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,
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 o f positive solution set possesses an unbounded connected branch which bifurcates from
0,θ. Our nonlinearity ft, u, v, w may be singular at u,v, t 0and/ort 1.
1. Introduction
Singular differential equations arise in the fields of gas dynamics, Newtonian fluid mechan-
ics, 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:
u
4
t
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 α
i
, β
i
,
N
i
, i 1, 2, 3 −∞ <α
1
≤ 0 ≤ β
1
< ∞, −∞ <α
2
≤ 0 ≤ β
2
< ∞, 0 ≤ α
t, u, v, w
, ∀0 <c≤ N
1
,
c
β
2
f
t, u, v, w
≤ f
t, u, cv, w
≤ c
α
2
f
t, u, v, w
, ∀0 <c≤ N
2
,
c
β
3
f
2
j1
m
3
k1
p
i,j,k
t
u
r
i
v
s
j
w
σ
k
,
1.3
where p
i,j,k
t ∈ C0, 1, 0, ∞, r
i
,s
j
∈ R,0 ≤ σ
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 ∈ C
2
0, 1 ∩ C
4
0, 1 is called
a positive solution of the BVP 1.1 if it satisfies the BVP 1.1ut > 0, −u
t > 0for
t ∈ 0, 1 and u
t > 0fort ∈ 0, 1.Forsomeλ ∈ 0, ∞,iftheBVP1.1 has a positive
solution u,thenλ 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:
u
4
t
f
t, u
t
,u
where E ⊂ 0, 1 is a closed subset and mesE 0, f ∈ C0, 1\E×0, ∞×0, ∞, 0, ∞.
In 15,heobtainedasufficient 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:
u
4
t
f
t, u
t
,t∈
0, 1
,
u
0
u
0
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 or ganized 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,andletΩ
1
, Ω
2
be bounded
open sets of E, θ ∈ Ω
1
⊂ Ω
1
⊂ Ω
2
. Suppose that T : K ∩ Ω
n
}
∞
n1
satisfy
a<···<a
n
< ···<a
1
<b
1
< ···<b
n
< ···<b,
lim
n → ∞
a
n
a, lim
n → ∞
b
n
b.
1.6
Suppose also that
{C
n
: n 1, 2, } is a family of connected subsets of R
1
C ∩
{
λ
}
× M
/
∅, ∀λ ∈
a, b
, 1.7
where lim sup
n → ∞
C
n
{x ∈ M: there exists a sequence x
n
i
∈ C
n
i
such that x
n
i
→ x, i →∞}.
2. Some Preliminaries and Lemmas
Let E {u ∈ C
2
0, 1 : u00,u
,u
t
≥
1
2
1 − t
u
, −u
t
≥ t
u
Let
G
0
t, s
⎧
⎨
⎩
s, 0 ≤ s ≤ t ≤ 1,
t, 0 ≤ t ≤ s ≤ 1,
G
t, s
1
0
G
0
t, r
G
0
r, s
1
0,
G
t, s
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
s
3
3
s
t
2
− s
2
2
t, s
⎧
⎪
⎪
⎨
⎪
⎪
⎩
s
1 − t
, 0 ≤ s ≤ t ≤ 1,
s
2
2
−
t
2
2
s
1 − s
, 0 ≤ t ≤ s ≤ 1,
G
2
t, s ≤ t or s, for all t, s ∈ 0, 1.
3 max
t∈0,1
Gt, s ≤ 1/2s, max
t∈0,1
G
i
t, s ≤ s, i 1, 2, for all s ∈ 0, 1.
4 Gt, s ≥ s/2t − t
2
/2, G
1
t, s ≥ s/21 − t, G
2
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,wehave
G
t, s
s
3
3
st
2
2
−
≤ t
or s
.
2.5
For 0 ≤ t ≤ s ≤ 1, by 2.4,wehave
G
t, s
t
3
3
−
t
3
2
ts −
ts
2
2
≤ st −
st
2
2
s
or s
.
2.6
Consequently, property 2 holds.
From property 2, it is easy to obtain property 3.
