Hindawi Publishing Corporation
Boundary Value Problems
Volume 2011, Article ID 594128, 21 pages
doi:10.1155/2011/594128
Research Article
Eigenvalue Problem and Unbounded Connected
Branch of Positive Solutions to a Class of Singular
Elastic Beam Equations
Huiqin Lu
School of Mathematical Sciences, Shandong Normal University, Jinan, 250014 Shandong, China
Correspondence should be addressed to Huiqin Lu, lhy@sdu.edu.cn
Received 16 October 2010; Revised 22 December 2010; Accepted 27 January 2011
Academic Editor: Kanishka Perera
Copyright q 2011 Huiqin Lu. 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.
This paper investigates the eigenvalue problem for a class of singular elastic beam equations where
one end is simply supported and the other end is clamped by sliding clamps. Firstly, we establish
a necessary and sufficient condition for the existence of positive solutions, then we prove that the
closure of positive solution set possesses an unbounded connected branch which bifurcates from
0, θ. Our nonlinearity f t, u, v, w may be singular at u,v, t 0 and/or t 1.
1. Introduction
Singular differential equations arise in the fields of gas dynamics, Newtonian fluid mechanics, the theory of boundary layer, and so on. Therefore, singular boundary value problems
have been investigated extensively in recent years see 1–4 and references therein.
This paper investigates the following fourth-order nonlinear singular eigenvalue
problem:
u4 t λft, ut, u t, ut,
t ∈ 0, 1,
u0 u 1 u 0 u 1 0,
1.1
where λ ∈ 0, ∞ is a parameter and f satisfies the following hypothesis:
H f ∈ C0, 1 × 0, ∞ × 0, ∞ × −∞, 0, 0, ∞, and there exist constants αi , βi ,
Ni , i 1, 2, 3 −∞ < α1 ≤ 0 ≤ β1 < ∞, −∞ < α2 ≤ 0 ≤ β2 < ∞, 0 ≤ α3 ≤ β3 < 1,
2
Boundary Value Problems
3
i1 βi < 1; 0 < Ni ≤ 1, i 1, 2, 3 such that for any t ∈ 0, 1, u, v ∈ 0, ∞,
w ∈ −∞, 0, f satisfies
cβ1 ft, u, v, w ≤ ft, cu, v, w ≤ cα1 ft, u, v, w,
∀0 < c ≤ N1 ,
c ft, u, v, w ≤ ft, u, cv, w ≤ c ft, u, v, w,
∀0 < c ≤ N2 ,
c ft, u, v, w ≤ ft, u, v, cw ≤ c ft, u, v, w,
∀0 < c ≤ N3 .
β2
α2
β3
α3
1.2
Typical functions that satisfy the above sublinear hypothesis H are those taking
the form
ft, u, v, w m3
m1 m2 pi,j,k turi vsj wσk ,
1.3
i1 j1 k1
where pi,j,k t ∈ C0, 1, 0, ∞, ri , sj ∈ R, 0 ≤ σk < 1, max{ri , 0} max{sj } σk < 1,
i 1, 2, . . . , m1 , j 1, 2, . . . , m2 , k 1, 2, . . . , m3 . The hypothesis H is similar to that in 5, 6.
Because of the extensive applications in mechanics and engineering, nonlinear fourthorder two-point boundary value problems have received wide attentions see 7–12 and
references therein. In mechanics, the boundary value problem 1.1 BVP 1.1 for short
describes the deformation of an elastic beam simply supported at left and clamped at right
by sliding clamps. The term u in f represents bending effect which is useful for the stability
analysis of the beam. BVP 1.1 has two special features. The first one is that the nonlinearity
f may depend on the first-order derivative of the unknown function u, and the second one is
that the nonlinearity ft, u, v, w may be singular at u, v, t 0 and/or t 1.
In this paper, we study the existence of positive solutions and the structure of positive
solution set for the BVP 1.1. Firstly, we construct a special cone and present a necessary and
sufficient condition for the existence of positive solutions, then we prove that the closure of
positive solution set possesses an unbounded connected branch which bifurcates from 0, θ.
Our analysis mainly relies on the fixed point theorem in a cone and the fixed point index
theory.
By singularity of f, we mean that the function f in 1.1 is allowed to be unbounded
at the points u 0, v 0, t 0, and/or t 1. A function ut ∈ C2 0, 1 ∩ C4 0, 1 is called
a positive solution of the BVP 1.1 if it satisfies the BVP 1.1 ut > 0, −u t > 0 for
t ∈ 0, 1 and u t > 0 for t ∈ 0, 1. For some λ ∈ 0, ∞, if the BVP 1.1 has a positive
solution u, then λ is called an eigenvalue and u is called corresponding eigenfunction of the
BVP 1.1.
The existence of positive solutions of BVPs has been studied by several authors in
the literature; for example, see 7–20 and the references therein. Yao 15, 18 studied the
following BVP:
u4 t ft, ut, u t,
t ∈ 0, 1 \ E,
u0 u 0 u 1 u 1 0,
1.4
where E ⊂ 0, 1 is a closed subset and mesE 0, f ∈ C0, 1\E×0, ∞×0, ∞, 0, ∞.
In 15, he obtained a sufficient condition for the existence of positive solutions of BVP 1.4
Boundary Value Problems
3
by using the monotonically iterative technique. In 13, 18, he applied Guo-Krasnosel’skii’s
fixed point theorem to obtain the existence and multiplicity of positive solutions of BVP 1.4
and the following BVP:
u4 t ft, ut,
t ∈ 0, 1,
1.5
u0 u 0 u1 u 1 0.
These differ from our problem because ft, u, v in 1.4 cannot be singular at u 0, v 0
and the nonlinearity f in 1.5 does not depend on the derivatives of the unknown functions.
In this paper, we first establish a necessary and sufficient condition for the existence
of positive solutions of BVP 1.1 for any λ > 0 by using the following Lemma 1.1. Efforts
to obtain necessary and sufficient conditions for the existence of positive solutions of BVPs
by the lower and upper solution method can be found, for example, in 5, 6, 21–23. In 5,
6, 22, 23 they considered the case that f depends on even order derivatives of u. Although
the nonlinearity f in 21 depends on the first-order derivative, where the nonlinearity f
is increasing with respect to the unknown function u. Papers 24, 25 derived the existence
of positive solutions of BVPs by the lower and upper solution method, but the nonlinearity
ft, u does not depend on the derivatives of the unknown functions, and ft, u is decreasing
with respect to u.
