Volume 2011, Article ID 594128,21pages 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,[email protected]
Received 16 October 2010; Revised 22 December 2010; Accepted 27 January 2011
Academic Editor: Kanishka Perera
Copyrightq2011 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 nonlinearityft, u, v, wmay be singular atu,v,t0 and/ort1.
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 yearssee1–4and references therein.
This paper investigates the following fourth-order nonlinear singular eigenvalue problem:
u4t λft, ut, ut, ut, t∈0,1,
u0 u1 u0 u1 0,
1.1
whereλ∈0, ∞is a parameter andfsatisfies the following hypothesis:
Hf ∈C0,1×0, ∞×0, ∞×−∞,0,0, ∞, and there exist constantsαi,βi,
3
i1βi < 1; 0 < Ni ≤ 1, i 1,2,3such that for anyt ∈ 0,1,u, v ∈ 0, ∞,
w∈−∞,0, fsatisfies
cβ1ft, u, v, w≤ft, cu, v, w≤cα1ft, u, v, w, ∀0< c≤N
1,
cβ2ft, u, v, w≤ft, u, cv, w≤cα2ft, u, v, w, ∀0< c≤N
2,
cβ3ft, u, v, w≤ft, u, v, cw≤cα3ft, u, v, w, ∀0< c≤N
3.
1.2
Typical functions that satisfy the above sublinear hypothesisHare those taking the form
ft, u, v, w
m1
i1
m2
j1
m3
k1
pi,j,kturivsjwσk, 1.3
where pi,j,kt ∈ C0,1,0, ∞,ri, sj ∈ R, 0 ≤ σk < 1, max{ri,0} max{sj} σk < 1,
i1,2, . . . , m1,j1,2, . . . , m2,k1,2, . . . , m3. The hypothesisHis similar to that in5,6.
Because of the extensive applications in mechanics and engineering, nonlinear fourth-order two-point boundary value problems have received wide attentions see 7–12 and references therein. In mechanics, the boundary value problem1.1 BVP 1.1for short describes the deformation of an elastic beam simply supported at left and clamped at right by sliding clamps. The termuinfrepresents bending effect which is useful for the stability analysis of the beam. BVP1.1has two special features. The first one is that the nonlinearity
fmay depend on the first-order derivative of the unknown functionu, and the second one is that the nonlinearityft, u, v, wmay be singular atu, v, t0 and/ort1.
In this paper, we study the existence of positive solutions and the structure of positive solution set for the BVP1.1. 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 from0, θ. Our analysis mainly relies on the fixed point theorem in a cone and the fixed point index theory.
By singularity off, we mean that the functionf in1.1is allowed to be unbounded at the pointsu 0,v 0,t 0, and/ort 1. A functionut ∈C20,1∩C40,1is called
a positive solution of the BVP1.1if it satisfies the BVP1.1 ut > 0, −ut > 0 for
t ∈ 0,1andut > 0 fort ∈ 0,1. For some λ ∈ 0, ∞, if the BVP1.1has a positive solutionu, thenλis called an eigenvalue anduis called corresponding eigenfunction of the BVP1.1.
The existence of positive solutions of BVPs has been studied by several authors in the literature; for example, see7–20 and the references therein. Yao15, 18studied the following BVP:
u4t ft, ut, ut, t∈0,1\E,
u0 u0 u1 u1 0,
1.4
by using the monotonically iterative technique. In13,18, he applied Guo-Krasnosel’skii’s fixed point theorem to obtain the existence and multiplicity of positive solutions of BVP1.4
and the following BVP:
u4t ft, ut, t∈0,1,
u0 u0 u1 u1 0.
1.5
These differ from our problem becauseft, u, vin1.4cannot be singular atu 0,v 0 and the nonlinearityfin1.5does 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 BVP1.1for anyλ > 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, in5,6,21–23. In5,
6,22,23they considered the case thatf depends on even order derivatives ofu. Although the nonlinearity f in 21 depends on the first-order derivative, where the nonlinearity f
is increasing with respect to the unknown functionu. Papers24,25derived the existence of positive solutions of BVPs by the lower and upper solution method, but the nonlinearity
ft, udoes not depend on the derivatives of the unknown functions, andft, uis decreasing with respect tou.