We next show that p roperty 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
G
t, s
s
t −
t
2
2
−
s
2
6
1
2
t −
t
2
>
1
2
t −
t
2
2
,
G
1
t, s
s
1 − t
≥
1
2
1 − t
,G
2
≥
1
2
t −
t
2
3
≥
1
2
t −
t
2
2
,
G
1
t, s
s
≥ 1 −
t
2
−
s
u
≤
u
,
1
2
u
≤
u
≤
u
2
,
1
4
1 − t
u
2
≤ u
t
≤
1 − t
u
2
,
t
u
2
≤−u
s
ds
1
0
u
s
ds ≤
u
,
u
max
t∈0,1
t
0
u
u
,
u
max
t∈0,1
1
t
−u
s
ds
1
0
−u
s
s
ds ≥
1
0
s
u
ds
1
2
u
.
2.11
Therefore, 2.9 holds. From 2.9,weget
u
By 2.9 and the definition of P, we can obtain that
u
t
1
0
G
0
t, s
−u
s
ds ≤
t
0
sds
1
t
2
,
∀t ∈
0, 1
,
u
t
≥
t −
t
2
2
u
≥
1
8
t −
t
2
2
u
1 − t
u
2
,
u
t
≥
1
2
1 − t
u
≥
≤
u
u
2
, ∀t ∈
0, 1
.
2.13
Thus, 2.10 holds.
For any fixed λ ∈ 0, ∞, define an operator T
λ
by
T
λ
u
t
: λ
Then, it is easy to know that
T
λ
u
t
λ
1
0
G
1
t, s
f
s, u
s
,u
s
f
s, u
s
,u
s
,u
s
ds, ∀u ∈ P \
{
θ
}
.
2.16
Lemma 2.3. Suppose that (H)and
0 <
1
0
sf
, ∀c ≥ N
−1
1
,
c
α
2
f
t, u, v, w
≤ f
t, u, cv, w
≤ c
β
2
f
t, u, v, w
, ∀c ≥ N
−1
2
,
c
α
3
≤
min{N
2
, 1/4N
2
u
2
}, c
3
≥ max{N
−1
3
,N
−1
3
u
2
}. It follows from H, 2.10, Lemma 2.1,and
2.17 that
T
λ
u
t
λ
1
0
s, c
1
u
s
c
1
s − s
2
/2
s −
s
2
2
,c
2
u
s
c
2
1
u
s
c
1
s − s
2
/2
β
1
c
α
2
2
u
s
c
2
1 − s
2
λ
1
0
sc
α
1
1
u
2
c
1
β
1
c
α
2
2
u
2
c
2
α
1
−β
1
1
c
α
2
−β
2
2
c
β
3
−α
3
3
u
β
1
β
2
α
3
2
λ
1
f
s, u
s
,u
s
,u
s
ds
≤ λ
1
0
sf
s, c
1
u
s
−1
c
3
u
s
−c
3
ds
≤ c
α
1
−β
1
1
c
α
2
−β
2
2
c
β
3
−α
u
t
λ
1
0
G
2
t, s
f
s, u
s
,u
s
,u
s
u
s
c
2
1 − s
1 − s
,
−1
c
3
u
s
−c
3
ds
≤ c
α
sf
s, s −
s
2
2
, 1 − s, −1
ds < ∞.
2.20
Thus, T
λ
is well defined on P \{θ}.
