Existence of Solutions for a Higher Order Multi-Point Boundary Value Problem

Abstract.

We study the n-th order multi-point boundary value problem

$$ u^{(n)} + f(t, u, u', \ldots ,u^{(n - 1)} ) = \lambda p(t), \quad t \in (0,1), $$

$$\left\{ \begin{array}{ll} u^{(i)} (0) = A_{i}, \, i = 0, \ldots , n - 3,\\ u^{(n - 2)} (0) - \sum\nolimits_{j = 1}^m {a_j u^{(n - 2)} (t_j ) = A_{n - 2}},\\ u^{(n - 2)} (1) - \sum \nolimits_{j = 1}^m {b_j u^{(n - 2)} (t_{j} ) = A_{n - 1}} \end{array} \right. $$

. Sufficient conditions are obtained for the existence of one and two solutions of the problem for different values of λ. Our results extend and improve some recent work in the literature. Our analysis mainly relies on the lower and upper solution method and topological degree theory.