Recently, the global structure of positive solutions of nonlinear boundary value problems has also been investigated see26–28and references therein. Ma and An26
and Ma and Xu27 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 theoremssee29,30. The termsfuin
26andft, u, uin27are not singular att0,1,u0,u 0. Yao14obtained 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 nonlinearityft, u, v, win BVP1.1may be singular atu, v, t0 and/ort 1, the global bifurcation theorems in29,30do not apply to our problem here. InSection 4, we also investigate the global structure of positive solutions for BVP1.1by applying the followingLemma 1.2.
The paper is organized as follows: in the rest of this section, two known results are stated. InSection 2, some lemmas are stated and proved. InSection 3, we establish a necessary and sufficient condition for the existence of positive solutions. InSection 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 1.1see31. LetXbe a real Banach space, letKbe a cone inX, and letΩ1,Ω2be bounded open sets ofE,θ∈Ω1 ⊂Ω1⊂Ω2. Suppose thatT :K∩Ω2\Ω1 → Kis completely continuous such that one of the following two conditions is satisfied:
1Tx ≤ x,x∈K∩∂Ω1;Tx ≥ x,x∈K∩∂Ω2. 2Tx ≥ x,x∈K∩∂Ω1;Tx ≤ x,x∈K∩∂Ω2.
Lemma 1.2see32. LetMbe a metric space anda, b⊂R1. Let{a
n}∞n1and{bn}∞n1satisfy
a <· · ·< an<· · ·< a1< b1<· · ·< bn<· · ·< b,
lim
n→ ∞ana, nlim→ ∞bnb.
1.6
Suppose also that {Cn :n1,2, . . .}is a family of connected subsets ofR1×M, satisfying the
following conditions:
1Cn∩{an} ×M/∅andCn∩{bn} ×M/∅for eachn.
2For any two given numbersαandβwitha < α < β < b,∞n1Cn∩α, β×Mis a
relatively compact set ofR1×M.
Then there exists a connected branchCoflim supn→ ∞Cnsuch that
C∩{λ} ×M/∅, ∀λ∈a, b, 1.7
wherelim supn→ ∞Cn{x∈M:there exists a sequencexni ∈Cni such thatxni → x, i → ∞}.
2. Some Preliminaries and Lemmas
LetE {u ∈ C20,1 : u0 0, u1 0, u0 0},u
2 max{u,u,u}, then E, · 2is aBanachspace, whereumaxt∈0,1|ut|. Define
P
u∈E:ut≥
t−t 2
2
u, ut≥ 1
21−tu
, −ut≥tu, t∈0,1 . 2.1
It is easy to conclude thatPis a cone ofE. Denote
Pr {u∈P:u2< r}; ∂Pr {u∈P :u2r}. 2.2
Let
G0t, s
⎧ ⎨ ⎩
s, 0≤s≤t≤1,
t, 0≤t≤s≤1,
Gt, s
1
0
G0t, rG0r, sdr.
ThenGt, sis the Green function of homogeneous boundary value problem
u4t 0, t∈0,1,
u0 u1 u0 u1 0,
Gt, s
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
s3
3
st2−s2
2 st1−t, 0≤s≤t≤1,
t3
3
ts2−t2
2 ts1−s, 0≤t≤s≤1,
G1t, s :Gtt, s
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
s1−t, 0≤s≤t≤1,
s2
2 −
t2
2 s1−s, 0≤t≤s≤1,
G2t, s :−Gtt, s
⎧ ⎨ ⎩
s, 0≤s≤t≤1,
t, 0≤t≤s≤1.
2.4
Lemma 2.1. Gt, s,G1t, s, andG2t, shave the following properties: 1Gt, s>0,Git, s>0,i1,2, for allt, s∈0,1.
2Gt, s≤st−t2/2,G1t, s≤s1−t,G2t, s≤tors, for allt, s∈0,1. 3maxt∈0,1Gt, s≤1/2s,maxt∈0,1Git, s≤s,i1,2, for alls∈0,1.
4Gt, s≥s/2t−t2/2,G1t, s≥s/21−t,G2t, s≥st, for allt, s∈0,1.
Proof. From2.4, it is easy to obtain the property2.18.
We now prove that property2is true. For 0≤s≤t≤1, by2.4, we have
Gt, s s 3
3
st2
2 −
s3
2 st−st
2≤st−st2
2 s
t−t 2
2
,
G1t, s s1−t, G2t, s≤tors.
