doi:10.1155/2009/609143

*Research Article*

**Existence of Solutions for**

*m*

**-point Boundary Value**

**Problems on a Half-Line**

**Changlong Yu, Yanping Guo, and Yude Ji**

*College of Sciences, Hebei University of Science and Technology, Shijiazhuang 050018, Hebei, China*

Correspondence should be addressed to Changlong Yu,changlongyu@126.com

Received 5 April 2009; Accepted 6 July 2009

Recommended by Paul Eloe

By using the Leray-Schauder continuation theorem, we establish the existence of solutions for
*m*-point boundary value problems on a half-line*xt* *ft, xt, xt* 0*,*0 *< t <* ∞*, x*0

*m*−2

*i*1 *αixηi,*lim*t*→∞*xt* 0, where*αi* ∈*R,im*−12*αi/*1 and 0*< η*1*< η*2*<*· · ·*< ηm*−2*<*∞are

given.

Copyrightq2009 Changlong Yu et al. 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.

**1. Introduction**

Multipoint boundary value problems BVPs for second-order diﬀerential equations in a finite interval have been studied extensively and many results for the existence of solutions, positive solutions, multiple solutions are obtained by use of the Leray-Schauder continuation theorem, Guo-Krasnosel’skii fixed point theorem, and so on; for details see 1–4 and the references therein.

In the last several years, boundary value problems in an infinite interval have been arisen in many applications and received much attention; see 5, 6. Due to the fact that an infinite interval is noncompact, the discussion about BVPs on the half-line is more complicated, see 5–14 and the references therein. Recently, in 15, Lian and Ge studied the following three-point boundary value problem:

*xt* *ft, xt, xt*0*,* 0*< t <*∞*,*

*x*0 *αxη,* lim

*t*→∞*x*

where *α* ∈ *R, α /*1*,*and *η* ∈ 0*,*∞ are given. In this paper, we will study the following

*m*-point boundary value problems:

*xt* *ft, xt, xt*0*,* 0*< t <*∞*,*

*x*0

*m*−2

*i*1
*αix*

*ηi*

*,* lim

*t*→∞*x*

_{}_{t}_{0}* _{,}* 1.2

where*αi* ∈ *R,*

_{m}_{−}_{2}

*i*1 *αi/*1*, αi*have the same signal, and 0 *< η*1 *< η*2 *<* · · · *< ηm*−2 *<* ∞are

given. We first present the Green function for second-order multipoint BVPs on the half-line and then give the existence results for1.2using the properties of this Green function and the Leray-Schauder continuation theorem.

We use the space*C*1

∞0*,*∞ {*x*∈*C*10*,*∞*,*lim*t*→∞*xt*exists, lim*t*→∞*xt*exists}

with the norm*x*max{*x*_{∞}*,x*_{∞}}, where · _{∞}is supremum norm on the half-line, and

*L*1_{}_{0}_{,}_{}_{∞} _{{}_{x}_{:} _{}_{0}_{,}_{}_{∞}_{} _{→} _{R}_{is absolutely integrable on}_{}_{0}_{,}_{}_{∞}_{}_{}}_{with the norm}_{}_{x}_{}

*L*1

_{∞}

0|*xt*|*dt*.

We set

*P*
_{}_{∞}

0

*psds,* *P*1
_{}_{∞}

0

*spsds,* *Q*
_{}_{∞}

0

*qsdt,* 1.3

and we suppose*αi, i*1*,*2*, . . . , m*−2 are the same signal in this paper and we always assume

*αmi*−12*αi.*

**2. Preliminary Results**

In this section, we present some definitions and lemmas, which will be needed in the proof of the main results.

*Definition 2.1*see15. It holds that*f* : 0*,*∞×*R*2 _{−→} _{R}_{is called an S-Carath´eodory}

function if and only if

ifor each*u, v*∈*R*2_{, t}_{→}_{f}_{}_{t, u, v}_{}_{is measurable on}_{}_{0}_{,}_{}_{∞}_{}_{,}

iifor almost every*t*∈0*,*∞*,u, v*→*ft, u, v*is continuous on*R*2_{,}

iiifor each*r >* 0, there exists*ϕrt* ∈*L*10*,*∞with*tϕrt* ∈*L*10*,*∞*, ϕrt* *>* 0 on
0*,*∞such that max{|*u*|*,*|*v*|} ≤*r*implies|*ft, u, v*| ≤*ϕrt*, for a.e.*t*∈0*,*∞.