From 2.4 and 2.14–2.16, it is easy to know that
T
λ
u
0
0,
T
λ
u
1
s
,u
s
,u
s
ds
≥
t −
t
2
2
λ
1
0
1
2
sf
τ, s
f
s, u
s
,u
s
,u
s
ds
t −
t
2
2
T
f
s, u
s
,u
s
,u
s
ds
≥
1
2
1 − t
λ
1
0
sf
τ, s
f
s, u
s
,u
s
,u
s
ds
1
2
1 − t
G
2
t, s
f
s, u
s
,u
s
,u
s
ds
≥ tλ
1
0
sf
s, u
s
,u
s
,u
s
ds
t
T
λ
u
, ∀t ∈
0, 1
,u∈ P \
Next, we show that T
λ
is bounded. In fact, for any u ∈ P
R
\ P
r
,by2.10 we can get
r
8
t −
t
2
2
≤ u
t
≤
t −
t
2
2
R,
r
4
≤ min{N
2
, r/4N
2
}, c
3
≥ max{N
−1
3
,
N
−1
3
R}. This, together with H, 2.22, 2.16,andLemma 2.1 yields that
T
λ
u
t
λ
1
s, c
1
u
s
c
1
s − s
2
/2
s −
s
2
2
,c
2
u
s
c
2
1 − s
s
c
1
s − s
2
/2
β
1
c
α
2
2
u
s
c
2
1 − s
β
2
c
α
2
−β
2
2
c
β
3
−α
3
3
R
β
1
β
2
α
3
λ
1
0
sf
s, s −
s
2
2
, 1 − s, −1
P
R
\ P
r
.
Since for each u ∈ V , 2.22 holds, we may choose still positive numbers c
1
≤ min{N
1
,
r/8N
1
}, c
2
≤ min{N
2
, r/4N
2
}, c
3
≥ max{N
−1
3
,N
−1
3
R}.Then
ds
≤ C
0
1
t
f
s, s −
s
2
2
, 1 − s, −1
ds
: H
t
,t∈
0, 1
,
2.24
Boundary Value Problems 11
where C
0
λc
t
dt C
0
1
0
1
t
f
s, s −
s
2
2
, 1 − s, −1
ds dt
C
0
1
0
s
0
f
s, s −
T
λ
u
t
2
−
T
λ
u
t
1
≤
t
2
t
Therefore, T
λ
V is relatively compact, that is, T
λ
is a compact operator on P
R
\ P
r
.
Finally, we show that T
λ
is continuous on P
R
\ P
r
. Suppose u
n
,u∈ P
R
\ P
r
, n 1, 2,
and u
n
− u
2
→ 0, n → ∞.Thenu
n
t → u
R}.Then
0 ≤ f
t, u
n
t
,u
n
t
,u
n
t
≤ C
0
f
t, t −
t
2
2
, 1 − t, −1
0
sf
s, s −
s
2
2
, 1 − s, −1
,t∈
0, 1
,s∈
0, 1
.
2.27
By 2.17, we know that sfs, s−s
2
/2, 1−s, −1 is integrable on 0, 1.Thus,fromtheLebesgue
dominated convergence theorem, it follows that
lim
n → ∞
T
λ
u
n
λ
1
0
s
f
s, u
n
s
,u
n
s
,u
n
s
− f
s, u
s, u
n
s
,u
n
s
,u
n
s
− f
s, u
s
,u
s
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 I
i
i 1, 2, 3, 4 with 0 <I
i
< 1 <I
i1
, i 1, 3suchthat
I
1
t −
t
2
2
≤ u
t
≤ I
2
t −
t
2
10, that u
t ≤ 0
for t ∈ 0, 1 and u
t ≤ 0, u
t ≥ 0fort ∈ 0, 1. By the concavity of ut and u
t,wehave
u
t
≥ tu
1
1 − t
u
0
t
u
1 − t
u
, ∀t ∈
0, 1
.
3.2
On the other hand,
u
t
1
0
G
0
t, s
ds
≤
t
2
2
u
t
1 − t
u
t −
t
2
2
, ∀t ∈
0, 1
.
3.3
Let I
1
min{u, 1/2}, let I
2
I
4
max{u
, 2}, and let I
3
min{u
, 1/2},then3.1
holds.