2.5
For 0≤t≤s≤1, by2.4, we have
Gt, s t 3
3 −
t3
2 ts−
ts2
2 ≤st−
st2
2 s
t−t 2
2
,
G1t, s s−t 2
2 −
s2
2 ≤s−tss1−t, G2t, s≤tors.
2.6
Consequently, property2holds.
From property2, it is easy to obtain property3.
For 0< s≤1, ifs≤t≤1, then
Gt, s
s t−
t2 2 − s2 6 1 2
t−t 2
2
t−t 2 2 − s2 3 ≥ 1 2
t−t 2
2
t−t 2 2 − t2 3 > 1 2
t−t 2
2
,
G1t, s
s 1−t≥
1
21−t, G2t, s≥st;
2.7
if 0≤t≤s, then
Gt, s
s ≥t−
t2 6 − ts 2 1 2
t−t 2
3 t−ts
≥ 1 2
t−t 2 3 ≥ 1 2
t−t 2
2
,
G1t, s
s ≥1−
t
2 −
s
2 ≥ 1
21−t, G2t, s≥st.
2.8
Therefore, property4holds.
Lemma 2.2. Assume thatu∈P\ {θ}, thenu2uand
1 4u
≤ u ≤u, 1 2u
≤u≤u. 2.9
1 8
t−t 2
2
u2≤ut≤
t−t 2
2
u2,
1
41−tu2≤u
t≤1−tu
2,
tu2≤ −ut≤ u2, ∀t∈0,1.
2.10
Proof. Assume thatu∈P\ {θ}, thenut≥0,−ut≥0,t∈0,1, so
umax
t∈0,1
t
0
usds
1
0
usds≤u,
u max
t∈0,1
t
0
usds
1
0
usds≥ 1
2u 1
0
1−sds 1
4u ,
u max
t∈0,1
1
t
−usds
1
0
−usds≤u,
u max
t∈0,1
1
t
−usds
1
0
−usds≥
1
0
suds 1
2u .
2.11
Therefore,2.9holds. From2.9, we get
u2max
By2.9and the definition ofP, we can obtain that
ut
1
0
G0t, s
−usds≤
t
0 sds
1
t
tds
u
t−t 2
2
u
t−t 2
2
u2,
∀t∈0,1,
ut≥
t−t 2
2
u ≥ 1
8
t−t 2
2
u2, ∀t∈0,1,
ut
1
t
−usds≤1−tu 1−tu2,
ut≥ 1
21−tu ≥ 1
41−tu2, ∀t∈0,1,
tu2tu≤ −ut≤uu2, ∀t∈0,1.
2.13
Thus,2.10holds.
For any fixedλ∈0, ∞, define an operatorTλby
Tλut :λ
1
0
Gt, sfs, us, us, usds, ∀u∈P\ {θ}. 2.14
Then, it is easy to know that
Tλut λ
1
0
G1t, sfs, us, us, usds, ∀u∈P\ {θ}, 2.15
Tλut −λ
1
0
G2t, sfs, us, us, usds, ∀u∈P\ {θ}. 2.16
Lemma 2.3. Suppose that (H) and
0<
1
0 sf
s, s−s 2
2,1−s,−1
ds < ∞ 2.17
Proof. FromH, for anyt∈0,1,u, v∈0, ∞,w∈−∞,0, we easily obtain the following inequalities:
cα1ft, u, v, w≤ft, cu, v, w≤cβ1ft, u, v, w, ∀c≥N−1
1 ,
cα2ft, u, v, w≤ft, u, cv, w≤cβ2ft, u, v, w, ∀c≥N−1
2 ,
cα3ft, u, v, w≤ft, u, v, cw≤cβ3ft, u, v, w, ∀c≥N−1
3 .
2.18
For everyu ∈P\ {θ},t ∈0,1, choose positive numbersc1 ≤min{N1,1/8N1u2},c2 ≤
min{N2,1/4N2u2},c3≥max{N3−1, N3−1u2}. It follows fromH,2.10,Lemma 2.1, and 2.17that
Tλut λ
1
0
Gt, sfs, us, us, usds
≤ 1 2λ 1 0 sf
s, c1 us c1s−s2/2
s−s 2
2
, c2 u
s
c21−s1−s,−1c3 us
−c3
ds ≤ 1 2λ 1 0 scα1
1
us c1s−s2/2
β1 cα2
2
us c21−s
β2 cβ3
3
us −c3
α3 f
s, s−s 2
2,1−s,−1
ds ≤ 1 2λ 1 0 scα1
1
u2 c1
β1 cα2
2
u2 c2
β2 cβ3
3
u2 c3
α3 f
s, s−s 2
2,1−s,−1
ds
≤ 1 2c
α1−β1
1 c
α2−β2
2 c
β3−α3
3 u
β1 β2 α3
2 λ
1
0 sf
s, s−s 2
2,1−s,−1
ds < ∞.