**Lemma 2.2.** *Supposem _{i}*

_{}−

_{1}2

*αi/*1

*,if for anyvt*∈

*L*10

*,*∞

*with*

*tvt*∈

*L*10

*,*∞

*, then the*

*BVP,*

*xt* *vt* 0*,* 0*< t <*∞*,*

*x*0

*m*−2

*i*1
*αix*

*ηi*

*,* lim

*t*→∞*x*

Therefore, the unique solution of 2.1 is *xt* _{0}∞*Gt, svsds,* which completes the
proof.

*Remark ofLemma 2.2.*Obviously*Gt, s*satisfies the properties of a Green function, so we call

*Gt, s*the Green function of the corresponding homogeneous multipoint BVP of2.1on the
half-line.

**Lemma 2.4.** *For the Green functionGt, s, it holds that*

lim

*t*→∞*Gt, s* *Gs*

_{Λ}1

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

*s,* *s*≤*η*1*,*

*i*

*k*1
*αkηk*

_{m}_{−}_{2}

*ki*1
*αk* Λ

*s, ηi* ≤*s*≤*ηi*1*<*∞*, i*1*,*2*, . . . , m*−3

*m*−2

*i*1

*αiηi* Λ*s,* *s*≥*ηm*−2*.*

2.13

**Lemma 2.5.** *For the functionx*∈*C*10*,*∞*,it is satisfied that*

*x*0

*m*−2

*i*1
*αix*

*ηi*

2.14

*andαi* *i*1*,*2*, . . . , m*−2*have the same signal,*0*< η*1 *< η*2 *<*· · ·*< ηm*−2*<*∞*, then there exists*
*η*∈*η*1*, ηm*−2*satisfying*

*x*0 *αxη,* 2.15

*whereαmi*−12*αi.*

*Proof.* Let*αi* *i*1*,*2*, . . . , m*−2are positive, and note*M*∗ max{*xt*|*t*∈*η*1*, ηm*−2}*, m*∗

min{*xt* | *t* ∈ *η*1*, ηm*−2}, then for every*ii* 1*,*2*, . . . , m*−2, we have*m*∗ ≤ *xηi* ≤ *M*∗*,*

so *m*∗*mi*−12*αi* ≤

_{m}_{−}_{2}

*i*1 *αixηi* ≤ *M*∗

_{m}_{−}_{2}

*i*1 *αi,* that is,*m*∗ ≤

_{m}_{−}_{2}

*i*1 *αixηi/*

_{m}_{−}_{2}

*i*1 *αix* ≤ *M*∗*.*

Because *xt* is continuous on the interval *η*1*, ηm*−2, there exists *η* ∈ *η*1*, ηm*−2 satisfying
*x*0 *αxη*, where*αm _{i}*

_{}−

_{1}2

*αi*.

**Theorem 2.6**see5. *LetM* ⊂*C*∞0*,*∞ {*x*∈*C*0*,*∞*,*lim*t*→∞*xtexists*}*. ThenMis*
*relatively compact inXif the following conditions hold:*

a*Mis uniformly bounded inC*∞0*,*∞*;*

b*the functions fromMare equicontinuous on any compact interval of*0*,*∞*;*

c*the functions fromMare equiconvergent, that is, for any given >* 0*, there exists aT*
*T>*0*such that*|*ft*−*f*∞|*< , for anyt > T, f* ∈*M.*

**3. Main Results**

Consider the space*X* {*x*∈*C*1

∞0*,*∞*, x*0 *mi*−12*αixηi,*lim*t*→∞*xt* 0}and define

the operator*T* :*X*×0*,*1 → *X*by

*Tx, λt* *λ*
_{}_{∞}

0

*Gt, sfs, xs, xsds,* 0≤*t <*∞*.* 3.1

**Theorem 3.1.** *Letf* :0*,*∞×*R*2 _{→}_{R}_{be an S-Carath´eodory function. Suppose further that there}*exists functionspt, qtrt*∈*L*1_{}_{0}_{,}_{}_{∞}_{}_{with}_{tp}_{}_{t}_{}_{, tq}_{}_{t}_{}_{tr}_{}_{t}_{}_{∈}* _{L}*1

_{}

_{0}

_{,}_{}

_{∞}

_{}

_{such that}_{f}_{}_{t, u, v}_{}_{≤}_{p}_{}_{t}_{}_{|}_{u}_{|}_{}_{q}_{}_{t}_{}_{|}_{v}_{|}_{}_{r}_{}_{t}_{} _{}3.2

*for almost everyt*∈0*,*∞*and allu, v*∈*R*2_{. Then}_{}_{1.2}_{}_{has at least one solution provided:}

