Boundary Value Problems
Volume 2008 (2008), Article ID 723828, 14 pages
doi:10.1155/2008/723828

Solvability for two classes of higher-order multi-point boundary value problems at resonance

Abstract

Using the theory of coincidence degree, we establish existence results of positive solutions for higher-order multi-point boundary value problems at resonance for ordinary differential equation u(n)(t)=f(t,u(t),u′(t),…,u(n−1)(t))+e(t),    t∈(0,1), with one of the following boundary conditions: u(i)(0)=0, i=1,2,…, n−2, u(n−1)(0)=u(n−1)(ξ), u(n−2)(1)=∑j=1m−2βju(n−2)(ηj), and u(i)(0)=0, i=1,2,…, n−1, u(n−2)(1)=∑j=1m−2βju(n−2)(ηj), where f:[0,1]×ℝn→ℝ=(−∞,+∞) is a continuous function, e(t)∈L1[0,1]βj∈ℝ (1≤j≤m−2, m≥4), 0<η1<η2<⋯<ηm−2<1, 0<ξ<1, all the β−j−s have not the same sign. We also give some examples to demonstrate our results.