Recently, the global structure of positive solutions of nonlinear boundary value
problems has also been investigated see 26–28 and references therein. Ma and An 26
and Ma and Xu 27 discussed the global structure of positive solutions for the nonlinear
eigenvalue problems and obtained the existence of an unbounded connected branch of
positive solution set by using global bifurcation theorems see 29, 30. The terms fu in
26 and ft, u, u in 27 are not singular at t 0, 1, u 0, u 0. Yao 14 obtained one or
two positive solutions to a singular elastic beam equation rigidly fixed at both ends by using
Guo-Krasnosel’skii’s fixed point theorem, but the global structure of positive solutions was
not considered. Since the nonlinearity ft, u, v, w in BVP 1.1 may be singular at u, v, t 0
and/or t 1, the global bifurcation theorems in 29, 30 do not apply to our problem here.
In Section 4, we also investigate the global structure of positive solutions for BVP 1.1 by
applying the following Lemma 1.2.
The paper is organized as follows: in the rest of this section, two known results are
stated. In Section 2, some lemmas are stated and proved. In Section 3, we establish a necessary
and sufficient condition for the existence of positive solutions. In Section 4, we prove that the
closure of positive solution set possesses an unbounded connected branch which comes from
0, θ.
Finally we state the following results which will be used in Sections 3 and 4,
respectively.
Lemma 1.1 see 31. Let X be a real Banach space, let K be a cone in X, and let Ω1 , Ω2 be bounded
open sets of E, θ ∈ Ω1 ⊂ Ω1 ⊂ Ω2 . Suppose that T : K ∩ Ω2 \ Ω1 → K is completely continuous
such that one of the following two conditions is satisfied:
1 Tx
≤ x
, x ∈ K ∩ ∂Ω1 ; Tx
≥ x
, x ∈ K ∩ ∂Ω2 .
2 Tx
≥ x
, x ∈ K ∩ ∂Ω1 ; Tx
≤ x
, x ∈ K ∩ ∂Ω2 .
Then, T has a fixed point in K ∩ Ω2 \ Ω1 .
4
Boundary Value Problems
∞
Lemma 1.2 see 32. Let M be a metric space and a, b ⊂ R1 . Let {an }∞
n1 and {bn }n1 satisfy
a < · · · < an < · · · < a1 < b1 < · · · < bn < · · · < b,
lim an a,
lim bn b.
n → ∞
1.6
n → ∞
Suppose also that
{Cn : n 1, 2, . . .} is a family of connected subsets of R1 × M, satisfying the
following conditions:
1 Cn ∩ {an } × M /
∅ and Cn ∩ {bn } × M /
∅ for each n.
2 For any two given numbers α and β with a < α < β < b, ∞
n1 Cn ∩ α, β × M is a
relatively compact set of R1 × M.
Then there exists a connected branch C of lim supn → ∞ Cn such that
C ∩ {λ} × M /
∅,
∀λ ∈ a, b,
1.7
where lim supn → ∞ Cn {x ∈ M : there exists a sequence xni ∈ Cni such that xni → x, i → ∞}.
2. Some Preliminaries and Lemmas
Let E {u ∈ C2 0, 1 : u0 0, u 1 0, u 0 0}, u
2 max{
u
, u
, u }, then
E, · 2 is a Banach space, where u
maxt∈0,1 |ut|. Define
P
u ∈ E : ut ≥
t2
t−
2
1
u
, u t ≥ 1 − t u , −u t ≥ t u , t ∈ 0, 1 .
2
2.1
It is easy to conclude that P is a cone of E. Denote
Pr {u ∈ P : u
2 < r};
∂Pr {u ∈ P : u
2 r}.
2.2
Let
G0 t, s Gt, s ⎧
⎨s,
0 ≤ s ≤ t ≤ 1,
⎩t,
0 ≤ t ≤ s ≤ 1,
1
0
G0 t, rG0 r, sdr.
2.3
Boundary Value Problems
5
Then Gt, s is the Green function of homogeneous boundary value problem
u4 t 0,
t ∈ 0, 1,
u0 u 1 u 0 u 1 0,
⎧ 3
s t2 − s 2
s
⎪
⎪
⎪
st1 − t, 0 ≤ s ≤ t ≤ 1,
⎨3 2
Gt, s ⎪
3
⎪
t s 2 − t2
⎪
⎩t ts1 − s, 0 ≤ t ≤ s ≤ 1,
3
2
⎧
⎪
0 ≤ s ≤ t ≤ 1,
⎪
⎨s1 − t,
G1 t, s : Gt t, s 2
2
⎪
⎪
⎩ s − t s1 − s, 0 ≤ t ≤ s ≤ 1,
2
2
⎧
⎨s, 0 ≤ s ≤ t ≤ 1,
G2 t, s : −Gt t, s ⎩t, 0 ≤ t ≤ s ≤ 1.
2.4
Lemma 2.1. Gt, s, G1 t, s, and G2 t, s have the following properties:
1 Gt, s > 0, Gi t, s > 0, i 1, 2, for all t, s ∈ 0, 1.
2 Gt, s ≤ st − t2 /2, G1 t, s ≤ s1 − t, G2 t, s ≤ t or s, for all t, s ∈ 0, 1.
3 maxt∈0,1 Gt, s ≤ 1/2s, maxt∈0,1 Gi t, s ≤ s, i 1, 2, for all s ∈ 0, 1.
4 Gt, s ≥ s/2t − t2 /2, G1 t, s ≥ s/21 − t, G2 t, s ≥ st, for all t, s ∈ 0, 1.
Proof. From 2.4, it is easy to obtain the property 2.18.
We now prove that property 2 is true. For 0 ≤ s ≤ t ≤ 1, by 2.4, we have
t2
s3 st2 s3
st2
2
−
st − st ≤ st −
s t−
,
Gt, s 3
2
2
2
2
G1 t, s s1 − t,
2.5
G2 t, s ≤ tor s.
For 0 ≤ t ≤ s ≤ 1, by 2.4, we have
t3 t3
ts2
st2
t2
Gt, s − ts −
≤ st −
s t−
,
3
2
2
2
2
G1 t, s s −
2
2
s
t
−
≤ s − ts s1 − t,
2
2
2.6
G2 t, s ≤ t or s.