*ηm*−2*PP*1*Q <*1*,* *α <*0*,*
*αηm*−2

1−*αPP*1*Q <*1*,* 0≤*α <*1*,*

max

*αηm*−2

*α*−1*PP*1*Q,*

*αP*1
*α*−1

*αηm*−2*P*
*α*−1

*<*1*,* *α >*1*.*

3.3

**Lemma 3.2.** *Let* *f* : 0*,*∞ ×*R*2 _{→} _{R}_{be an S-Carath´eodory function. Then, for each}_{λ}_{∈}

0*,*1*, Tx, λis completely continuous inX.*

*Proof.* First we show*T*is well defined. Let*x*∈*X*; then there exists*r >*0 such that*x* ≤*r*. For
each*λ*∈0*,*1, it holds that

*Tx, λt* *λ*
_{}_{∞}

0

*Gt, sfs, xs, xsds*

≤

_{}_{∞}

0

|*Gt, s*|*ϕrsds <*∞*,* ∀*t*∈0*,*∞*.*

3.4

Further,*Gt, s*is continuous in*t*so the Lebesgue dominated convergence theorem implies
that

|*Tx, λt*1−*Tx, λt*2| ≤*λ*
_{}_{∞}

0

|*Gt*1*, s*−*Gt*2*, s*|*f*

*s, xs, xsds*

≤*λ*
_{}_{∞}

0

|*Gt*1*, s*−*Gt*2*, s*|*ϕrsds*

−→0*,* as*t*1 −→*t*2*,*

3.5

_{T}_{}_{x, λ}_{}_{}_{t}

1−*Tx, λt*2≤*λ*
*t*2

*t*1

_{f}

*s, xs, xsds*

≤

_{t}_{2}

*t*1

*ϕrsds*−→0 as*t*1−→*t*2*,*

3.6

Obviously,*Tx, λ*0 _{i}m_{}−_{1}2*αiTx, ληi*. Notice that

lim

*t*→∞*Tx, λ*

_{}_{t}_{lim}

*t*→∞

_{}_{∞}

*t*

*fs, xs, xsds*0*,* 3.7

so we can get*Tx, λt*∈*X*.

We claim that*Tx, λ*is completely continuous in*X*, that is, for each*λ*∈0*,*1,*Tx, λ*

is continuous in*X*and maps a bounded subset of*X*into a relatively compact set.

Let*xn* → *x*as*n* → ∞in*X*. Next we prove that for each*λ*∈0*,*1,*Txn, λ* → *Tx, λ*

as*n* → ∞in*X*. Because*f*is a S-Carath´eodory function and

∞

0

*Gsfs, xns, xns*

−*fs, xs, xsds*≤2

_{}_{∞}

0

G*s*ϕ*r*0*sds <*∞*,* 3.8

where*r*0*>*0 is a real number such that max{max*n*∈*N*\{0}*xn,x*} ≤*r*0, we have

|*Txn, λ*∞−*Tx, λ*∞| ≤*λ*

_{}_{∞}

0

G*sfs, xns, xns*

−*fs, xs, xsds*

−→0*,* as*n*−→∞*.*

3.9

Also, we can get

|*Txn, λt*−*Txn, λ*∞| ≤*λ*

_{}_{∞}

0

G*t, s*−*Gsfs, xns, xnsds*

≤

_{}_{∞}

0

G*t, s*−*Gs*ϕ*r*0*sds*

−→0*,* as*t*−→∞*,*

3.10

_{T}_{}_{x}_{n}_{, λ}_{}_{}_{t}_{}_{−}_{T}_{}_{x}

*n, λ*∞≤

_{}_{∞}

*t*

_{f}

*s, xns, xnsds*

≤

_{}_{∞}

*t*

*ϕr*0*sds*−→0*,* as*t*−→∞*.*

3.11

Similarly, we have

|*Tx, λt*−*Tx, λ*∞| −→0*,* as*t*−→∞*,*

For any positive number*T*0*<*∞, when*t*∈0*, T*0, we have

|*Txn, λt*−*Tx, λt*| ≤

_{}_{∞}

0

|*Gt, s*|*fs, xns, xns*

−*fs, xs, xsds*

−→0*,* as*n*−→∞*,*
_{T}_{}_{x}_{n}_{, λ}_{}_{}_{t}_{}_{−}_{T}_{}_{x, λ}_{}_{}_{t}_{}_{≤}∞

*t*

_{f}

*s, xns, xns*

−*fs, xs, xsds*

−→0*,* as*n*−→∞*.*

3.13

Combining3.9–3.13, we can see that*T*·*, λ*is continuous. Let*B*⊂*X*be a bounded
subset; it is easy to prove that *TB* is uniformly bounded. In the same way, we can prove

3.5,3.6, and3.12, we can also show that*TB*is equicontinuous and equiconvergent. Thus,
byTheorem 2.6,*T*·*, λ*:*X*×0*,*1 → *X*is completely continuous. The proof is completed.

