Existence of positive solutions for multi-point boundary value problems - B. Karna (Electronic Journal of Qualitative Theory of Differential Equations, 2007)
The existence of positive solutions are established for the multi-point boundary value problems $$ \left\{ \begin{array}{ll} (-1)^nu^{(2n)}(x)=\lambda p(x)f(u(x)),\quad 0<x<1 \\ u^{(2i)}(0)=\sum_{j=1}^{m}a_ju^{(2i)}(\eta _j), \quad u^{(2i+1)}(1)=\sum_{j=1}^{m}b_ju^{(2i+1)}(\eta _j), \quad i=0, 1, \ldots , n-1 \end{array} \right. $$ where $a_j,b_j\in[0,\infty), \ j=1, 2, \ldots, m,$ with $0<\sum_{j=1}^{m}a_j<1, \ 0<\sum_{j=1}^{m}b_j<1,$ and $ \eta_j \in(0,1)$ with $0<\eta_1<\eta_2<\ldots <\eta_m<1,$ under certain conditions on $f$ and $p$ using the Krasnosel'skii fixed point theorem for certain values of $\lambda$. We use the positivity of the Green's function and cone theory to prove our results.