Consequently, property 2 holds.
From property 2, it is easy to obtain property 3.
We next show that property 4 is true. From 2.4, we know that property 4 holds
for s 0.
6
Boundary Value Problems
For 0 < s ≤ 1, if s ≤ t ≤ 1, then
t2 s 2 1
t2
t2 s 2
1
t2
t2 t2
1
t2
Gt, s
t− −
t− t− −
≥
t− t− −
>
t−
,
s
2
6
2
2
2
3
2
2
2
3
2
2
G1 t, s
1
1 − t ≥ 1 − t,
s
2
G2 t, s ≥ st;
2.7
if 0 ≤ t ≤ s, then
1
t2 ts 1
t2
t2
1
t2
Gt, s
≥t− −
t − t − ts ≥
t−
≥
t−
,
s
6
2
2
3
2
3
2
2
G1 t, s
t s 1
≥ 1 − − ≥ 1 − t,
s
2 2 2
2.8
G2 t, s ≥ st.
Therefore, property 4 holds.
Lemma 2.2. Assume that u ∈ P \ {θ}, then u
2 u and
1
u ≤ u
≤ u ,
4
1
t2
t2
t−
u
2 ≤ ut ≤ t −
u
2 ,
8
2
2
1
u ≤ u ≤ u .
2
2.9
1
1 − t
u
2 ≤ u t ≤ 1 − t
u
2 ,
4
t
u
2 ≤ −u t ≤ u
2 ,
2.10
∀t ∈ 0, 1.
Proof. Assume that u ∈ P \ {θ}, then u t ≥ 0, −u t ≥ 0, t ∈ 0, 1, so
u
max
t
t∈0,1
u
max
t
t∈0,1
u max
t∈0,1
u max
t∈0,1
1
u sds 1
0
u sds 0
1
1
0
u sds ≥
0
−u sds t
−u sds t
u sds ≤ u ,
1
0
1
u 2
1
1 − sds 0
1
u ,
4
−u sds ≤ u ,
0
1
−u sds ≥
1
0
2.11
1 su ds u .
2
Therefore, 2.9 holds. From 2.9, we get
u
2 max u
, u , u u .
2.12
Boundary Value Problems
7
By 2.9 and the definition of P , we can obtain that
ut 1
G0 t, s −u s ds ≤
t
0
sds 1
0
t
2
2
t
t
u t −
tds u t −
u
2 ,
2
2
∀t ∈ 0, 1,
ut ≥
t−
t2
2
1
u t t
u
≥
1
8
t−
t2
2
u
2 ,
∀t ∈ 0, 1,
−u sds ≤ 1 − tu 1 − t
u
2 ,
1
1
1 − tu ≥ 1 − t
u
2 , ∀t ∈ 0, 1,
2
4
t
u
2 tu ≤ −u t ≤ u u
2 , ∀t ∈ 0, 1.
u t ≥
2.13
Thus, 2.10 holds.
For any fixed λ ∈ 0, ∞, define an operator Tλ by
Tλ ut : λ
1
Gt, sf s, us, u s, u s ds,
∀u ∈ P \ {θ}.
2.14
∀u ∈ P \ {θ},
2.15
0
Then, it is easy to know that
Tλ u t λ
1
G1 t, sf s, us, u s, u s ds,
0
Tλ u t −λ
1
G2 t, sf s, us, u s, u s ds,
∀u ∈ P \ {θ}.
2.16
0
Lemma 2.3. Suppose that (H) and
1
0<
0
hold. Then Tλ P \ {θ} ⊂ P .
s2
sf s, s − , 1 − s, −1 ds < ∞
2
2.17
8
Boundary Value Problems
Proof. From H, for any t ∈ 0, 1, u, v ∈ 0, ∞, w ∈ −∞, 0, we easily obtain the following
inequalities:
cα1 ft, u, v, w ≤ ft, cu, v, w ≤ cβ1 ft, u, v, w,
∀c ≥ N1−1 ,
cα2 ft, u, v, w ≤ ft, u, cv, w ≤ cβ2 ft, u, v, w,
∀c ≥ N2−1 ,
cα3 ft, u, v, w ≤ ft, u, v, cw ≤ cβ3 ft, u, v, w,
∀c ≥ N3−1 .
2.18
For every u ∈ P \ {θ}, t ∈ 0, 1, choose positive numbers c1 ≤ min{N1 , 1/8N1 u
2 }, c2 ≤
min{N2 , 1/4N2 u
2 }, c3 ≥ max{N3−1 , N3−1 u
2 }. It follows from H, 2.10, Lemma 2.1, and
2.17 that
Tλ ut λ
1
Gt, sf s, us, u s, us ds
0
1
≤ λ
2
1
≤ λ
2
1
≤ λ
2
1
0
1
0
1
0
us
sf s, c1
c1 s − s2 /2
sc1α1
sc1α1
s2
s−
2
u s
u s
, c2
1 − s, −1c3
c2 1 − s
−c3
ds
β2 α3 β1 us
u s
s2
β3 u s
α2
c2
c3
f s, s − , 1 − s, −1 ds
c2 1 − s
−c3
2
c1 s − s2 /2
u
2
c1
β1
c2α2
u
2
c2
1 α −β α −β β −α
β β α
≤ c1 1 1 c2 2 2 c3 3 3 u
21 2 3 λ
2
β2
1
0
β
c3 3
u
2
c3
α3 s2
f s, s − , 1 − s, −1 ds
2
s2
sf s, s − , 1 − s, −1 ds < ∞.
2
2.19
Similar to 2.19, from H, 2.10, Lemma 2.1, and 2.17, for every u ∈ P \ {θ}, t ∈
0, 1, we have
Tλ u t λ
1
G1 t, sf s, us, u s, us ds
0
≤λ
1
0
≤
us
sf s, c1
c1 s − s2 /2
α −β α −β β −α
β β α
c1 1 1 c2 2 2 c3 3 3 u
21 2 3 λ
s2
s−
2
1
0
u s
u s
, c2
1 − s, −1c3
c2 1 − s
−c3
s2
sf s, s − , 1 − s, −1 ds < ∞.
