doi:10.1155/2011/894135

*Research Article*

**Solutions to a Three-Point Boundary Value Problem**

**Jin Liang**

**1**

_{and Zhi-Wei Lv}

_{and Zhi-Wei Lv}

**2, 3**

*1 _{Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China}*

*2 _{Department of Mathematics and Physics, Anyang Institute of Technology, Anyang, Henan 455000, China}*

*3*

_{Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China}Correspondence should be addressed to Jin Liang,jinliang@sjtu.edu.cn

Received 25 November 2010; Accepted 19 January 2011

Academic Editor: Toka Diagana

Copyrightq2011 J. Liang and Z.-W. Lv. 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.

By using the fixed-point index theory and Leggett-Williams fixed-point theorem, we study the
exis-tence of multiple solutions to the three-point boundary value problem*u*t *atft, ut, u*t
0, 0*< t <*1;*u0 u*0 0;*u*1−*αu*η *λ, whereη*∈0,1/2,*α*∈1/2η,1/ηare constants,
*λ*∈0,∞is a parameter, and*a,f*are given functions. New existence theorems are obtained, which
extend and complement some existing results. Examples are also given to illustrate our results.

**1. Introduction**

It is known that when diﬀerential equations are required to satisfy boundary conditions at more than one value of the independent variable, the resulting problem is called a multipoint boundary value problem, and a typical distinction between initial value problems and multipoint boundary value problems is that in the former case one is able to obtain the solutions depend only on the initial values, while in the latter case, the boundary conditions at the starting point do not determine a unique solution to start with, and some random choices among the solutions that satisfy these starting boundary conditions are normally not to satisfy the boundary conditions at the other specified points. As it is noticed elsewhere see, e.g., Agarwal 1, Bisplinghoﬀ and Ashley 2, and Henderson3, multi point boundary value problem has deep physical and engineering background as well as realistic mathematical model. For the development of the research of multi point boundary value problems for diﬀerential equations in last decade, we refer the readers to, for example,

In this paper, we study the existence of multiple solutions to the following three-point boundary value problem for a class of third-order diﬀerential equations with inhomogeneous three-point boundary values,

*ut* *atft, ut, ut*0*,* 0*< t <*1*,*

*u*0 *u*0 0*,* *u*1−*αuηλ,*

1.1

where*η* ∈0*,*1*/*2,*α*∈1*/*2*η,*1*/η*,*λ*∈0*,*∞, and*a*,*f*are given functions. To the authors’
knowledge, few results on third-order diﬀerential equations with inhomogeneous three-point
boundary values can be found in the literature. Our purpose is to establish new existence
theorems for1.1which extend and complement some existing results.

Let *X* be an Banach space, and let *Y* be a cone in *X*. A mapping *β* is said to be a
nonnegative continuous concave functional on*Y* if*β*:*Y* → 0*,*∞is continuous and

*βtx* 1−*ty*≥*tβx* 1−*tβy,* *x, y*∈*Y, t*∈0*,*1*.* 1.2

Assume that

H

*a*∈*C*0*,*1*,*0*,*∞*,* 0*<*

_{1}

0

1−*ssasds <*∞*,*

*f*∈*C*0*,*1×0*,*∞×0*,*∞*,*0*,*∞*.*

1.3

Define

max*f*0 lim

*v*→0*t*max∈0*,*1_{u}_{∈}sup_{}_{0}_{,}_{}_{∞}_{}

*ft, u, v*
*v* *,*

min*f*0 lim

*v*→0*t*min∈0*,*1*u*∈inf0*,*∞

*ft, u, v*
*v* *,*

max*f*∞* _{v}*lim

_{→}

_{}

_{∞}max

*t*∈0

*,*1

_{u}_{∈}sup

_{}

_{0}

_{,}_{}

_{∞}

_{}

*ft, u, v*
*v* *,*

min*f*∞* _{v}*lim

_{→}

_{}

_{∞}min

*t*∈0

*,*1

*u*∈inf0

*,*∞

*ft, u, v*
*v* *.*

1.4

**2. Lemmas**

Let*EC*1_{}_{0}_{,}_{1}_{}_{be a Banach Space with norm}

*u*1max

*u,u,* 2.1

where

*u* max

*t*∈0*,*1|*ut*|*,* *u*

_{} _{max}

*t*∈0*,*1*u*

_{}_{t}_{}_{.}

2.2

It is not hard to see Lemmas2.1and2.2.