2.19
Similar to2.19, fromH,2.10,Lemma 2.1, and2.17, for everyu ∈ P\ {θ},t ∈ 0,1, we have
Tλut λ
1
0
G1t, sfs, us, us, usds
≤λ
1
0 sf
s, c1 us c1s−s2/2
s−s 2
2
, c2 u
s
c21−s1−s,−1c3 us
−c3
ds
≤cα1−β1
1 c
α2−β2
2 c
β3−α3
3 u
β1 β2 α3
2 λ
1
0 sf
s, s− s 2
2 ,1−s,−1
−Tλut λ
1
0
G2t, sfs, us, us, usds
≤λ
1
0 sf
s, c1 us c1s−s2/2
s−s 2
2
, c2 u
s
c21−s1−s,−1c3 us
−c3
ds
≤cα1−β1
1 c
α2−β2
2 c
β3−α3
3 u
β1 β2 α3
2 λ
1
0 sf
s, s− s 2
2 ,1−s,−1
ds < ∞.
2.20
Thus,Tλis well defined onP\ {θ}.
From2.4and2.14–2.16, it is easy to know that
Tλu0 0, Tλu1 0, Tλu0 0,
Tλut λ
1
0
Gt, sfs, us, us, usds
≥
t−t 2
2
λ
1
0
1 2sf
s, us, us, usds
≥
t−t 2
2
λ
1
0
max
τ∈0,1Gτ, sf
s, us, us, usds
t−t 2
2
Tλu, ∀t∈0,1, u∈P\ {θ},
Tλut λ
1
0
G1t, sfs, us, us, usds
≥ 1
21−tλ
1
0
sfs, us, us, usds
≥ 1
21−tλ
1
0
max
τ∈0,1G1τ, sf
s, us, us, usds
1
21−tTλu
, ∀t∈0,1, u∈P\ {θ},
−Tλut λ
1
0
G2t, sfs, us, us, usds
≥tλ
1
0
sfs, us, us, usds
≥tλ
1
0
max
τ∈0,1G2τ, sf
s, us, us, usds
tTλu, ∀t∈0,1, u∈P\ {θ}.
2.21
Obviously,u∗ is a positive solution of BVP 1.1 if and only ifu∗ is a positive fixed point of the integral operatorTλinP.
Lemma 2.4. Suppose that (H) and 2.17hold. Then for anyR > r > 0,Tλ : PR \Pr → P is
completely continuous.
Proof. First of all, notice thatTλmapsPR\Pr intoP byLemma 2.3.
Next, we show thatTλis bounded. In fact, for anyu∈PR\Pr, by2.10we can get
r
8
t−t 2
2
≤ut≤
t−t 2
2
R, r
41−t≤u
t≤1−tR, rt≤ −ut≤R, ∀t∈0,1.
2.22
Choose positive numbersc1 ≤ min{N1,r/8N1},c2 ≤ min{N2,r/4N2},c3 ≥ max{N3−1, N3−1R}. This, together withH,2.22,2.16, andLemma 2.1yields that
Tλutλ1
0
G2t, sfs, us, us, usds
≤λ
1
0 sf
s, c1 us c1s−s2/2
s−s 2
2
, c2 u
s
c21−s1−s,−1c3 us
−c3
ds
≤λ
1
0 scα1
1
us c1s−s2/2
β1 cα2
2
us c21−s
β2 cβ3
3
us −c3
α3 f
s, s−s 2
2,1−s,−1
ds
≤cα1−β1
1 c
α2−β2
2 c
β3−α3
3 Rβ1 β2 α3λ
1
0 sf
s, s− s 2
2,1−s,−1
ds
< ∞, ∀t∈0,1, u∈PR\Pr.
2.23
Thus,Tλis bounded onPR\Pr.
Now we show thatTλ is a compact operator onPR\Pr. By2.23and Ascoli-Arzela
theorem, it suffices to show thatTλV is equicontinuous for arbitrary bounded subsetV ⊂
PR\Pr.
