Boundary Value Problems
Volume 2008 (2008), Article ID 723828, 14 pages
doi:10.1155/2008/723828
Research Article
Solvability for Two Classes of Higher-Order Multi-Point
Boundary Value Problems at
Resonance
Department of Mathematics, Beihua University, Jilin City 132013, China
Received 13 July 2007; Revised 14 December 2007; Accepted 29 January 2008
Academic Editor: Ivan Kiguradze
Copyright © 2008 Yunzhu Gao and Minghe Pei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.