**Lemma 2.1.** *Letu*∈*C*1_{}_{0}_{,}_{1}_{}_{be the unique solution of}_{}_{1.1}_{}_{. Then}

*ut*

_{1}

0

*Gt, sasfs, us, usds*

*αt*2

21−*αη*

1

0

*G*1

*η, sasfs, us, usds* *λt*

2

21−*αη,*

2.3

*where*

*Gt, s* 1

2 ⎧ ⎨ ⎩

2*t*−*t*2_{−}_{s}_{s,}_{s}_{≤}_{t,}

1−*st*2_{,}_{t}_{≤}_{s,}

*G*1*t, s* *∂Gt, s*

*∂t*

⎧ ⎨ ⎩

1−*ts,* *s*≤*t,*

1−*st,* *t*≤*s.*

2.4

**Lemma 2.2.** *One has the following.*

i0≤*G*1*t, s*≤1−*ss,* 1*/*2*t*21−*ss*≤*Gt, s*≤*G*1*, s* 1*/*21−*ss.*

ii*G*1*t, s*≥1*/*4*G*1*s, s* 1*/*41−*ss,* for*t*∈1*/*4*,*3*/*4*, s*∈0*,*1*.*

iii *G*11*/*2*, s*≥1*/*21−*ss,* for*s*∈0*,*1*.*

**Lemma 2.3.** *Letu*∈*C*1_{}_{0}_{,}_{1}_{}_{be the unique solution of}_{}_{1.1}_{}_{. Then}_{u}_{}_{t}_{}_{is nonnegative and satisfies}*u*1*u.*

iFor*t*≤*η*,

*ut* 1

21−*αη*

*t*

0

2*ts*−*s*21−*αηt*2*sα*−1*asfs, us, usds*

_{η}

*t*

*t*21−*αηt*2*sα*−1*asfs, us, usds*

_{1}

*η*

*t*21−*sasfs, us, usdsλt*2

*,*

*ut* 1

21−*αη*

_{t}

0

2*s*1−*αη*2*tsα*−1*asfs, us, usds*

_{η}

*t*

2*t*1−*αη*2*tsα*−1*asfs, us, usds*

_{1}

*η*

2*t*1−*sasfs, us, usds*2*λt*

*,*

2.5

that is,*ut*≤*ut*.

iiFor*t*≥*η*,

*ut* 1

21−*αη*

*η*

0

2*ts*−*s*21−*αηt*2*sα*−1*asfs, us, usds*

*t*

*η*

2*ts*−*s*21−*αηt*2*αη*−*sasfs, us, usds*

_{1}

*t*

*t*2_{}_{1}_{−}_{s}_{}_{a}_{}_{s}_{}_{f}_{s, u}_{}_{s}_{}_{, u}_{}_{s}_{}_{ds}_{}* _{λt}*2

*,*

*ut* 1

21−*αη*

*η*

0

2*s*1−*αη*2*tsα*−1*asfs, us, usds*

_{t}

*η*

2*s*1−*αη*2*tαη*−*sasfs, us, usds*

_{1}

*t*

2*t*1−*sasfs, us, usds*2*λt*

*.*

On the other hand, for*η*≤*s*≤*t*, we have

2*s*1−*αη*2*tαη*−*s*−2*ts*−*s*21−*αηt*2*αη*−*s*

*αηt*−*s*2*s*−*t* *s*1−*t*2−*t* *ss*−*t*

*αηt*−*s*

2*s*−*t*− *s*
*αη*

*s*1−*t*2−*t.*

2.7

Since*α*∈1*/*2*η,*1*/η*,

2*s*1−*αη*2*tαη*−*s*≥2*ts*−*s*2_{1}_{−}_{αη}_{}* _{t}*2

_{αη}_{−}

_{s}_{.}_{}

_{2.8}

_{}

So,*ut*≤*ut*. Therefore,*ut*≤*ut*, which means

*u* ≤*u,* *u*1*u.* 2.9

The proof is completed.