Since for eachu∈V,2.22holds, we may choose still positive numbersc1≤min{N1, r/8N1},c2≤min{N2,r/4N2},c3≥max{N3−1, N3−1R}. Then
Tλutλ1
t
fs, us, us, usds
≤C0
1
t
f
s, s−s 2
2,1−s,−1
ds
:Ht, t∈0,1,
whereC0λcα11−β1c
α2−β2
2 c
β3−α3
3 Rβ1 β2 α3. Notice that
1
0
HtdtC0
1
0
1
t
f
s, s−s 2
2,1−s,−1
ds dt
C0
1
0
s
0 f
s, s−s 2
2,1−s,−1
dt ds
C0
1
0 sf
s, s−s 2
2,1−s,−1
ds < ∞.
2.25
Thus for any givent1, t2∈0,1witht1≤t2and for anyu∈V, we get
Tλut2−T
λut1≤
t2
t1
Tλutdt≤t2
t1
Htdt. 2.26
From 2.25,2.26, and the absolute continuity of integral function, it follows that TλV is
equicontinuous.
Therefore,TλVis relatively compact, that is,Tλis a compact operator onPR\Pr.
Finally, we show thatTλis continuous onPR\Pr. Supposeun, u∈PR\Pr,n1,2, . . .
andun−u2 → 0, n → ∞. Thenunt → ut,unt → utandunt → utas
n → ∞uniformly, with respect tot ∈0,1. FromH, choose still positive numbersc1 ≤
min{N1,r/8N1},c2≤min{N2,r/4N2},c3≥max{N3−1, N3−1R}. Then
0≤ft, unt, unt, unt≤C0f
t, t−t 2
2,1−t,−1
, t∈0,1,
0≤G2t, sfs, uns, uns, uns≤C0sf
s, s−s 2
2,1−s,−1
, t∈0,1, s∈0,1.
2.27
By2.17, we know thatsfs, s−s2/2,1−s,−1is integrable on0,1. Thus, from theLebesgue
dominated convergence theorem, it follows that
lim
n→ ∞Tλun−Tλu2nlim→ ∞Tλun −T
λu
≤ lim
n→ ∞λ
1
0
sfs, uns, uns, uns
−fs, us, us, usds
λ
1
0 s lim
n→ ∞
fs, uns, uns, uns
−fs, us, us, unsds
0.
2.28
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 BVP1.1.
Theorem 3.1. Suppose (H) holds, then BVP1.1has at least one positive solution for anyλ >0if and only if the integral inequality2.17holds.
Proof. Suppose first thatutbe a positive solution of BVP1.1for any fixedλ >0. Then there exist constantsIii1,2,3,4with 0< Ii<1< Ii 1,i1,3 such that
I1
t−t 2
2
≤ut≤I2
t−t 2
2
, I31−t≤ut≤I41−t, t∈0,1. 3.1
In fact, it follows fromu4t≥0,t∈0,1andu0 u1 u0 u1 0, thatut≤0 fort∈0,1andut≤0,ut≥0 fort∈0,1. By the concavity ofutandut, we have
ut≥tu1 1−tu0 tu ≥
t−t 2
2
u,
ut≥tu1 1−tu0 1−tu, ∀t∈0,1.
3.2
On the other hand,
ut
1
0
G0t, s−usds
t
0
s−usds
1
t
t−usds
≤ t2 2u
t1−tu
t−t 2
2
u,
ut
1
t
−usds≤1−tu, ∀t∈0,1.