2
ds
Boundary Value Problems
−Tλ u t λ
1
0
1
9
G2 t, sf s, us, u s, us ds
u s
u s
≤λ
, c2
1 − s, −1c3
c2 1 − s
−c3
0
1 s2
α1 −β1 α2 −β2 β3 −α3
β1 β2 α3
≤ c1
c2
c3
λ sf s, s − , 1 − s, −1 ds < ∞.
u
2
2
0
us
sf s, c1
c1 s − s2 /2
s2
s−
2
ds
2.20
Thus, Tλ is well defined on P \ {θ}.
From 2.4 and 2.14–2.16, it is easy to know that
Tλ u 1 0,
Tλ u0 0,
Tλ u 0 0,
1
Tλ ut λ Gt, sf s, us, u s, u s ds
0
1
1 t2
sf s, us, u s, u s ds
λ
≥ t−
2
0 2
1
t2
≥ t−
max Gτ, sf s, us, u s, u s ds
λ
2
0 τ∈0,1
t2
t−
Tλ u
, ∀t ∈ 0, 1, u ∈ P \ {θ},
2
Tλ u t λ
1
G1 t, sf s, us, u s, u s ds
0
1
≥ 1 − tλ
2
1
sf s, us, u s, u s ds
0
1
1
max G1 τ, sf s, us, u s, u s ds
1 − tλ
2
0 τ∈0,1
1
1 − tTλ u , ∀t ∈ 0, 1, u ∈ P \ {θ},
2
1
−Tλ u t λ G2 t, sf s, us, u s, us ds
≥
0
≥ tλ
1
sf s, us, u s, us ds
0
≥ tλ
1
max G2 τ, sf s, us, u s, u s ds
0 τ∈0,1
tTλ u ,
∀t ∈ 0, 1, u ∈ P \ {θ}.
Therefore, TP \ {θ} ⊂ P follows from 2.21.
2.21
10
Boundary Value Problems
Obviously, u∗ is a positive solution of BVP 1.1 if and only if u∗ is a positive fixed
point of the integral operator Tλ in P .
Lemma 2.4. Suppose that (H) and 2.17 hold. Then for any R > r > 0, Tλ : PR \ Pr → P is
completely continuous.
Proof. First of all, notice that Tλ maps PR \ Pr into P by Lemma 2.3.
Next, we show that Tλ is bounded. In fact, for any u ∈ PR \ Pr , by 2.10 we can get
r
8
t2
t−
2
t2
t−
2
≤ ut ≤
R,
r
1 − t ≤ u t ≤ 1 − tR,
4
rt ≤ −u t ≤ R,
∀t ∈ 0, 1.
2.22
Choose positive numbers c1 ≤ min{N1 , r/8N1 }, c2 ≤ min{N2 , r/4N2 }, c3 ≥ max{N3−1 ,
N3−1 R}. This, together with H, 2.22, 2.16, and Lemma 2.1 yields that
Tλ u t λ
1
0
G2 t, sf s, us, u s, u s ds
u s
u s
≤λ
ds
, c2
1 − s, −1c3
c2 1 − s
−c3
0
1
β1 β2 α3 us
u s
s2
β3 u s
α1
α2
≤ λ sc1
c2
c3
f s, s − , 1 − s, −1 ds
c2 1 − s
−c3
2
c1 s − s2 /2
0
1
s2
α −β α −β β −α
≤ c1 1 1 c2 2 2 c3 3 3 Rβ1 β2 α3 λ sf s, s − , 1 − s, −1 ds
2
0
1
< ∞,
us
sf s, c1
c1 s − s2 /2
s2
s−
2
∀t ∈ 0, 1, u ∈ PR \ Pr .
2.23
Thus, Tλ is bounded on PR \ Pr .
Now we show that Tλ is a compact operator on PR \ Pr . By 2.23 and Ascoli-Arzela
theorem, it suffices to show that Tλ V is equicontinuous for arbitrary bounded subset V ⊂
PR \ Pr .
Since for each u ∈ V , 2.22 holds, we may choose still positive numbers c1 ≤ min{N1 ,
r/8N1 }, c2 ≤ min{N2 , r/4N2 }, c3 ≥ max{N3−1 , N3−1 R}. Then
Tλ u t λ
1
f s, us, u s, us ds
t
1 s2
≤ C0 f s, s − , 1 − s, −1 ds
2
t
: Ht,
t ∈ 0, 1,
2.24
Boundary Value Problems
11
α −β1 α2 −β2 β3 −α3 β1 β2 α3
c2
c3
R
.
where C0 λc1 1
1
0
Notice that
1 1 s2
Htdt C0
f s, s − , 1 − s, −1 ds dt
2
0 t
1 s s2
C0
f s, s − , 1 − s, −1 dt ds
2
0 0
1
s2
C0 sf s, s − , 1 − s, −1 ds < ∞.
2
0
2.25
Thus for any given t1 , t2 ∈ 0, 1 with t1 ≤ t2 and for any u ∈ V , we get
Tλ u t2 − Tλ u t1 ≤
t2
Tλ u tdt ≤
t1
t2
Htdt.
2.26
t1
From 2.25, 2.26, and the absolute continuity of integral function, it follows that Tλ V is
equicontinuous.
Therefore, Tλ V is relatively compact, that is, Tλ is a compact operator on PR \ Pr .
Finally, we show that Tλ is continuous on PR \ Pr . Suppose un , u ∈ PR \ Pr , n 1, 2, . . .
and un − u
2 → 0, n → ∞. Then un t → u t, un t → u t and un t → ut as
n → ∞ uniformly, with respect to t ∈ 0, 1. From H, choose still positive numbers c1 ≤
min{N1 , r/8N1 }, c2 ≤ min{N2 , r/4N2 }, c3 ≥ max{N3−1 , N3−1 R}. Then
t2
0≤
≤ C0 f t, t − , 1 − t, −1 ,
2
s2
0 ≤ G2 t, sfs, un s, un s, un s ≤ C0 sf s, s − , 1 − s, −1 ,
2
ft, un t, un t, un t
t ∈ 0, 1,
t ∈ 0, 1, s ∈ 0, 1.
