Positivity 13 (2009), 443–457
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020443-15, published online July 5, 2008
The existence of three positive solutions
for some nonlinear boundary value problems
on the half-line
Sihua Liang and Jihui Zhang
Abstract. In this paper, by introducing a new operator, improving and gener-
ating a p-Laplace operator for some p>1, we consider the existence of triple
positive solutions for some nonlinear m-point boundary value problems on
+ a(t)f(t, u(t)) = 0, 0 <t<+∞,
where ϕ : R → R is the increasing homeomorphism and positive homomor-
phism and ϕ(0) = 0. We show the existence of at least three positive solutions
with suitable growth conditions imposed on the nonlinear term by using the
ﬁve functionals ﬁxed-point theorem.
Mathematics Subject Classiﬁcation (2000). 34B15, 34B40.
Keywords. The ﬁve functionals ﬁxed-point theorem, boundary-value prob-
lems, positive solutions, half-line, cone.
In this paper, we consider the existence of positive solutions of the following
boundary value problem on a half-line
+ a(t)f (t, u(t)) = 0, 0 <t<+∞, (1.1)
Project supported by Foundation of Major Project of Science and Technology of Chinese Educa-
tion Ministry, SRFDP of Higher Education, NSF of Education Committee of Jiangsu Province
and Project of Graduate Education Innovation of Jiangsu Province.