3.3
Let I1 min{u,1/2},let I2 I4 max{u,2},and letI3 min{u,1/2}, then3.1
Choose positive numbers c1 ≤ N1I2−1,c2 ≤ N2I4−1,c3 ≥ max{N3−1, N3−1u2}. This,
together withH,1.2, and2.18yields that
f
t, t−t 2
2,1−t,−1
f
t, c1t−t 2/2 c1ut ut, c2
1−t c2utu
t, 1
c3 c3
−utu
t
≤cα1
1
t−t2/2 c1ut
β1 cα2
2
1−t c2ut
β21
c3
α3 c
3
−ut
β3
ft, ut, ut, ut
≤cα1
1
1
c1I1
β1 cα2
2
1
c2I3
β21
c3
α3 − c3
ut
β3
ft, ut, ut, ut
C∗−ut−β3
ft, ut, ut, ut, t∈0,1,
3.4
whereC∗cα1−β1
1 c
α2−β2
2 c
β3−α3
3 I
−β1
1 I
−β2
3 . Hence, integrating3.4fromtto 1, we obtain
λ
1
t
−usβ3 f
s, s−s 2
2,1−s,−1
ds≤C∗−ut, t∈0,1. 3.5
Since−utincreases on0,1, we get
−utβ3 λ
1
t
f
s, s−s 2
2,1−s,−1
ds≤C∗−ut, t∈0,1, 3.6
that is, λ 1 t f
s, s−s 2
2,1−s,−1
ds≤C∗ −u
t
−utβ3, t∈0,1. 3.7
Notice thatβ3<1, integrating3.7from 0 to 1, we have
λ 1 0 1 t f
s, s−s 2
2,1−s,−1
ds dt≤C∗1−β3
−1−
u11−β3
. 3.8
That is, λ 1 0 s 0 f
s, s−s 2
2,1−s,−1
dt ds≤C∗1−β3
−1−
u11−β3
. 3.9
Thus,
1
0 sf
s, s−s 2
2,1−s,−1
By an argument similar to the one used in deriving3.5, we can obtain
λ
1
t
−usα3 f
s, s− s 2
2,1−s,−1
ds≥C∗−ut, t∈0,1, 3.11
whereC∗c1β1−α1c
β2−α2
2 c
α3−β3
3 I−
α1
2 I−
α2
4 . So,
λ
1
t
f
s, s−s 2
2 ,1−s,−1
ds≥C∗u−2α3
−ut, t∈0,1. 3.12
Integrating3.12from 0 to 1, we have
λ
1
0
1
t
f
s, s−s 2
2,1−s,−1
ds dt≥C∗u−2α3
−u1. 3.13
That is,
λ
1
0
s
0 f
s, s−s 2
2,1−s,−1
dt ds≥C∗u−2α3
−u1. 3.14
So,
1
0 sf
s, s−s 2
2,1−s,−1
ds >0. 3.15
This and3.10imply that2.17holds.
Now assume that2.17holds, we will show that BVP1.1has at least one positive solution for anyλ >0. By2.17, there exists a sufficient smallδ >0 such that
1−δ
δ
sf
s, s−s 2
2,1−s,−1
ds >0. 3.16
For any fixedλ >0, first of all, we prove
Tλu2≥ u2, ∀u∈∂Pr, 3.17
where 0< r≤min{N1, N2, N3,λδ1 β32−3β1 β2
1−δ
δ sfs, s−s2/2,1−s,−1ds
1/1−β1 β2 β3}. Letu∈∂Pr, then
r
8
t− t 2
2
≤ut≤r
t− t 2
2
≤N1
t−t 2
2
, r
41−t≤u
t≤r1−t≤N21−t,
δr≤rt≤ −ut≤r ≤N3, ∀t∈δ,1−δ.
FromLemma 2.1,3.18, andH, we get
Tλu2Tλu≥λ max t∈δ,1−δ
1
0
G2t, sfs, us, us, usds
≥δλ
1−δ
δ
sf
s, us s−s2/2
s− s 2
2
,u
s
1−s1−s,−1
−us
ds
≥δλ
1−δ
δ
s
us s−s2/2
β1us
1−s
β2
−usβ3 f
s, s−s 2
2 ,1−s,−1
ds
≥δ
r
8
β1r
4
β2
δrβ3λ
1−δ
δ
sf
s, s−s 2
2,1−s,−1
ds
≥δ1 β32−3β1 β2rβ1 β2 β3λ
1−δ
δ
sf
s, s− s 2
2,1−s,−1
ds
≥r u2, u∈∂Pr.
3.19
Thus,3.17holds.
Next, we claim that
Tλu2≤ u2, ∀u∈∂PR, 3.20
whereR≥max{8N−1
1 ,4N−21,λN
α3−β3
3
1
0sfs, s−s2/2,1−s,−1ds
1/1−β1 β2 β3 }. LetcN3/R, then foru∈∂PR, we get
N−1 1
t−t 2
2
≤ R 8
t−t 2
2
≤ut≤R
t−t 2
2
, N−1
2 1−t≤ R
41−t≤u
t≤R1−t,
−cut≤cu2cRN3, ∀t∈0,1.