Boundary Value Problems 13
Choose positive numbers c
1
≤ N
1
I
−1
2
, c
2
2
/2
c
1
u
t
u
t
,c
2
1 − t
c
2
u
t
u
t
,
1
c
1
c
α
2
2
1 − t
c
2
u
t
β
2
1
c
3
α
3
c
3
−u
I
1
β
1
c
α
2
2
1
c
2
I
3
β
2
1
c
3
α
3
−
c
3
u
t
−β
3
f
t, u
t
,u
t
,u
t
,t∈
0, 1
,
3.4
where C
∗
t
−u
s
β
3
f
s, s −
s
2
2
, 1 − s, −1
ds ≤ C
∗
−u
t
,t∈
0, 1
t
,t∈
0, 1
, 3.6
that is,
λ
1
t
f
s, s −
s
2
2
, 1 − s, −1
ds ≤ C
∗
−u
t
ds dt ≤ C
∗
1 − β
3
−1
−u
1
1−β
3
. 3.8
That is,
λ
1
0
s
0
f
s, s −
s
2
, 1 − s, −1
ds < ∞. 3.10
14 Boundary Value Problems
By an argument similar to the one used in deriving 3.5,wecanobtain
λ
1
t
−u
s
α
3
f
s, s −
s
2
2
, 1 − s, −1
ds ≥ C
∗
3
I
−α
1
2
I
−α
2
4
.So,
λ
1
t
f
s, s −
s
2
2
, 1 − s, −1
ds ≥ C
∗
u
−α
3
2
u
−α
3
2
−u
1
. 3.13
That is,
λ
1
0
s
0
f
s, s −
s
2
2
, 1 − s, −1
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,thereexistsasufficient small δ>0suchthat
1−δ
δ
sf
s, s −
s
2
2
, 1 − s, −1
ds > 0. 3.16
For any fixed λ>0, first of all, we prove
T
λ
u
2
≥
u
2
, ∀u ∈ ∂P
r
, 3.17
where 0 <r≤ min{N
,then
r
8
t −
t
2
2
≤ u
t
≤ r
t −
t
2
2
≤ N
1
t −
t
2
2
,
r
.
3.18
Boundary Value Problems 15
From Lemma 2.1, 3.18,andH,weget
T
λ
u
2
T
λ
u
≥ λ max
t∈δ,1−δ
1
0
G
2
t, s
s − s
2
/2
s −
s
2
2
,
u
s
1 − s
1 − s
,
−1
−u
s
s
β
3
f
s, s −
s
2
2
, 1 − s, −1
ds
≥ δ
r
8
β
1
r
4
β
2
2
β
3
λ
1−δ
δ
sf
s, s −
s
2
2
, 1 − s, −1
ds
≥ r
u
2
,u∈ ∂P
r
.
3.19
Thus, 3.17 holds.
Next, we claim that
T
λ
1
β
2
β
3
}.
Let c N
3
/R,thenforu ∈ ∂P
R
,weget
N
−1
1
t −
t
2
2
≤
R
8
t −
t
2
2
1 − t
,
−cu
t
≤ c
u
2
cR N
3
, ∀t ∈
0, 1
.
3.21
Therefore, by Lemma 2.1 and H, it follows that
T
λ
u
ds
≤ λ
1
0
sf
s,
u
s
s − s
2
/2
s −
s
2
2
,
u
s
1 − s
2
/2
β
1
u
s
1 − s
β
2
1
c
β
3
−cu
s
α
3
sf
s, s −
s
2
2
, 1 − s, −1
ds
R
β
1
β
2
β
3
N
3
α
3
−β
3
λ
1
0
sf
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)and2.17 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
/
R 2max
⎧
⎨
⎩
8N
−1
1
, 4N
−1
2
,
λ
2
N
α
3
−β
3
3
1
0
sf
s, s −
s
2
2
, 1 − s, −1
t −
t
2
2
≤ u
t
≤
t −
t
2
2
u
2
,
N
−1
2
1 − t
≤
R
4
T
λ
2
u
2
≤ λ
2
1
0
sf
s, u
s
,u
s
,u
s
ds
,
−1
u
2
N
3
N
3
u
2
−u
s
ds
≤ λ
2
u
s
2
2
, 1 − s, −1
ds
λ
2
u
β
1
β
2
β
3
2
N
3
α
3
−β
3
1
0
sf
N
1
,N
2
,N
3
,
λ
1
δ
1β
3
2
−3β
1
β
2
1−δ
δ
sf
s, s −
s
2
2
, 1 − s, −1
t
≤ r
t −
t
2
2
≤N
1
t −
t
2
2
;
u
2
4
1 − t
≤ u
t
δ, 1 − δ
.
