Volume 2010, Article ID 708376,13pages doi:10.1155/2010/708376
Research Article
Multiple Positive Solutions for nth Order Multipoint Boundary Value Problem
Yaohong Li
1, 2and Zhongli Wei
2, 31Department of Mathematics, Suzhou University, Suzhou, Anhui 234000, China
2School of Mathematics, Shandong University, Jinan, Shandong 250100, China
3School of Sciences, Shandong Jianzhu University, Jinan, Shandong 250101, China
Correspondence should be addressed to Yaohong Li,[email protected] Received 22 January 2010; Revised 9 April 2010; Accepted 3 June 2010 Academic Editor: Ivan T. Kiguradze
Copyrightq2010 Y. Li and Z. Wei. 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.
We study the existence of multiple positive solutions fornth-order multipoint boundary value problem.unt atfut 0,t ∈0,1,uj−10 0j 1,2, . . . , n−1,u1 m
i1αiuηi, wheren≥2, 0< η1< η2<· · ·< ηm<1,αi>0, i1,2, . . . , m. We obtained the existence of multiple positive solutions by applying the fixed point theorems of cone expansion and compression of norm type and Leggett-Williams fixed-point theorem. The results obtained in this paper are different from those in the literature.
1. Introduction
The existence of positive solutions fornth-order multipoint boundary problems has been studied by some authorssee1,2. In1, Pang et al. studied the expression and properties of Green’s funtion and obtained the existence of at least one positive solution fornth-order differential equations by applying means of fixed point index theory:
unt atfut 0, t∈0,1, uj−10 0
j 1,2, . . . , n−1
, u1 m
i1
αiu ηi
, 1.1
wheren≥2, 0< η1< η2<· · ·< ηm<1, αi>0, i1,2, . . . , m.
By using the fixed point theorems of cone expansion and compression of norm type, Yang and Wei in2also obtained the existence of at least one positive solutions for the BVP 1.1if m ≥ 2. This work is motivated by Masee 3. This method is simpler than that
of1. In addition, Eloe and Ahmad in4had solved successfully the existence of positive solutions to the BVP1.1ifm1. Hao et al. in5had discussed the existence of at least two positive solutions for the BVP1.1by applying the Krasonse’skii-Guo fixed point theorem on cone expansion and compression ifm1. However, there are few papers dealing with the existence of multiple positive solutions fornth-order multipoint boundary value problem.
In this paper, we study the existence of at least two positive solutions associated with the BVP1.1by applying the fixed point theorems of cone expansion and compression of norm type ifm≥2 and the existence of at least three positive solutions for BVP1.1by using Leggett-Williams fixed-point theorem. The results obtained in this paper are different from those in the literature and essentially improve and generalize some well-known resultssee 1–8.
The rest of the paper is organized as follows. InSection 2, we present several lemmas.
InSection 3, we give some preliminaries and the fixed point theorems of cone expansion and compression of norm type. The existence of at least two positive solutions for the BVP1.1 is formulated and proved in Section 4. In Section 5, we give Leggett-Williams fixed-point theorem and obtain the existence of at least three positive solutions for the BVP1.1.
2. Several Lemmas
Definition 2.1. A function ut is said to be a position of the BVP 1.1 if utsatisfies the following:
1ut∈C0,1∩Cn0,1;
2ut>0 fort∈0,1and satisfies boundary value conditions1.1;
3unt −atfuthold fort∈0,1.
Lemma 2.2see1. Suppose that
Dm
i1
αiηn−1i /1; 2.1
then for anyy∈C0,1, the problem
unt yt 0, t∈0,1, uj−10 0
j 1,2, . . . , n−1
, u1 m
i1
αiu ηi
2.2
has a unique solution:
ut − 1
n−1!
t
0
t−sn−1ysds tn−1 n−1!1−D
1
0
1−sn−1ysds
− tn−1 n−1!1−D
m−2
i1
αi
ηi
0
ηi−sn−1
ysds
1
0
Kt, sysds,
2.3
where
Kt, s K1t, s K2t, s,
K1t, s 1 n−1!