*Proof ofTheorem 3.1.* In view ofLemma 2.2, it is clear that*x*∈*X*is a solution of the BVP1.2if
and only if*x*is a fixed point of*T*·*,*1*.*Clearly,*Tx,*0 0 for each*x*∈*X.*If for each*λ*∈0*,*1
the fixed points*T*·*, λ*in*X* belong to a closed ball of*X* independent of*λ,* then the
Leray-Schauder continuation theorem completes the proof. We have known*T*·*, λ*is completely
continuous byLemma 3.2. Next we show that the fixed point of*T*·*, λ*has a priori bound*M*

independently of*λ*. Assume*xTx, λ*and set

*Q*1
_{}_{∞}

0

*sqsds,* *R*
_{}_{∞}

0

*rsds,* *R*1
_{}_{∞}

0

*srsdt.* 3.14

According toLemma 2.5, we know that for any*x*∈ *X*, there exists*η* ∈*η*1*, ηm*−2satisfying
*x*0 *αxη*. Hence, there are three cases as follow.

*Case 1α <*0. For any*x*∈*X, x*0*xη*≤0 holds and, therefore, there exists a*t*0∈0*, η*such

that*xt*0 0. Then, we have

|*xt*|_{}
*t*

*t*0

*xsds*_{}≤*tηx*_{∞}≤*tηm*−2*x*_{∞}*,* *t*∈0*,*∞*,* 3.15

and so it holds that

_{x}

∞≤*λft, x, xL*1≤*ft, x, x _{L}*1

≤*pt*|*xt*|*qt*|*xt*|*rt _{L}*1

≤*ηm*−2*PP*1*Qx*_{∞}*R,*

3.16

therefore,

_{x}

∞≤ _{1}_{−}_{η}*R*

*m*−2*P*−*P*1−*Q* *M*

At the same time, we have

|*xt*| ≤*λ*
_{∞}

0

*Gt, sfs, xs, xsds*

≤

_{∞}

0
_{sf}

*s, xs, xsds*

≤*P*1*x*∞*Q*1*M*1 *R*1*,* *t*∈0*,*∞*,*

3.18

and so

*x*_{∞}≤ *Q*1*M*

1*R*1

1−*P*1 *M*1*.*

3.19

Set*M*max{*M*_{1}*, M*1}, which is independent of*λ*.

*Case 2*0≤*α <*1. For any*x*∈*X*, we have

|*xt*|_{}*αxη*
_{t}

0

*xsds*_{}≤*αxηtx*_{∞}*,* *t*∈0*,*∞*,* 3.20

which implies that|*xt*| ≤*αη/*1−*α* *tx*_{∞}≤*αηm*−2*/*1−*α* *tx*_{∞}for all*t*∈0*,*∞.

In the same way as for Case1, we can get

_{x}

∞≤

1−*αR*

1−*α*1−*P*1−*Q*−*αηm*−2*P* *M*

2*,*

*x*_{∞}≤ *Q*1*M*

2*R*1

1−*α*−*P*1 *M*2*.*

3.21

Set*M*max{*M*_{2}*, M*2}, which is independent of*λ*and is what we need.

*Case 3α >*1. For*x*∈*X*, we have

|*xt*|_{}*xη*
*t*

*η*

*xsds*_{}≤ 1

*α*|*x*0|*t*−*ηx*

∞*,* *t*∈0*,*∞*,* 3.22

and so|*xt*| ≤*αη/α*−1 *tx*_{∞}≤*αηm*−2*/α*−1 *tx*∞for all*t*∈0*,*∞.

Similarly, we obtain

* _{x}*
∞≤

*α*−1*R*

*α*−11−*P*1−*Q*−*αηm*−2*P* *M*

3*,*

*x*_{∞}≤ *α*

*Q*1*M*_{3}*R*1

*αηm*−2

*QM*_{3} *R*
*α*−1−*αηm*−2*P* *M*3*.*

3.23

**Acknowledgment**

The Natural Science Foundation of Hebei ProvinceA2009000664 and the Foundation of Hebei University of Science and TechnologyXL200759are acknowledged.

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