**Lemma 2.4.** *Letu*∈*C*1_{}_{0}_{,}_{1}_{}_{be the unique solution of}_{}_{1.1}_{}_{. Then}

min

*t*∈1*/*4*,*3*/*4*u*

_{}_{t}_{}_{≥} 1

4*u*1*.* 2.10

*Proof.* From2.3, it follows that

*ut*

_{1}

0

*G*1*t, sasf*

*s, us, usds*

*αt*

1−*αη*

1

0

*G*1

*η, sasfs, us, usds* *λt*

1−*αη*

≤ 1

0

1−*ssasfs, us, usds*

*α*

1−*αη*

_{1}

0

*G*1

*η, sasfs, us, usds* *λ*

1−*αη.*

2.11

Hence,

*u*1*u*≤
1

0

1−*ssasfs, us, usds*

*α*

1−*αη*

_{1}

0

*G*1

*η, sasfs, us, usd*s *λ*
1−*αη.*

By Lemmas2.2and2.3, we get, for any*t*∈1*/*4*,*3*/*4,

min

*t*∈1*/*4*,*3*/*4*u*

_{}_{t}_{min}

*t*∈1*/*4*,*3*/*4
1

0

*G*1*t, sasf*

*s, us, usds*

*αt*

1−*αη*

1

0

*G*1

*η, sasfs, us, usds* *λt*

1−*αη*

≥ 1 4

1

0

1−*ssasfs, us, usds*

*α*

1−*αη*

1

0

*G*1

*η, sasfs, us, usds* *λ*

1−*αη*

*.*

2.13

Thus,

min

*t*∈1*/*4*,*3*/*4*u*

_{}_{t}_{}_{≥} 1

4*u*1*.* 2.14

Define a cone by

*K*

*u*∈*E*:*u*≥0*,* min

*t*∈1*/*4*,*3*/*4*u*

_{}_{t}_{}_{≥} 1

4*u*1

*.* 2.15

Set

*Kr* {*u*∈*K*:*u*1*< r*}*,* *∂Kr*{*u*∈*K*:*u*1*r*}*,* *r >*0*,*

*Kr* {*u*∈*K*:*u*1≤*r*}*,* *K*

*β, r, su*∈*K*:*r*≤*βu,* *u*_{1}≤*s,* *s > r >*0*.*

2.16

Define an operator*T*by

*Tut*

_{1}

0

*Gt, sasfs, us, usds*

*αt*2

21−*αη*

_{1}

0

*G*1

*η, sasfs, us, usds* *λt*

2

21−*αη.*

2.17

Lemma2.1implies that1.1has a solution*uut*if and only if*u*is a fixed point of*T*.

From Lemmas2.1and2.2and the Ascoli-Arzela theorem, the following follow.

**Theorem 2.6**see10. *LetEbe a real Banach Space, letK*⊂*Ebe a cone, and*Ω*r* {*u*∈*K*:*u* ≤

*r*}*. Let operatorT* :*K*∩Ω*r* → *Kbe completely continuous and satisfyTx /x, for allx*∈*∂*Ω*r. Then*
i*ifTx* ≤ *x,* *for all* *x*∈*∂*Ω*r, theniT,*Ω*r, K* 1*,*

ii*ifTx* ≥ *x,* *for all* *x*∈*∂*Ω*r, theniT,*Ω*r, K* 0*.*

**Theorem 2.7**see8. *LetT* :*Pc* → *Pcbe a completely continuous operator andβa nonnegative*
*continuous concave functional onP* *such thatβx* ≤ *x* *for allx* ∈ *Pc. Suppose that there exist*

0*< d*0*< a*0 *< b*0≤*csuch that*

a{*x*∈*Pβ, a*0*, b*0:*βx> a*0}*/*∅*andβTx> a*0*forx*∈*Pβ, a*0*, b*0*,*

b*Tx< d*0*forx* ≤*d*0,

c*βTx> a*0*forx*∈*Pβ, a*0*, cwithTx> b*0.