3.21
Therefore, byLemma 2.1andH, it follows that
Tλutλ1
0
G2t, sfs, us, us, usds
≤λ
1
0 sf
s, us s−s2/2
s−s 2
2
, u
s
1−s1−s,−1
1
c
−cus
≤λ
1
0
us s−s2/2
β1us
1−s
β21
c
β3
−cusα3 f
s, s− s 2
2,1−s,−1
ds
≤Rβ1 β2
N3 R
α3−β3 Rα3λ
1
0 sf
s, s−s 2
2,1−s,−1
ds
Rβ1 β2 β3N
3α3−β3λ
1
0 sf
s, s−s 2
2,1−s,−1
ds
≤Ru2, u∈∂PR.
3.22
This implies that3.20holds.
By Lemmas1.1and2.4,3.17, and3.20, we obtain thatTλhas a fixed point inPR\Pr.
Therefore, BVP1.1has a positive solution inPR\Pr for anyλ >0.
4. Unbounded Connected Branch of Positive Solutions
In this section, we study the global continua results under the hypothesesHand2.17. Let
L{λ, u∈0, ∞×P\ {θ}:λ, u satisfies BVP1.1}, 4.1
then, byTheorem 3.1,L∩{λ} ×P/∅for anyλ >0.
Theorem 4.1. Suppose (H) and2.17hold, then the closureLof positive solution set possesses an unbounded connected branchCwhich comes from0, θsuch that
ifor anyλ >0, C∩{λ} ×P/∅, and
iilimλ,uλ∈C,λ→0 uλ20, limλ,uλ∈C,λ→ ∞uλ2 ∞.
Proof. We now prove our conclusion by the following several steps.
First, we prove that for arbitrarily given 0< λ1< λ2< ∞, L∩λ1, λ2×Pis bounded.
In fact, let
R2 max
⎧ ⎨
⎩8N1−1,4N2−1,
λ2N3α3−β3
1
0 sf
s, s−s 2
2,1−s,−1
ds
1/1−β1 β2 β3⎫⎬
⎭, 4.2
then foru∈P\ {θ}andu2≥R, we get
N1−1
t− t 2
2
≤ R 8
t−t 2
2
≤ut≤
t−t 2
2
u2,
N2−11−t≤ R
41−t≤u
t≤1−tu
2, ∀t∈0,1.
Therefore, byLemma 2.1andH, it follows that
Tλu2≤ Tλ2u2≤λ2
1
0
sfs, us, us, usds
≤λ2
1
0 sf
s, us s−s2/2
s−s 2
2
, u
s
1−s1−s,−1 u2
N3 N3
u2
−us
ds
≤λ2uβ21 β2
N3
u2
α3−β3 uα3
2
1
0 sf
s, s−s 2
2,1−s,−1
ds
λ2u2β1 β2 β3N3α3−β3
1
0 sf
s, s−s 2
2,1−s,−1
ds
<u2, ∀λ∈λ1, λ2.
4.4
Let
r 1
2min
⎧ ⎨
⎩N1, N2, N3,
λ1δ1 β32−3β1 β2
1−δ
δ
sf
s, s−s 2
2,1−s,−1
ds
1/1−β1 β2 β3⎫⎬
⎭,
4.5
whereδis given by3.16. Then foru∈P\ {θ}andu2≤r, we get
u2
8
t− t 2
2
≤ut≤r
t−t 2
2
≤N1
t− t 2
2
; u2
4 1−t≤u
t≤r1−t≤N21−t,
δu2≤tu2≤ −ut≤r≤N3, ∀t∈δ,1−δ.
4.6
Therefore, byLemma 2.1andH, it follows that
Tλu ≥ Tλ1u ≥λ1 max
t∈δ,1−δ
1
0
G2t, sfs, us, us, usds
≥δλ1
1−δ
δ
sf
s, us s−s2/2
s−s 2
2
, u
s
1−s1−s,−1
−us
ds
≥δλ1
1−δ
δ
s
us s−s2/2
β1us
1−s
β2
−usβ3 f
s, s−s 2
2,1−s,−1
≥δ
u2
8
β1u
2
4
β2
δu2β3λ1
1−δ
δ
sf
s, s− s 2
2,1−s,−1
ds
≥δ1 β32−3β1 β2uβ1 β2 β3
2 λ1
1−δ
δ
sf
s, s−s 2
2 ,1−s,−1
ds
>u2, u∈∂Pr.
4.7
Therefore,u Tλuhas no positive solution in λ1, λ2×P \PR∪λ1, λ2×Pr. As a
consequence,L∩λ1, λ2×Pis bounded.
