Boundary Value Problems
Volume 2008 (2008), Article ID 254593, 10 pages
doi:10.1155/2008/254593
Research Article
Global Behavior of the Components for the Second Order m-Point Boundary Value Problems
1Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
2Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China
Received 9 October 2007; Accepted 16 December 2007
Academic Editor: Kanishka Perera
Copyright © 2008 Yulian An and Ruyun Ma. 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
We consider the nonlinear eigenvalue problems
u″+rf(u)=0,
0<t<1,
u(0)=0,
u(1)=∑i=1m−2αiu(ηi),
where
m≥3,
ηi∈(0,1),
and
αi>0
for
i=1,…,m−2,
with
∑i=1m−2αi<1;
r∈ℝ;
f∈C1(ℝ,ℝ).
There exist two constants
s2<0<s1
such that
f(s1)=f(s2)=f(0)=0
and
f0:=limu→0(f(u)/u)∈(0,∞),
f∞:=lim|u|→∞(f(u)/u)∈(0,∞).
Using the global bifurcation techniques, we study the global
behavior of the components of nodal solutions of the above
problems.