⎧⎨
⎩
tn−11−sn−1−t−sn−1, 0≤s < t≤1, tn−11−sn−1, 0≤t≤s≤1, K2t, s D
n−1!1−Dtn−11−sn−1− 1 n−1!1−D
s≤ηi
αitn−1
ηi−sn−1 .
2.4
Lemma 2.3see1. LetD <1; Green’s functionKt, sdefined by2.4satisfies 0≤Kt, s≤Ks, ∀t, s∈0,1,
t∈ηmin1,1Kt, s≥γKs, ∀s∈0,1, 2.5
whereγηn−11 :
Ks max
t∈0,1K1t, s max
t∈0,1K2t, s sn−11−sn−1 n−1!
1−1−sn−1/n−22−n
K21, s. 2.6
We omit the proofLemma 2.3here and you can see the detail of Theorem 2.2 in1.
Lemma 2.4see2. LetD <1, y∈C0,1, andy≥0; the unique solutionutof the BVP2.2 satisfies
t∈ηmin1,1ut≥γu, 2.7
whereγis defined byLemma 2.3,umaxt∈0,1|ut|.
3. Preliminaries
In this section, we give some preliminaries for discussing the existence of multiple positive solutions of the BVP1.1in the next. In real Banach spaceC0,1in which the norm is defined by
umax
t∈0,1|ut|, 3.1
set
P
u∈C0,1|u0 0, ut>0 for 0< t≤1, min
t∈η1,1ut≥γu
. 3.2
Obviously,Pis a positive cone inC0,1, whereγis fromLemma 2.3.
For convenience, we make the following assumptions:
A1a : 0,1 → 0,∞is continuous andatdoes not vanish identically, fort ∈ η1,1;
A2f : 0,∞ → 0,∞is continuous;
A3Dm
i1αiηn−1i <1.
Let
Aut 1
0
Kt, sasfusds, ∀t∈0,1, 3.3
whereKt, sis defined by2.4.
From Lemmas2.2–2.4, we have the following result.
Lemma 3.1see2. Suppose thatA1–A3are satisfied, then A : C0,1 → C0,1 is a completely continuous operator,AP⊂P, and the fixed points of the operatorAinPare the positive solutions of the BVP1.1.
For convenience, one introduces the following notations. Let
r 1
n−1!1−D 1
0
1−sn−1asds,
R γm
i2αi
n−1!1−D ηi
η1
ηi−ηisn−1
−
ηi−sn−1
asds m≥2.
3.4
Problem 1. Inspired by the work of the paper2, whether we can obtain a similar conclusion or not, if
ulim→0inffu
u > R−1, lim
u→∞inffu
u > R−1; 3.5
or
ulim→0supfu
u < r−1, lim
u→∞supfu
u < r−1. 3.6
The aim of the following section is to establish some simple criteria for the existence of multiple positive solutions of the BVP1.1, which gives a positive answer to the questions stated above. The key tool in our approach is the following fixed point theorem, which is a useful method to prove the existence of positive solutions for differential equations, for example2–5,9.
Lemma 3.2see10,11. Suppose thatEis a real Banach space andP is cone inE, and letΩ1,Ω2
be two bounded open sets inEsuch that 0∈Ω1, Ω1⊂Ω2. Let operatorA:P∩Ω2\Ω1 → Pbe completely continuous. Suppose that one of two conditions holds:
iAu ≤ u,for allu∈P∩∂Ω1;Au ≥ u,for allu∈P∩∂Ω2; iiAu ≥ u,for allu∈P∩∂Ω1;Au ≤ u,for allu∈P∩∂Ω2. thenAhas at least one fixed point inP∩Ω2\Ω1.
4. The Existence of Two Positive Solutions
Theorem 4.1. Suppose that the conditions A1–A3are satisfied and the following assumptions hold:
B1limu→0inffu/u> R−1; B2limu→∞inffu/u> R−1;
B3There exists a constantρ >0 such thatfu≤r−1ρ, u∈0, ρ.
