Positivity 12 (2008), 547–554
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/030547-8, published online March 12, 2008
Existence of positive solutions for a class
of second-order two-point boundary
Guowei Zhang, Jingxian Sun and Tie Zhang
Abstract. By using different convex functionals to compute ﬁxed point index,
the existence of positive solutions for a class of second-order two-point bound-
ary value problem
(t)+h(t)f(ϕ(t)) = 0, 0 <t<1,
αϕ(0) − βϕ
(0) = 0,γϕ(1) + δϕ
(1) = 0,
is obtained under some conditions of growth, where α, β, γ, δ ≥ 0,ρ= αγ +
γβ + δα > 0, and h(t) is allowed to be singular at t = 0 and t =1.
Mathematics Subject Classiﬁcation (2000). Primary 34B15, Secondary 34B18.
Keywords. Positive solution, cone, ﬁxed point index.
Let E be a real Banach space, θ be the zero element in E and P be a cone in E.For
the theory and properties of cone and ﬁxed point index in Banach spaces we refer
to . Krasnosel’skii ﬁxed point theorem about cone compression and expansion of
norm type is extensively applied to many second-order two-point boundary value
problem of various kinds, see [2–4,6,7,9,10,12], for example. In [1,8], the authors
dealt with modiﬁcations of the classical Krasnosel’skii ﬁxed point theorem about
cone compression and expansion of norm type. Very recently, the authors in 
extend Krasnosel’skii ﬁxed point theorem by replacing the norm with some convex
functional on P .
Now we state two Lemmas in  which will be applied to the existence of
positive solutions for a class of second-order two-point boundary value problem in
this paper by using different convex functionals. ρ : P → R is said to be a convex
functional on P ,ifρ(tx +(1− t)y) ≤ tρ(x)+(1− t)ρ(y) for all x, y ∈ P and
t ∈ [0, 1].
Supported by the National Natural Science Foundation of China(10771031,10671167).