*Then,T* *has at least three fixed pointsx*1*, x*2*,* *and* *x*3*inPcsatisfying*

*x*1*< d*0*,* *a*0*< βx*2*,* *x*3*> d*0*,* *βx*3*< a*0*.* 2.18

**3. Main Results**

In this section, we give new existence theorem about two positive solutions or three positive solutions for1.1.

Write

Λ1

1

0

1−*ssasds* *α*

1−*αη*

1

0

*G*1

*η, sasds*

_{−}1

*,*

Λ2

3*/*4

1*/*4

1−*ssasds* *α*

1−*αη*

3*/*4

1*/*4

*G*1

*η, sasds*

_{−}1

*.*

3.1

**Theorem 3.1.** *Assume that*

*H*1min*f*0min*f*∞ ∞*;*

*H*2*there exists a constantρ*1 *>*0*such thatft, u, v*≤1*/*2Λ1*ρ*1*, fort*∈0*,*1*,u*∈0*, ρ*1
*andv*∈0*, ρ*1*.*

*Then, the problem*1.1*has at least two positive solutionsu*1*andu*2*such that*

0*<u*11*< ρ*1*<u*21*,* 3.2

*forλsmall enough.*

*Proof.* Since

min*f*0 lim

*v*→0*t*min∈0*,*1*u*∈inf0*,*∞

*ft, u, v*

there is*ρ*0∈0*, ρ*1such that

By Theorem2.6, we have

*iT,*Ω*ρ*0*, K*

0*.* 3.9

On the other hand, since

min*f*∞* _{v}*lim

_{→}

_{}

_{∞}min

*t*∈0

*,*1

*u*∈inf0

*,*∞

*ft, u, v*

there exist*ρ*_{0}∗,*ρ*∗_{0}*> ρ*1such that

*ft, u, v*≥8Λ2*v,* for*t*∈0*,*1*, u*∈0*,*∞*, v*≥ 1
4*ρ*

∗

0*.* 3.11

LetΩ*ρ*∗_{0}{*u*∈*K*:*u*1 *< ρ*∗_{0}}. Then, by a argument similar to that above, we obtain

*Tu*1≥ *u*1*,* ∀*u*∈*∂*Ω*ρ*_{0}∗*.* 3.12

By Theorem2.6,

*iT,*Ω*ρ*∗_{0}*, K*

0*.* 3.13

Finally, letΩ*ρ*1 {*u*∈*K* :*u*1 *< ρ*1}, and let*λ*satisfy 0*< λ*≤1*/*21−*αηρ*1for any

*u*∈*∂*Ω*ρ*1. Then,*H*2implies

*Tut*

_{1}

0

*G*1*t, sasf*

*s, us, usds*

*αt*

1−*αη*

_{1}

0

*G*1

*η, sasfs, us, usds* *λt*

1−*αη*

≤ 1

0

1−*ssas*1

2Λ1*ρ*1*ds*

*α*

1−*αη*

1

0

*G*1

*η, sas*1

2Λ1*ρ*1*ds*

*λ*

1−*αη*

≤ 1 2Λ1

1

0

1−*ssasds* *α*

1−*αη*

_{1}

0

*G*1

*η, sasds*

*ρ*1

1*/*21−*αηρ*1
1−*αη*

1

2*ρ*1
1
2*ρ*1

*u*_{1}*,*

3.14

which means that*Tu* ≤ *u*1. Thus,*Tu*1 ≤ *u*1, for all*u*∈*∂*Ω*ρ*1.
Using Theorem2.6, we get

*iT,*Ω*ρ*1*, K*

1*.* 3.15

From3.9–3.15and*ρ*0*< ρ*1 *< ρ*∗_{0}, it follows that

*iT,*Ω*ρ*∗_{0}\Ω*ρ*1*, K*

−1*,* *iT,*Ω*ρ*1\Ω*ρ*0*, K*

Therefore,*T*has fixed point*u*1 ∈Ω*ρ*1\Ω*ρ*0and fixed point*u*2 ∈Ω*ρ*∗0\Ω*ρ*1. Clearly,*u*1,*u*2are
both positive solutions of the problem1.1and

0*<u*11*< ρ*1*<u*21*.* 3.17

The proof of Theorem3.1is completed.

