Existence of positive solutions for multi-point boundary value problems

B. Karna, Marshall University, West Virginia, USA
B. Lawrence, Marshall University, West Virginia, USA

E. J. Qualitative Theory of Diff. Equ., No. 26. (2007), pp. 1-11.

Abstract: The existence of positive solutions are established for the multi-point boundary value problems
$$
\left\{ \begin{array}{ll}
(-1)^nu^{(2n)}(x)=\lambda p(x)f(u(x)), \hspace*{.2in} 0<x<1 \\
u^{(2i)}(0)=\sum_{j=1}^{m}a_ju^{(2i)}(\eta _j), \quad
u^{(2i+1)}(1)=\sum_{j=1}^{m}b_ju^{(2i+1)}(\eta _j), \quad i=0, 1,
\ldots , n-1
\end{array} \right.
$$
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 $\l$. We use the positivity of the Green's function and cone theory to prove our results.

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