4.6
Therefore, by Lemma 2.1 and H, it follows that
T
λ
u
≥
T
λ
1
u
≥ λ
1
max
t∈δ,1−δ
1
0
G
2
t, s
f
2
/2
s −
s
2
2
,
u
s
1 − s
1 − s
,
−1
−u
s
s
β
3
f
s, s −
s
2
2
, 1 − s, −1
ds
18 Boundary Value Problems
≥ δ
u
2
8
β
1
≥ δ
1β
3
2
−3β
1
β
2
u
β
1
β
2
β
3
2
λ
1
1−δ
δ
sf
s, s −
s
2
2
2
× P is bounded.
By the complete continuity of T
λ
, L ∩ λ
1
,λ
2
× P is compact.
Second, we choose sequences {a
n
}
∞
n1
and {b
n
}
∞
n1
satisfy
0 < ···<a
n
< ···<a
1
<b
1
< ···<b
n
< ···,
lim
× P
/
∅. 4.9
Let n be fixed, suppose that for any b
n
,u ∈ L ∩ {b
n
}×P, the connected branch C
u
of
L ∩ a
n
,b
n
× P, passing through b
n
,u,leadstoC
u
∩ {a
n
}×P∅.SinceC
u
is compact,
there exists a bounded open subset Ω
1
of a
n
,b
n
u
and L ∩ ∂Ω
1
are two disjoint closed subsets
of L ∩
Ω
1
.SinceL ∩ Ω
1
is a compact metric space, there are two disjoint compact subsets M
1
and M
2
of L ∩ Ω
1
such that L ∩ Ω
1
M
1
∪ M
2
, C
u
⊂ M
1
,andL ∩ ∂Ω
1
⊂ M
2
.Evidently,
,b
n
×
{
θ
}
∅,L∩ ∂Ω
u
∅.
4.10
If L ∩ ∂Ω
1
∅, then taking Ω
u
Ω
1
.
It is obvious that in {b
n
}×P, the family of {Ω
u
∩ {b
n
}×P : b
n
,u ∈ L} makes up
an open covering of L ∩ {b
n
}×P.SinceL ∩ {b
× P
⊂ Ω,
Ω ∩
{
a
n
}
× P
∪
a
n
,b
n
×
{
θ
}
∅,L∩ ∂Ω∅.
4.11
Boundary Value Problems 19
Hence, by the homotopy invariance of the fixed point index, we obtain
i
T
b
n
Ω ∩
{
b
n
}
× P
∩
{
b
n
}
× P
r
n
∅,
Ω ∩
{
b
n
}
× P
⊂
{
However, by the excision property and additivity of the fixed point index, we have
from 4.12 and 4.14 that iT
b
n
,P
R
n
,P0, which contradicts 4.15.Hence,thereexists
some b
n
,u ∈ L ∩ {b
n
}×P such that the connected branch C
u
of L ∩ a
n
,b
n
× P containing
b
n
,u satisfies that C
u
∩{a
n
}×P
/
∅.LetC
n
be the connected branch of L including C
∈ C,wehave,byH, 4.2, 4.3, 4.5, 4.6,andLemma 2.1,
u
λ
2
T
λ
u
λ
2
≤ λ
1
0
sf
s, u
λ
s
,u
λ
s
−β
3
u
λ
α
3
2
1
0
sf
s, s −
s
2
2
, 1 − s, −1
ds
λ
u
λ
β
1
β
2
N
3
α
3
−β
3
1
0
sf
s, s −
s
2
2
, 1 − s, −1
ds,
4.16
u
λ
2
T
λ
s
ds
≥ λδ
u
λ
2
8
β
1
u
λ
2
4
β
2
δ
u
β
1
β
2
β
3
2
1−δ
δ
sf
s, s −
s
2
2
, 1 − s, −1
ds
≥ λδ
1β
3
2
−3β
1
β
2
r
λ
2
0, lim
λ,u
λ
∈C,λ → ∞
u
λ
2
∞.
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 Math-
ematics 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.
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