2.27
By 2.17, we know that sfs, s − s2 /2, 1 − s, −1 is integrable on 0, 1. Thus, from the Lebesgue
dominated convergence theorem, it follows that
lim Tλ un − Tλ u
2 lim Tλ un − Tλ u n → ∞
n → ∞
≤ lim λ
n → ∞
1
0
sf s, un s, un s, un s − f s, us, u s, u s ds
1 λ s lim f s, un s, un s, un s − f s, us, u s, un s ds
0
2.28
n → ∞
0.
Thus, Tλ is continuous on PR \ Pr . Therefore, Tλ : PR \ Pr → P is completely continuous.
12
Boundary Value Problems
3. A Necessary and Sufficient Condition for Existence of
Positive Solutions
In this section, by using the fixed point theorem of cone, we establish the following necessary
and sufficient condition for the existence of positive solutions for BVP 1.1.
Theorem 3.1. Suppose (H) holds, then BVP 1.1 has at least one positive solution for any λ > 0 if
and only if the integral inequality 2.17 holds.
Proof. Suppose first that ut be a positive solution of BVP 1.1 for any fixed λ > 0. Then there
exist constants Ii i 1, 2, 3, 4 with 0 < Ii < 1 < Ii1 , i 1, 3 such that
I1
t2
t−
2
≤ ut ≤ I2
t2
t−
2
,
I3 1 − t ≤ u t ≤ I4 1 − t,
t ∈ 0, 1.
3.1
In fact, it follows from u4 t ≥ 0, t ∈ 0, 1 and u0 u 1 u 0 u1 0, that u t ≤ 0
for t ∈ 0, 1 and u t ≤ 0, u t ≥ 0 for t ∈ 0, 1. By the concavity of ut and u t, we have
ut ≥ tu1 1 − tu0 t
u
≥
t2
t−
2
u t ≥ tu 1 1 − tu 0 1 − tu ,
u
,
3.2
∀t ∈ 0, 1.
On the other hand,
ut 1
G0 t, s −u s ds 0
1
s −u s ds 0
t2 ≤ u t1 − tu 2
u t t
−u sds ≤ 1 − tu ,
1
t −u s ds
t
t2
t−
2
u ,
3.3
∀t ∈ 0, 1.
t
Let I1 min{
u
, 1/2}, let I2 I4 max{
u
, 2}, and let I3 min{
u , 1/2}, then 3.1
holds.
Boundary Value Problems
13
Choose positive numbers c1 ≤ N1 I2−1 , c2 ≤ N2 I4−1 , c3 ≥ max{N3−1 , N3−1 u
2 }. This,
together with H, 1.2, and 2.18 yields that
t2
f t, t − , 1 − t, −1
2
f
≤
c1α1
t − t2 /2
1−t 1 c3
t, c1
ut, c2
u t,
u t
c1 ut
c2 u t
c3 −u t
β1 β3
t − t2 /2
1 − t β2 1 α3
c3
α2
c2
f t, ut, u t, u t
c1 ut
c2 u t
c3
−u t
α3 β3
c3
≤
− f t, ut, u t, ut
u t
−β C∗ −u t 3 f t, ut, u t, u t , t ∈ 0, 1,
3.4
c1α1
1
c1 I1
β1
α −β1 α2 −β2 β3 −α3 −β1 −β2
c2
c3
I1 I3 .
where C∗ c1 1
1
λ
t
c2α2
1
c2 I3
β2 1
c3
Hence, integrating 3.4 from t to 1, we obtain
β3
s2
−u s f s, s − , 1 − s, −1 ds ≤ C∗ −u t ,
2
t ∈ 0, 1.
3.5
t ∈ 0, 1,
3.6
Since −u t increases on 0, 1, we get
β3
−u t λ
1 s2
f s, s − , 1 − s, −1 ds ≤ C∗ −u t ,
2
t
that is,
1 −u t
s2
λ f s, s − , 1 − s, −1 ds ≤ C∗
,
2
t
−u tβ3
t ∈ 0, 1.
3.7
Notice that β3 < 1, integrating 3.7 from 0 to 1, we have
1 1 −1 1−β3
s2
λ
−u 1
f s, s − , 1 − s, −1 ds dt ≤ C∗ 1 − β3
.
2
0 t
3.8
1 s −1 1−β3
s2
λ
−u 1
f s, s − , 1 − s, −1 dt ds ≤ C∗ 1 − β3
.
2
0 0
3.9
That is,
Thus,
1
0
s2
sf s, s − , 1 − s, −1 ds < ∞.
2
3.10
14
Boundary Value Problems
By an argument similar to the one used in deriving 3.5, we can obtain
1
λ
t
α3
s2
−u s f s, s − , 1 − s, −1 ds ≥ C∗ −u t ,
2
β −α1 β2 −α2 α3 −β3 −α1 −α2
c2
c3
I2 I4 .
where C∗ c1 1
t ∈ 0, 1,
3.11
So,
1 s2
3
−u t ,
λ f s, s − , 1 − s, −1 ds ≥ C∗ u
−α
2
2
t
t ∈ 0, 1.
3.12
Integrating 3.12 from 0 to 1, we have
1 1 s2
3
λ
f s, s − , 1 − s, −1 ds dt ≥ C∗ u
−α
−u 1 .
2
2
0 t
3.13
1 s s2
3
−u 1 .
λ
f s, s − , 1 − s, −1 dt ds ≥ C∗ u
−α
2
2
0 0
3.14
That is,
So,
1
0
s2
sf s, s − , 1 − s, −1 ds > 0.
2
3.15
This and 3.10 imply that 2.17 holds.
Now assume that 2.17 holds, we will show that BVP 1.1 has at least one positive
solution for any λ > 0. By 2.17, there exists a sufficient small δ > 0 such that
1−δ
δ
s2
sf s, s − , 1 − s, −1 ds > 0.
2
3.16
For any fixed λ > 0, first of all, we prove
Tλ u
2 ≥ u
2 ,
where 0 < r ≤ min{N1 , N2 , N3 , λδ1β3 2−3β1 β2 Let u ∈ ∂Pr , then
r
8
t2
t−
2
t2
≤ ut ≤ r t −
2
≤ N1
t2
t−
2
∀u ∈ ∂Pr ,
1−δ
δ
,
δr ≤ rt ≤ −u t ≤ r ≤ N3 ,
3.17
sfs, s − s2 /2, 1 − s, −1ds1/1−β1 β2 β3 }.
r
1 − t ≤ u t ≤ r1 − t ≤ N2 1 − t,
4
∀t ∈ δ, 1 − δ.
