutions if $f$ is either superlinear or sublinear by a simple application of a fixed point theorem in cones. Our result extends a result established by Erbe and Wang for two-point BVPs and a result established by the author for three-point BVPs.