**Theorem 3.2.** *Assume that*

*H*3max*f*0max*f*∞0*;*

*H*4*there exists a constantρ*2 *>*0*such thatft, u, v*≥2Λ2*ρ*2*, fort*∈0*,*1*, u*∈0*, ρ*2*and*

*v*∈1*/*4*ρ*2*, ρ*2*.*

*Then, the problem*1.1*has at least two positive solutionsu*1*andu*2*such that*

0*<u*11*< ρ*2*<u*21 3.18

*forλsmall enough.*

*Proof.* By

max*f*0 lim

*v*→0*t*max∈0*,*1_{u}_{∈}sup_{}_{0}_{,}_{}_{∞}_{}

*ft, u, v*

*v* 0*,* 3.19

we see that there exists*ρ*∗∈0*, ρ*2such that

*ft, u, v*≤ 1

2Λ1*v,* for*t*∈0*,*1*, u*∈0*,*∞*, v*∈

0*, ρ*∗*.* 3.20

Put

Ω*ρ∗*

*u*∈*K*:*u*_{1} *< ρ*∗*,* 3.21

and let*λ*satisfy

0*< λ*≤ 1

2

1−*αηρ*∗*.* 3.22

Then Lemmas2.2and2.3and3.20implies that for any*u*∈*∂*Ω*ρ∗*,

*Tut*

_{1}

0

*G*1*t, sasf*

*s, us, usds*

*αt*

1−*αη*

_{1}

0

*G*1

*η, sasfs, us, usds* *λt*

≤
_{1}

0

1−*ssas*1

2Λ1*u*

_{}_{s}_{}_{ds}_{} *α*

1−*αη*

_{1}

0

*G*1

*η, sas*1

2Λ1*u*

_{}_{s}_{}_{ds}_{} *λ*

1−*αη*

≤ 1 2Λ1

1

0

1−*ssasds* *α*

1−*αη*

_{1}

0

*G*1

*η, sasds*

*u*1

1−*αηρ*∗

21−*αη*

≤ 1
2*u*1

1
2*ρ*∗

*u*_{1}*.*

3.23

So*Tu* ≤ *u*1. Hence,*Tu*1≤ *u*1, for all*u*∈*∂*Ω*ρ∗*.

Applying Theorem2.6, we have

*iT,*Ω*ρ∗, K*

1*.* 3.24

Next, by

max*f*∞* _{v}*lim

_{→}

_{}

_{∞}max

*t*∈0

*,*1

_{u}_{∈}sup

_{}

_{0}

_{,}_{}

_{∞}

_{}

*ft, u, v*

*v* 0*,* 3.25

we know that there exists*r*0 *> ρ*2such that

*ft, u, v*≤ 1

2Λ1*v,* for*t*∈0*,*1*, u*∈0*,*∞*, v*≥*r*0*.* 3.26

*Case 1.* max*t*∈0*,*1*ft, u, v*is unbounded.