3.18
Boundary Value Problems
15
From Lemma 2.1, 3.18, and H, we get
1
Tλ u
2 Tλ u ≥ λ max
t∈δ,1−δ
≥ δλ
1−δ
δ
G2 t, sf s, us, u s, u s ds
0
us
sf s,
s − s2 /2
s2
s−
2
u s
,
1 − s, −1 −u s ds
1−s
1−δ β1 β2
β3
u s
us
s2
≥ δλ
−u s f s, s − , 1 − s, −1 ds
s
1−s
2
s − s2 /2
δ
1−δ β1 β
r
r 2
s2
β3
≥δ
sf s, s − , 1 − s, −1 ds
δr λ
8
4
2
δ
≥δ
1β3 −3β1 β2 β1 β2 β3
2
r
1−δ
λ
δ
≥ r u
2 ,
3.19
s2
sf s, s − , 1 − s, −1 ds
2
u ∈ ∂Pr .
Thus, 3.17 holds.
Next, we claim that
Tλ u
2 ≤ u
2 ,
∀u ∈ ∂PR ,
3.20
1/1−β1 β2 β3 α −β 1
where R ≥ max{8N1−1 , 4N2−1 , λN3 3 3 0 sfs, s − s2 /2, 1 − s, −1ds
}.
Let c N3 /R, then for u ∈ ∂PR , we get
N1−1
t2
t−
2
R
t2
t2
≤
t−
≤ ut ≤ R t −
,
8
2
2
N2−1 1 − t ≤
−cu t ≤ c
u
2 cR N3 ,
R
1 − t ≤ u t ≤ R1 − t,
4
∀t ∈ 0, 1.
Therefore, by Lemma 2.1 and H, it follows that
Tλ u t λ
1
G2 t, sf s, us, u s, u s ds
0
≤λ
1
0
us
sf s,
s − s2 /2
s2
s−
2
u s
1 ,
−cu s ds
1 − s, −1
1−s
c
3.21
16
Boundary Value Problems
1 β1 β2 β3
α3
us
u s
1
s2
−cu s f s, s − , 1 − s, −1 ds
≤λ
1−s
c
2
s − s2 /2
0
β1 β2
≤R
N3
R
β1 β2 β3
R
1
α3 −β3
α3
R λ
0
α3 −β3
N3 1
λ
0
≤ R u
2 ,
s2
sf s, s − , 1 − s, −1 ds
2
s2
sf s, s − , 1 − s, −1 ds
2
u ∈ ∂PR .
3.22
This implies that 3.20 holds.
By Lemmas 1.1 and 2.4, 3.17, and 3.20, we obtain that Tλ has a fixed point in PR \ Pr .
Therefore, BVP 1.1 has a positive solution in PR \ Pr for any λ > 0.
4. Unbounded Connected Branch of Positive Solutions
In this section, we study the global continua results under the hypotheses H and 2.17. Let
L {λ, u ∈ 0, ∞ × P \ {θ} : λ, u satisfies BVP 1.1},
4.1
then, by Theorem 3.1, L ∩ {λ} × P / ∅ for any λ > 0.
Theorem 4.1. Suppose (H) and 2.17 hold, then the closure L of positive solution set possesses an
unbounded connected branch C which comes from 0, θ such that
i for any λ > 0, C ∩ {λ} × P /
∅, and
ii limλ,uλ ∈C,λ → 0 uλ 2 0, limλ,uλ ∈C,λ → ∞ uλ 2 ∞.
Proof. We now prove our conclusion by the following several steps.
First, we prove that for arbitrarily given 0 < λ1 < λ2 < ∞, L∩λ1 , λ2 ×P is bounded.
In fact, let
⎧
1/1−β1 β2 β3 ⎫
1 ⎨
⎬
2
s
α −β
R 2 max 8N1−1 , 4N2−1 , λ2 N3 3 3
sf s, s − , 1 − s, −1 ds
,
⎩
⎭
2
0
4.2
then for u ∈ P \ {θ} and u
2 ≥ R, we get
N1−1
t2
t−
2
N2−1 1 − t ≤
R
≤
8
t2
t−
2
≤ ut ≤
t2
t−
2
R
1 − t ≤ u t ≤ 1 − t
u
2 ,
4
u
2 ,
∀t ∈ 0, 1.
4.3
Boundary Value Problems
17
Therefore, by Lemma 2.1 and H, it follows that
Tλ u
2 ≤ Tλ2 u
2 ≤ λ2
≤ λ2
1
0
≤
1
sf s, us, u s, u s ds
0
us
sf s,
s − s2 /2
β β
λ2 u
21 2
N3
u
2
α3 −β3
β β β
λ2 u
21 2 3 N3 α3 −β3
< u
2 ,
s2
s−
2
u
α2 3
1
0
u
2 N3 u s
−u s ds
,
1 − s, −1
1−s
N3 u
2
1
0
s2
sf s, s − , 1 − s, −1 ds
2
4.4
s2
sf s, s − , 1 − s, −1 ds
2
∀λ ∈ λ1 , λ2 .
Let
⎧
1/1−β1 β2 β3 ⎫
1−δ ⎬
⎨
2
s
1
,
sf s, s − , 1 − s, −1 ds
r min N1 , N2 , N3 , λ1 δ1β3 2−3β1 β2 ⎭
⎩
2
2
δ
4.5
where δ is given by 3.16. Then for u ∈ P \ {θ} and u
2 ≤ r, we get
u
2
t2
t2
t2
t−
≤ ut ≤ r t −
≤ N1 t −
;
8
2
2
2
u
2
1 − t ≤ u t ≤ r1 − t ≤ N2 1 − t,
4
δ
u
2 ≤ t
u
2 ≤ −u t ≤ r ≤ N3 ,
∀t ∈ δ, 1 − δ.
Therefore, by Lemma 2.1 and H, it follows that
Tλ u
≥ Tλ1 u
≥ λ1 max
t∈δ,1−δ
≥ δλ1
1−δ
δ
1
G2 t, sf s, us, u s, us ds
0
us
sf s,
s − s2 /2
s2
s−
2
u s
,
1 − s, −1 −u s ds
1−s
1−δ β1 β2
β3
u s
us
s2
≥ δλ1
−u s f s, s − , 1 − s, −1 ds
s
1−s
2
s − s2 /2
δ
4.6
18
u
2 ≥δ
8
≥δ
β1 1β3 −3β1 β2 2
u
2
4
β2
β3
δ
u
2 λ1
β β β
u
21 2 3 λ1
1−δ
δ
Boundary Value Problems
1−δ
sf
s, s −
δ
s2
, 1 − s, −1 ds
2
s2
sf s, s − , 1 − s, −1 ds
2
u ∈ ∂Pr .