Then the BVP1.1has at least two positive solutionsu1andu2such that
0<u1< ρ <u2. 4.1
Proof. At first, it follows from the conditionB1that we may chooseρ1∈0, ρsuch that
fu> R−1u, 0< u≤ρ1. 4.2
SetΩ1 {u∈C0,1:u< ρ1}, andu∈P∩∂Ω1; from3.3and2.4andLemma 2.4, for 0< t≤1, we have
Au1 1
n−1!1−D 1
0
D1−sn−1asfusds−m−2
i1
αi
ηi
0
ηi−sn−1
asfusds
≥
m
i1αi
n−1!1−D ηi
0
ηi−ηisn−1
−
ηi−sn−1
asfusds
> R−1m
i1αi
n−1!1−D ηi
0
ηi−ηisn−1
−
ηi−sn−1
asusds
> R−1m
i2αi
n−1!1−D ηi
η1
ηi−ηisn−1
−
ηi−sn−1
asusds
> R−1γum
i2αi
n−1!1−D ηi
η1
ηi−ηisn−1
−
ηi−sn−1 asds R−1Ruu.
4.3
Therefore, we have
Au ≥ Au1>u, u∈P∩∂Ω1. 4.4
Further, it follows from the conditionB2that there existsρ2 > ρsuch that
fu> R−1u, u≥ρ2. 4.5
Letρ∗max{2ρ, γ−1ρ2}, setΩ2 {u∈C0,1:u< ρ∗}, thenu∈P∩∂Ω2 andLemma 2.4 imply
ηmin1≤t≤1ut≥γu ≥ρ2, 4.6
and by the conditionB2,2.4,3.3, andLemma 2.4, we have
Au1 1
n−1!1−D 1
0
D1−sn−1asfusds−m
i1
αi
ηi
0
ηi−sn−1
asfusds
≥
m
i1αi
n−1!1−D ηi
0
ηi−ηisn−1
−
ηi−sn−1
asfusds
> R−1m
i1αi
n−1!1−D ηi
0
ηi−ηisn−1
−
ηi−sn−1
asusds
> R−1m
i2αi
n−1!1−D ηi
η1
ηi−ηisn−1
−
ηi−sn−1
asusds
> R−1γum−2
i2 αi
n−1!1−D ηi
η1
ηi−ηisn−1
−
ηi−sn−1 asds R−1Ruu.
4.7 Therefore, we have
Au ≥ Au1>u, u∈P∩∂Ω2. 4.8 Finally, letΩ3 {u ∈C0,1 :u < ρ}andu∈ P∩∂Ω3. By2.3,3.3, and the condition B3, we have
Aut≤ tn−1
n−1!1−D 1
0
1−sn−1asfusds
≤ r−1ρ n−1!1−D
1
0
1−sn−1asdsr−1rρu,
4.9
which implies
Au ≤ u, u∈P∩∂Ω3. 4.10
Thus from4.4–4.10and Lemmas3.1and3.2,Ahas a fixed pointu1inP∩Ω3\Ω1and a fixedu2inP∩Ω2\Ω3. Both are positive solutions of BVP1.1and satisfy
0<u1< ρ <u2. 4.11
The proof is complete.
Corollary 4.2. Suppose that the conditionsA1–A3are satisfied and the following assumptions hold:
B1limu→0inffu/u ∞;
B2limu→∞inffu/u ∞;
B3there exists a constantρ>0 such thatfu≤r−1ρ, u∈0, ρ. Then the BVP1.1has at least two positive solutionsu1andu2such that
0<u1< ρ<u2. 4.12
Proof. From the conditionsBi i1,2, there exist sufficiently big positive constantsMii 1,2such that
ulim→0supfu
u > M2, lim
u→∞supfu
u > M1 4.13
by the conditionB3; so all the conditions ofTheorem 4.1are satisfied; by an application of Theorem 4.1, the BVP1.1has two positive solutionsu1andu2such that
0<u1< ρ<u2. 4.14
Theorem 4.3. Suppose that the conditions A1–A3are satisfied and the following assumptions hold:
C1limu→0supfu/u< r−1; C2limu→∞supfu/u< r−1;
C3there exists a constantl >0 such thatfu≥R−1l, u∈γl, l.