Define a function*f*∗:0*,*∞ → 0*,*∞by

*f*∗*ρ*max*ft, u, v*:*t*∈0*,*1*, u, v*∈0*, ρ.* 3.27

Clearly,*f*∗is nondecreasing and lim*ρ*→∞*f*∗*ρ/ρ*0, and

*f*∗*ρ*≤ 1

2Λ1*ρ,* for*ρ > r*0*.* 3.28

Taking*ρ*∗≥max{2*r*0*,*2*λ/*1−*αη,*2*ρ*2}, it follows from3.26–3.28that

*ft, u, v*≤*f*∗*ρ*∗≤ 1

2Λ1*ρ*

By Lemmas2.2and2.3and3.28, we have

In this case, there exists an*M >*0 such that

max Therefore, in both cases, taking

we get

*Tu*1≤ *u*1*,* ∀*u*∈*∂*Ω*ρ*∗*.* 3.34

By Theorem2.6, we have

*iT,*Ω*ρ*∗*, K*

1*.* 3.35

Finally, putΩ*ρ*2{*u*∈*K*:*u*1*< ρ*2}. Then*H*4implies that

*Tu*

1 2

_{1}

0

*G*1

1
2*, s*

*asfs, us, usds*

*α*

21−*αη*

_{1}

0

*G*1

*η, sasfs, us, usds* *λ*

21−*αη*

≥
_{3}_{/}_{4}

1*/*4
1

21−*ssas*2Λ2*ρ*2*ds*

*α*

21−*αη*

_{3}_{/}_{4}

1*/*4

*G*1

*η, sas*2Λ2*ρ*2*ds*

≥Λ2*ρ*2
3*/*4

1*/*4

1−*ssasds* *α*

1−*αη*

3*/*4

1*/*4

*G*1

*η, sasds*

*ρ*2*,*

3.36

that is,*Tu* ≥*ρ*2, and then*Tu*1≥ *u*1, for all*u*∈*∂*Ω*ρ*2. By virtue of Theorem2.6, we have

*iT,*Ω*ρ*2*, K*

0*.* 3.37

From3.24,3.35,3.37, and*ρ*∗*< ρ*2*< ρ*∗, it follows that

*iT,*Ω*ρ*∗\Ω*ρ*2*, K*

1*,* *iT,*Ω*ρ*2\Ω*ρ∗, K*

−1*.* 3.38

Hence,*T* has fixed point*u*1 ∈Ω*ρ*2\Ω*ρ∗*and fixed point*u*2∈Ω*ρ*∗\Ω*ρ*2. Obviously,*u*1,*u*2are
both positive solutions of the problem1.1and

0*<u*11*< ρ*2*<u*21*.* 3.39

The proof of Theorem3.2is completed.

**Theorem 3.3.** *Let there existd*0*,a*0*,b*0*, andcwith*

*such that*

*ft, u, v*≤ 1

4Λ1*d*0*,* *t*∈0*,*1*, u*∈0*, d*0*, v*∈0*, d*0*,* 3.41

*ft, u, v*≥35Λ2*a*0*,* *t*∈0*,*1*, u*∈*a*0*, b*0*, v*∈*a*0*, b*0*,* 3.42

*ft, u, v*≤1

2Λ1*c,* *t*∈0*,*1*, u*∈0*, c, v*∈0*, c.* 3.43

*Then problem*1.1*has at least three positive solutionsu*1,*u*2,*u*3*satisfying*

*u*11*< d*0*,* *a*0*< βu*2*,* *u*31*> d*0*,* *βu*3*< a*0*,* 3.44

*forλ*≤1*/*21−*αηd*0*.*

*Proof.* Let

*βu* min

*t*∈1*/*4*,*3*/*4|*ut*|*,* *u*∈*K.* 3.45

Then,*β*is a nonnegative continuous concave functional on*K*and*βu*≤ *u*1for each*u*∈*K*.
Let*u*be in*Kc*. Equation3.43implies that

*Tut*

1

0

*G*1*t, sasf*

*s, us, usds*

*αt*

1−*αη*

1

0

*G*1

*η, sasfs, us, usds* *λt*

1−*αη*

≤ 1 2Λ1

1

0

1−*ssasds* *α*

1−*αη*

_{1}

0

*G*1

*η, sasds*

*c*1*/*2

1−*αηc*

1−*αη*

*c.*

3.46

Hence,*Tu*1≤*c*. This means that*T* :*Kc* → *Kc*.

Take

*u*0*t* 1

2*a*0*b*0*,* *t*∈0*,*1*.* 3.47

Then,

*u*0∈

*u*∈*Kβ, a*0*, b*0

:*βu> a*0

*/*

By3.42, we have, for any*u*∈*Kβ, a*0*, b*0,

Therefore,ain Theorem2.7holds.

≥ min

So,cin Theorem2.7holds. Thus, by Theorem2.7, we know that the operator*T* has at least
three positive fixed points*u*1*, u*2*, u*3∈*Kc*satisfying

*u*11*< d*0*,* *a*0*< βu*2*,* *u*31*> d*0*,* *βu*3*< a*0*.* 3.53

**4. Examples**

In this section, we give three examples to illustrate our results.