By the complete continuity ofTλ,L∩λ1, λ2×Pis compact.
Second, we choose sequences{an}∞n1and{bn}∞n1satisfy
0<· · ·< an<· · ·< a1< b1<· · ·< bn<· · ·,
lim
n→ ∞an0, nlim→ ∞bn ∞.
4.8
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 nbe fixed, suppose that for anybn, u ∈ L∩{bn} ×P, the connected branchCu of
L∩an, bn×P, passing throughbn, u, leads toCu∩{an} ×P ∅. SinceCuis compact,
there exists a bounded open subsetΩ1ofan, bn×Psuch thatCu⊂Ω1,Ω1∩{an} ×P ∅,
andΩ1∩an, bn× {θ} ∅, whereΩ1and later∂Ω1denote the closure and boundary ofΩ1
with respect toan, bn×P. IfL∩∂Ω1/∅, thenCuandL∩∂Ω1are two disjoint closed subsets
ofL∩Ω1. SinceL∩Ω1is a compact metric space, there are two disjoint compact subsetsM1
and M2 of L∩Ω1 such thatL∩Ω1 M1 ∪M2,Cu ⊂ M1, and L∩∂Ω1 ⊂ M2. Evidently, γ: distM1, M2>0. Denoting byV theγ/3-neighborhood ofM1and lettingΩu Ω1∩V,
then it follows that
Cu⊂Ωu, Ωu∩{an} ×P∪an, bn× {θ} ∅, L∩∂Ωu∅. 4.10
IfL∩∂Ω1∅, then takingΩu Ω1.
It is obvious that in{bn} ×P, the family of{Ωu∩{bn} ×P: bn, u ∈ L}makes up
an open covering ofL∩{bn} ×P. SinceL∩{bn} ×Pis a compact set, there exists a finite
subfamily{Ωui∩{bn} ×P:bn, ui∈L}
k
i1which also coversL∩{bn} ×P. LetΩ
k
i1Ωui, then
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 ofΩ,4.4, and4.7, it follows easily that there exist 0< rn< Rnsuch 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 4.12and 4.14 that iTbn, PRn, P 0, which contradicts 4.15. Hence, there exists somebn, u∈L∩{bn} ×Psuch that the connected branchCuofL∩an, bn×Pcontaining
bn, usatisfies thatCu∩{an}×P/∅. LetCnbe the connected branch ofLincludingCu, then
thisCnsatisfies4.9.
ByLemma 1.2, there exists a connected branchC∗ of lim supn→ ∞Cn such thatC∗ ∩
{λ} ×P/∅ for any λ > 0. Noticing lim supn→ ∞Cn ⊂ L, we have C∗ ⊂ L. Let Cbe the
connected branch ofLincludingC∗, thenC∩{λ} ×P/∅for anyλ >0. Similar to4.4and
4.7, for anyλ >0,λ, uλ∈C, we have, byH,4.2,4.3,4.5,4.6, andLemma 2.1,
uλ2Tλuλ2≤λ
1
0
sfs, uλs, uλs, uλs
ds
≤λuλβ21 β2
N3
uλ2
α3−β3 uλα23
1
0 sf
s, s−s 2
2,1−s,−1
ds
λuλ2β1 β2 β3N3α3−β3
1
0 sf
s, s−s 2
2,1−s,−1
ds
≤λRβ1 β2 β3N3α3−β3
1
0 sf
s, s−s 2
2,1−s,−1
ds,
4.16
uλ2Tλuλ2≥λt∈maxδ,1−δ
1
0
G2t, sf
s, uλs, uλs, uλs
ds
≥λδ
uλ2
8
β1u
λ2
4
β2
δuλ2β3
1−δ
δ
sf
s, s−s 2
2,1−s,−1
ds
≥λδ1 β32−3β1 β2u
λβ21 β2 β3
1−δ
δ
sf
s, s−s 2
2,1−s,−1
ds
≥λδ1 β32−3β1 β2rβ1 β2 β3
1−δ
δ
sf
s, s−s 2
2,1−s,−1
ds,
whereδis given by3.16. Letλ → 0 in4.16andλ → ∞in4.17, we have
lim λ,uλ∈C,λ→0
uλ20, λ,u lim
λ∈C,λ→ ∞
uλ2 ∞. 4.18
Therefore,Theorem 4.1holds 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 NSFC10871120and HESTPSPJ09LA08.
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