Then the BVP1.1has at least two positive solutionsu1andu2such that
0<u1< l <u2. 4.15
Proof. It follows from the conditionC1that we may chooseρ3∈0, lsuch that
fu< r−1u, 0< u≤ρ3. 4.16
SetΩ4{u∈C0,1:u< ρ3},andu∈P∩∂Ω4; from3.2and2.4, for 0< t≤1, we have
Aut≤ tn−1
n−1!1−D 1
0
1−sn−1asfusds
< r−1u n−1!1−D
1
0
1−sn−1asdsr−1ruu.
4.17
Therefore, we have
Au<u, u∈P∩∂Ω4. 4.18
It follows from the conditionC2that there existsρ4> lsuch thatfu< r−1uforu≥ρ4,and we consider two cases.
Case i. Suppose thatfis unbounded; there existsl∗> ρ4such thatfu≤fl∗for 0< u≤l∗. Then foru∈Pandul∗, we have
Aut≤ tn−1
n−1!1−D 1
0
1−sn−1asfusds
≤ tn−1 n−1!1−D
1
0
1−sn−1asfl∗ds
< r−1l∗ n−1!1−D
1
0
1−sn−1asdsr−1rl∗l∗u.
4.19
Case ii. Iff is bounded, that is,fu ≤ N for allu ∈ 0,∞, takingl∗ ≥ max{2l, Nr}, for u∈Pandul∗, we have
Aut≤ tn−1
n−1!1−D 1
0
1−sn−1asfusds
≤ N
n−1!1−D 1
0
1−sn−1asds≤Nr≤l∗u.
4.20
Hence, in either case, we always may setΩ5{u∈C0,1:u< l∗}such that
Au ≤ u, u∈P∩∂Ω5. 4.21
Finally, setΩ6{u∈C0,1:u< l}; thenu∈P∩∂Ω6andLemma 2.4imply
t∈ηmin1,1ut≥γuγl, 4.22
and by the conditionC3,2.4, and3.3, we have
Au1 1
n−1!1−D 1
0
D1−sn−1asfusds−m
i1
αi
ηi
0
ηi−sn−1
asfusds
≥
m
i1αi
n−1!1−D ηi
0
ηi−ηisn−1
−
ηi−sn−1
asfusds
≥ R−1lm
i2αi
n−1!1−D ηi
η1
ηi−ηisn−1
−
ηi−sn−1 asds
≥ R−1lγm
i2αi
n−1!1−D ηi
η1
ηi−ηisn−1
−
ηi−sn−1 asds
R−1lRu.
4.23
Hence, we have
Au ≥ u, u∈P∩∂Ω6. 4.24
From4.18–4.24and Lemmas3.1and3.2,Ahas a fixed pointu1inP∩Ω6\Ω4and a fixed u2inP∩Ω5\Ω6. Both are positive solutions of the BVP1.1and satisfy
0<u1< l <u2. 4.25
The proof is complete.
Corollary 4.4. Suppose that the conditionsA1–A3are satisfied and the following assumptions hold:
C1limu→0supfu/u 0;
C2limu→∞supfu/u 0;
C3there exists a constantρ>0 such thatfu≥R−1ρ, u∈γρ, ρ. Then BVP1.1has at least two positive solutionsu1andu2such that
0<u1< ρ<u2. 4.26
The proof ofCorollary 4.4is similar to that ofCorollary 4.2; so we omit it.
5. The Existence of Three Positive Solutions
LetEbe a real Banach space with coneP. A mapβ:P → 0,∞is said to be a nonnegative continuous concave functional onPifβis continuous and
β
tx 1−ty
≥tβx 1−tβ y
5.1 for allx, y ∈ P andt ∈ 0,1.Leta, b be two numbers such that 0 < a < band letβ be a nonnegative continuous concave functional onP. We define the following convex sets:
Pa{x∈P:x< a}, ∂Pa{x∈P:xa}, Pa{x∈P :x ≤a}, P
β, a, b
x∈P :a≤βx,x ≤b
. 5.2
Lemma 5.1see12. LetA : Pc → Pc be completely continuous and letβbe a nonnegative continuous concave functional on P such that βx ≤ x for x ∈ Pc. Suppose that there exist 0< d < a < b≤csuch that
i{x∈Pβ, a, b:βx> a}/∅and βAx> aforx∈Pβ, a, b, iiAx< dforx ≤d,
iiiβAx> aforx∈Pβ, a, cwithAx> b.