> u
2 ,
4.7
Therefore, u Tλ u has no positive solution in λ1 , λ2 × P \ PR ∪ λ1 , λ2 × Pr . As a
consequence, L ∩ λ1 , λ2 × P is bounded.
By the complete continuity of Tλ , L ∩ λ1 , λ2 × P is compact.
∞
Second, we choose sequences {an }∞
n1 and {bn }n1 satisfy
0 < · · · < an < · · · < a1 < b1 < · · · < bn < · · ·,
lim an 0,
n → ∞
4.8
lim bn ∞.
n → ∞
We are to prove that for any positive integer n, there exists a connected branch Cn of L
satisfying
Cn ∩ {an } × P / ∅,
Cn ∩ {bn } × P / ∅.
4.9
Let n be fixed, suppose that for any bn , u ∈ L ∩ {bn } × P , the connected branch Cu of
L ∩ an , bn × P , passing through bn , u, leads to Cu ∩ {an } × P ∅. Since Cu is compact,
there exists a bounded open subset Ω1 of an , bn × P such that Cu ⊂ Ω1 , Ω1 ∩ {an } × P ∅,
and Ω1 ∩ an , bn × {θ} ∅, where Ω1 and later ∂Ω1 denote the closure and boundary of Ω1
with respect to an , bn × P . If L ∩ ∂Ω1 /
∅, then Cu and L ∩ ∂Ω1 are two disjoint closed subsets
of L ∩ Ω1 . Since L ∩ Ω1 is a compact metric space, there are two disjoint compact subsets M1
and M2 of L ∩ Ω1 such that L ∩ Ω1 M1 ∪ M2 , Cu ⊂ M1 , and L ∩ ∂Ω1 ⊂ M2 . Evidently,
γ : distM1 , M2 > 0. Denoting by V the γ /3-neighborhood of M1 and letting Ωu Ω1 ∩ V ,
then it follows that
Cu ⊂ Ωu ,
Ωu ∩ {an } × P ∪ an , bn × {θ} ∅,
L ∩ ∂Ωu ∅.
4.10
If L ∩ ∂Ω1 ∅, then taking Ωu Ω1 .
It is obvious that in {bn } × P , the family of {Ωu ∩ {bn } × P : bn , u ∈ L} makes up
an open covering of L ∩ {bn } × P . Since L ∩ {bn } × P is a compact set, there exists a finite
subfamily {Ωui ∩ {bn } × P : bn , ui ∈ L}ki1 which also covers L ∩ {bn } × P . Let Ω ki1 Ωui ,
then
L ∩ {bn } × P ⊂ Ω,
Ω ∩ {an } × P ∪ an , bn × {θ} ∅,
L ∩ ∂Ω ∅.
4.11
Boundary Value Problems
19
Hence, by the homotopy invariance of the fixed point index, we obtain
iTbn , Ω ∩ {bn } × P , P iTan , Ω ∩ {an } × P , P 0.
4.12
By the first step of this proof, the construction of Ω, 4.4, and 4.7, it follows easily that there
exist 0 < rn < Rn such that
Ω ∩ {bn } × P ∩ {bn } × Prn ∅,
Ω ∩ {bn } × P ⊂ {bn } × PRn ,
4.13
iTbn , Prn , P 0,
4.14
iTbn , PRn , P 1.
4.15
However, by the excision property and additivity of the fixed point index, we have
from 4.12 and 4.14 that iTbn , PRn , P 0, which contradicts 4.15. Hence, there exists
some bn , u ∈ L ∩ {bn } × P such that the connected branch Cu of L ∩ an , bn × P containing
∅. Let Cn be the connected branch of L including Cu , then
bn , u satisfies that Cu ∩ {an } × P /
this Cn satisfies 4.9.
By Lemma 1.2, there exists a connected branch C∗ of lim supn → ∞ Cn such that C∗ ∩
{λ} × P / ∅ for any λ > 0. Noticing lim supn → ∞ Cn ⊂ L, we have C∗ ⊂ L. Let C be the
connected branch of L including C∗ , then C ∩ {λ} × P / ∅ for any λ > 0. Similar to 4.4 and
4.7, for any λ > 0, λ, uλ ∈ C, we have, by H, 4.2, 4.3, 4.5, 4.6, and Lemma 2.1,
1
uλ 2 Tλ uλ 2 ≤ λ
≤
β β
λ
uλ 21 2
0
N3
uλ 2
β β2 β3
λ
uλ 21
β1 β2 β3
≤ λR
sf s, uλ s, uλ s, uλ s ds
α3 −β3
N3 α3 −β3
α
uλ 2 3
s2
sf s, s − , 1 − s, −1 ds
2
N3 1
sf
0
uλ 2 Tλ uλ 2 ≥ λ max
t∈δ,1−δ
1
0
0
s2
sf s, s − , 1 − s, −1 ds
2
1
0
α3 −β3
1
s2
s, s − , 1 − s, −1 ds,
2
G2 t, sf s, uλ s, uλ s, uλ s ds
s2
sf s, s − , 1 − s, −1 ds
δ
uλ 2 2
δ
1−δ s2
β1 β2 β3
1β3 −3β1 β2 ≥ λδ
2
sf s, s − , 1 − s, −1 ds
uλ 2
2
δ
1−δ s2
1β3 −3β1 β2 β1 β2 β3
≥ λδ
2
r
sf s, s − , 1 − s, −1 ds,
2
δ
uλ 2
≥ λδ
8
β1 uλ 2
4
4.16
β2
β3
1−δ
4.17
20
Boundary Value Problems
where δ is given by 3.16. Let λ → 0 in 4.16 and λ → ∞ in 4.17, we have
lim
λ,uλ ∈C,λ → 0
uλ 2 0,
lim
uλ 2 ∞.
λ,uλ ∈C,λ → ∞
4.18
Therefore, Theorem 4.1 holds and the proof is complete.