*Example 4.1.* Consider the problem

where*η*1*/*2,*α*1. Set

So, the condition*H*1is satisfied. Observe

Λ1

Thus, condition*H*2is satisfied.

Therefore, by Theorem3.1, the problem4.1has at least two positive solutions*u*1and

*Example 4.2.* Consider the problem

*ut* 2×581*t*2sin*utut*25−*ut*0*,* 0*< t <*1*,*

*u*0 *u*0 0*,* *u*1−*u*

1 2

*λ,*

4.9

where*η*1*/*2,*α*1. Set

*at* 2*,* *ft, u, v* 581*t*2sin*utut*25−*ut.* 4.10

Then,

max*f*0max*f*∞0*,* 4.11

that is, the condition*H*3is satisfied. Moreover,

Λ2

3*/*4

1*/*4

1−*ssasds* *α*

1−*αη*

_{3}_{/}_{4}

1*/*4

*G*1

*η, sasds*

−1

3*/*4

1*/*4

1−*ss*2*ds* 1

1−1*/*2
1*/*2

1*/*4

1−1 2

*s*2*ds*

_{3}_{/}_{4}

1*/*2
1

21−*s*2*ds*
−1

48

29*.*

4.12

Taking

*ρ*28*,* for*t*∈0*,*1*, u*∈

0*, ρ*2

*, v*∈

1
4*ρ*2*, ρ*2

*,* 4.13

we get

*ft, u, v*≥58825−8828*ρ*2*>*2Λ2*ρ*2 96

29*ρ*2*.* 4.14

Thus, condition*H*4is satisfied.

Consequently, by Theorem3.2, we see that for

0*< λ*≤ 1

2

1−*αηρ*∗≤ 1_{4}*ρ*2*,* 4.15

the problem4.9has at least two positive solutions*u*1and*u*2such that

That is, the conditions of Theorem3.3are satisfied. Consequently, the problem1.1has at
least three positive solutions*u*1*, u*2*, u*3∈*Kc*for

*λ*≤ 1

2

1−*αηd*0 1

4 4.21

satisfying

*u*11*<*1*,* 2*< βu*2*,* *u*31*>*1*,* *βu*3*<*2*.* 4.22

**Acknowledgments**

This paper was supported partially by the NSF of China 10771202and the Specialized Research Fund for the Doctoral Program of Higher Education of China2007035805.

**References**

1 R. P. Agarwal, *Focal Boundary Value Problems for Diﬀerential and Diﬀerence Equations*, vol. 436 of

*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.
2 R. L. Bisplinghoﬀand H. Ashley,*Principles of Aeroelasticity*, Dover Publications, Mineola, NY, USA,

2002.

3 J. Henderson, Ed.,*Boundary Value Problems for Functional-Diﬀerential Equations*, World Scientific, River
Edge, NJ, USA, 1995.

4 D. R. Anderson, “Green’s function for a third-order generalized right focal problem,” *Journal of*
*Mathematical Analysis and Applications*, vol. 288, no. 1, pp. 1–14, 2003.

5 R. I. Avery and A. C. Peterson, “Three positive fixed points of nonlinear operators on ordered Banach
spaces,”*Computers & Mathematics with Applications*, vol. 42, no. 3–5, pp. 313–322, 2001.

6 A. Boucherif and N. Al-Malki, “Nonlinear three-point third-order boundary value problems,”*Applied*
*Mathematics and Computation*, vol. 190, no. 2, pp. 1168–1177, 2007.

7 G. L. Karakostas, K. G. Mavridis, and P. C. Tsamatos, “Triple solutions for a nonlocal functional
boundary value problem by Leggett-Williams theorem,”*Applicable Analysis*, vol. 83, no. 9, pp. 957–
970, 2004.

8 R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered
Banach spaces,”*Indiana University Mathematics Journal*, vol. 28, no. 4, pp. 673–688, 1979.

9 Y. Sun, “Positive solutions for third-order three-point nonhomogeneous boundary value problems,”

*Applied Mathematics Letters*, vol. 22, no. 1, pp. 45–51, 2009.