ThenAhas at least three fixed pointsx1, x2, x3inPcsuch that
x1< d, a < βx2, and x3> d withβx3< a. 5.3
Now, we establish the existence conditions of three positive solutions for the BVP1.1.
Theorem 5.2. Suppose thatA1–A3hold and there exist numbersaanddwith 0< d < asuch that the following conditions are satisfied:
D1limu→ ∞fu/u<1/G, D2fu< d/G, u∈0, d, D3fu> a/F, u∈a, a/γ, where
F min
t∈η1,1 1
η1
Kt, sasds, G max
t∈0,1
1
0
Kt, sasds, 5.4
Then the boundary value problem1.1has at least three positive solutions.
Proof. LetP be defined by3.2and letAbe defined by3.3. Foru∈P, let
βu min
t∈η1,1ut. 5.5
Then it is easy to check that βis a nonnegative continuous concave functional on P with βu≤ uforu∈PandA:P → Pis completely continuous.
First, we prove that ifD1holds, then there exists a numberc > a/γandA:Pc → Pc. To do this, byD1, there existM >0 andλ <1/Gsuch that
fu< λu, foru > M. 5.6
Set
δ max
u∈0,Mfu; 5.7
it follows thatfu< λuδfor allu∈0,∞. Take c >max
δG 1−λG,a
γ
. 5.8
Ifu∈Pc, then Aut≤ max
t∈0,1
1
0
Kt, sasfusds < max
t∈0,1
1
0
Kt, sasdsλuδ<λcδG < c, 5.9
that is,
Au< c. 5.10
Hence5.10show that ifD1holds, then there exists a numberc > a/γ such thatAmaps PcintoPc.
Now we show that {u ∈ Pβ, a, a/γ : βu > a}/∅ and βAu > a for all u ∈ Pβ, a, a/γ. In fact, takext ≡ a a/γ/2 > a, so x ∈ {u ∈ Pβ, a, a/γ : βu > a}.
Moreover, foru∈Pβ, a, a/γ, thenβu> a, and we have a
γ ≥ u ≥βu> a. 5.11 Therfore, byD3we obtain
βAu min
t∈η1,1 1
0
Kt, sasfusds > a F min
t∈η1,1
1
η1
Kt, sasdsa. 5.12
Next, we assert thatAu< dforu ≤d. In fact, ifu∈Pd, byD2we have Au< d
G
t∈0,1max 1
0
Kt, sasds
d. 5.13
Hence,A:Pd → Pdforu∈Pd.
Finally, we assert that ifu∈Pβ, a, candAu> a/γ, thenβAu> a. To see this, if u∈Pβ, a, candAu> a/γ,then we have fromLemma 2.3that
βAu min
t∈η1,1
1
0
Kt, sasfusds
≥ 1
0
t∈minη1,1Kt, sasfusds≥γ 1
0
Ksasfusds
≥γ 1
0
t∈0,1maxKt, sasfusds≥γmax
t∈0,1
1
0
Kt, sasfusdsγAu.
5.14
So we have
βAu≥γAu> γ·a
γ a. 5.15
To sum up5.10∼5.15, all the conditions of Lemma 5.1are satisfied by taking b a/γ. Hence, A has at least three fixed points; that is, BVP1.1has at least three positive solutions u1, u2, andu3such that
u1< d, a < βu2, and u3> d with βu3< a. 5.16
The proof is complete.
Acknowledgments
The authors are grateful to the referee’s valuable comments and suggestions. The project is supported by the Natural Science Foundation of Anhui ProvinceKJ2010B226, The Excellent Youth Foundation of Anhui Province Office of Education 2009SQRZ169, and the Natural Science Foundation of Suzhou University2009yzk17
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