Acknowledgments
This work is carried out while the author is visiting the University of New England. The
author thanks Professor Yihong Du for his valuable advices and the Department of Mathematics for providing research facilities. The author also thanks the anonymous referees
for their careful reading of the first draft of the manuscript and making many valuable
suggestions. Research is supported by the NSFC 10871120 and HESTPSP J09LA08.
References
1 R. P. Agarwal and D. O’Regan, “Nonlinear superlinear singular and nonsingular second order
boundary value problems,” Journal of Differential Equations, vol. 143, no. 1, pp. 60–95, 1998.
2 L. Liu, P. Kang, Y. Wu, and B. Wiwatanapataphee, “Positive solutions of singular boundary value
problems for systems of nonlinear fourth order differential equations,” Nonlinear Analysis: Theory,
Methods & Applications, vol. 68, no. 3, pp. 485–498, 2008.
3 D. O’Regan, Theory of Singular Boundary Value Problems, World Scientific, River Edge, NJ, USA, 1994.
4 Y. Zhang, “Positive solutions of singular sublinear Emden-Fowler boundary value problems,” Journal
of Mathematical Analysis and Applications, vol. 185, no. 1, pp. 215–222, 1994.
5 Z. Wei, “Existence of positive solutions for 2nth-order singular sublinear boundary value problems,”
Journal of Mathematical Analysis and Applications, vol. 306, no. 2, pp. 619–636, 2005.
6 Z. Wei and C. Pang, “The method of lower and upper solutions for fourth order singular m-point
boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 322, no. 2, pp. 675–
692, 2006.
7 A. R. Aftabizadeh, “Existence and uniqueness theorems for fourth-order boundary value problems,”
Journal of Mathematical Analysis and Applications, vol. 116, no. 2, pp. 415–426, 1986.
8 R. P. Agarwal, “On fourth order boundary value problems arising in beam analysis,” Differential and
Integral Equations, vol. 2, no. 1, pp. 91–110, 1989.
9 Z. Bai, “The method of lower and upper solutions for a bending of an elastic beam equation,” Journal
of Mathematical Analysis and Applications, vol. 248, no. 1, pp. 195–202, 2000.
10 D. Franco, D. O’Regan, and J. Perán, “Fourth-order problems with nonlinear boundary conditions,”
Journal of Computational and Applied Mathematics, vol. 174, no. 2, pp. 315–327, 2005.
11 C. P. Gupta, “Existence and uniqueness theorems for the bending of an elastic beam equation,”
Applicable Analysis, vol. 26, no. 4, pp. 289–304, 1988.
12 Y. Li, “On the existence of positive solutions for the bending elastic beam equations,” Applied
Mathematics and Computation, vol. 189, no. 1, pp. 821–827, 2007.
13 Q. Yao, “Positive solutions of a nonlinear elastic beam equation rigidly fastened on the left and simply
supported on the right,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 5-6, pp. 1570–
1580, 2008.
14 Q. Yao, “Existence and multiplicity of positive solutions to a singular elastic beam equation rigidly
fixed at both ends,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2683–2694,
2008.
15 Q. Yao, “Monotonically iterative method of nonlinear cantilever beam equations,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 432–437, 2008.
16 Q. Yao, “Solvability of singular cantilever beam equation,” Annals of Differential Equations, vol. 24, no.
1, pp. 93–99, 2008.
Boundary Value Problems
21
17 Q. L. Yao, “Positive solution to a singular equation for a beam which is simply supported at left and
clamped at right by sliding clamps,” Journal of Yunnan University. Natural Sciences, vol. 31, no. 2, pp.
109–113, 2009.
18 Q. L. Yao, “Existence and multiplicity of positive solutions to a class of nonlinear cantilever beam
equations,” Journal of Systems Science & Mathematical Sciences, vol. 29, no. 1, pp. 63–69, 2009.
19 Q. L. Yao, “Positive solutions to a class of singular elastic beam equations rigidly fixed at both ends,”
Journal of Wuhan University. Natural Science Edition, vol. 55, no. 2, pp. 129–133, 2009.
20 Q. Yao, “Existence of solution to a singular beam equation fixed at left and clamped at right by sliding
clamps,” Journal of Natural Science. Nanjing Normal University, vol. 9, no. 1, pp. 1–5, 2007.
21 J. R. Graef and L. Kong, “A necessary and sufficient condition for existence of positive solutions of
nonlinear boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no.
11, pp. 2389–2412, 2007.
22 Y. Xu, L. Li, and L. Debnath, “A necessary and sufficient condition for the existence of positive
solutions of singular boundary value problems,” Applied Mathematics Letters, vol. 18, no. 8, pp. 881–
889, 2005.
23 J. Zhao and W. Ge, “A necessary and sufficient condition for the existence of positive solutions
to a kind of singular three-point boundary value problem,” Nonlinear Analysis: Theory, Methods &
Applications, vol. 71, no. 9, pp. 3973–3980, 2009.
24 Z. Q. Zhao, “Positive solutions of boundary value problems for nonlinear singular differential
equations,” Acta Mathematica Sinica, vol. 43, no. 1, pp. 179–188, 2000.
25 Z. Zhao, “On the existence of positive solutions for 2n-order singular boundary value problems,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2553–2561, 2006.
26 R. Ma and Y. An, “Global structure of positive solutions for nonlocal boundary value problems
involving integral conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp.
4364–4376, 2009.
27 R. Ma and J. Xu, “Bifurcation from interval and positive solutions of a nonlinear fourth-order
boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 1, pp. 113–
122, 2010.
28 R. Y. Ma and B. Thompson, “Nodal solutions for a nonlinear fourth-order eigenvalue problem,” Acta
Mathematica Sinica, vol. 24, no. 1, pp. 27–34, 2008.
29 E. Dancer, “Global solutions branches for positive maps,” Archive for Rational Mechanics and Analysis,
vol. 55, pp. 207–213, 1974.
30 P. H. Rabinowitz, “Some aspects of nonlinear eigenvalue problems,” The Rocky Mountain Journal of
Mathematics, vol. 3, no. 2, pp. 161–202, 1973.
31 D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in
Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.
32 J. X. Sun, “A theorem in point set topology,” Journal of Systems Science & Mathematical Sciences, vol. 7,
no. 2, pp. 148–150, 1987.