Existence of Solutions for point Boundary Value Problems on a Half Line

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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-linext ft, xt, xt 0,0 < t <, x0

m−2

i1 αixηi,limt→∞xt 0, whereαiR,im−12αi/1 and 0< η1< η2<· · ·< 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, xt0, 0< t <,

x0 αxη, lim

t→∞x

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

m-point boundary value problems:

xt ft, xt, xt0, 0< t <,

x0

m−2

i1 αix

ηi

, lim

t→∞x

t 0, 1.2

whereαiR,

m2

i1 αi/1, αihave the same signal, and 0 < η1 < η2 < · · · < 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 spaceC1

∞0,∞ {xC10,,limt→∞xtexists, limt→∞xtexists}

with the normxmax{x,x}, where · is supremum norm on the half-line, and

L10, {x : 0, Ris absolutely integrable on0,}with the normx

L1

0|xt|dt.

We set

P

0

psds, P1

0

spsds, Q

0

qsdt, 1.3

and we supposeαi, i1,2, . . . , m−2 are the same signal in this paper and we always assume

αmi−12αi.

2. Preliminary Results

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

Definition 2.1see15. It holds thatf : 0,∞×R2 −→ Ris called an S-Carath´eodory

function if and only if

ifor eachu, vR2, tft, u, vis measurable on0,,

iifor almost everyt∈0,,u, vft, u, vis continuous onR2,

iiifor eachr > 0, there existsϕrtL10,∞withtϕrtL10,, ϕrt > 0 on 0,∞such that max{|u|,|v|} ≤rimplies|ft, u, v| ≤ϕrt, for a.e.t∈0,∞.

Lemma 2.2. Supposemi12αi/1,if for anyvtL10,with tvtL10,, then the

BVP,

xt vt 0, 0< t <,

x0

m−2

i1 αix

ηi

, lim

t→∞x

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Therefore, the unique solution of 2.1 is xt 0Gt, svsds, which completes the proof.

Remark ofLemma 2.2.ObviouslyGt, ssatisfies the properties of a Green function, so we call

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

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

lim

t→∞Gt, s Gs

Λ1

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

s, sη1,

i

k1 αkηk

m2

ki1 αk Λ

s, ηisηi1<, i1,2, . . . , m−3

m−2

i1

αiηi Λs, sηm−2.

2.13

Lemma 2.5. For the functionxC10,,it is satisfied that

x0

m−2

i1 αix

ηi

2.14

andαi i1,2, . . . , m−2have the same signal,0< η1 < η2 <· · ·< ηm−2<, then there exists ηη1, ηm−2satisfying

x0 αxη, 2.15

whereαmi−12αi.

Proof. Letαi i1,2, . . . , m−2are positive, and noteM∗ max{xt|tη1, ηm−2}, m

min{xt | tη1, ηm−2}, then for everyii 1,2, . . . , m−2, we havem∗ ≤ xηiM,

so mmi−12αi

m2

i1 αixηiM

m2

i1 αi, that is,m∗ ≤

m2

i1 αixηi/

m2

i1 αixM.

Because xt is continuous on the interval η1, ηm−2, there exists ηη1, ηm−2 satisfying x0 αxη, whereαmi12αi.

Theorem 2.6see5. LetMC∞0,∞ {xC0,,limt→∞xtexists}. ThenMis relatively compact inXif the following conditions hold:

aMis uniformly bounded inC∞0,;

bthe functions fromMare equicontinuous on any compact interval of0,;

cthe functions fromMare equiconvergent, that is, for any given > 0, there exists aT T>0such that|ftf∞|< , for anyt > T, fM.

3. Main Results

Consider the spaceX {xC1

∞0,, x0 mi−12αixηi,limt→∞xt 0}and define

the operatorT :X×0,1 → Xby

Tx, λt λ

0

Gt, sfs, xs, xsds, 0≤t <. 3.1

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Theorem 3.1. Letf :0,∞×R2 Rbe an S-Carath´eodory function. Suppose further that there exists functionspt, qtrtL10,withtpt, tqttrtL10,such that

ft, u, vpt|u|qt|v|rt 3.2

for almost everyt∈0,and allu, vR2. Then1.2has at least one solution provided:

ηm−2PP1Q <1, α <0, αηm−2

1−αPP1Q <1, 0≤α <1,

max

αηm−2

α−1PP1Q,

αP1 α−1

αηm−2P α−1

<1, α >1.

3.3

Lemma 3.2. Let f : 0,∞ ×R2 R be an S-Carath´eodory function. Then, for each λ

0,1, Tx, λis completely continuous inX.

Proof. First we showTis well defined. LetxX; then there existsr >0 such thatxr. For eachλ∈0,1, it holds that

Tx, λt λ

0

Gt, sfs, xs, xsds

0

|Gt, s|ϕrsds <,t∈0,.

3.4

Further,Gt, sis continuous intso the Lebesgue dominated convergence theorem implies that

|Tx, λt1−Tx, λt2| ≤λ

0

|Gt1, sGt2, s|f

s, xs, xsds

λ

0

|Gt1, sGt2, s|ϕrsds

−→0, ast1 −→t2,

3.5

Tx, λt

1−Tx, λt2≤λ t2

t1

f

s, xs, xsds

t2

t1

ϕrsds−→0 ast1−→t2,

3.6

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Obviously,Tx, λ0 im12αiTx, ληi. Notice that

lim

t→∞Tx, λ

t lim

t→∞

t

fs, xs, xsds0, 3.7

so we can getTx, λtX.

We claim thatTx, λis completely continuous inX, that is, for eachλ∈0,1,Tx, λ

is continuous inXand maps a bounded subset ofXinto a relatively compact set.

Letxnxasn → ∞inX. Next we prove that for eachλ∈0,1,Txn, λTx, λ

asn → ∞inX. Becausefis a S-Carath´eodory function and

0

Gsfs, xns, xns

fs, xs, xsds≤2

0

Gsϕr0sds <, 3.8

wherer0>0 is a real number such that max{maxnN\{0}xn,x} ≤r0, we have

|Txn, λ∞−Tx, λ∞| ≤λ

0

Gsfs, xns, xns

fs, xs, xsds

−→0, asn−→∞.

3.9

Also, we can get

|Txn, λtTxn, λ∞| ≤λ

0

Gt, sGsfs, xns, xnsds

0

Gt, sGsϕr0sds

−→0, ast−→∞,

3.10

Txn, λtTx

n, λ∞≤

t

f

s, xns, xnsds

t

ϕr0sds−→0, ast−→∞.

3.11

Similarly, we have

|Tx, λtTx, λ∞| −→0, ast−→∞,

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For any positive numberT0<∞, whent∈0, T0, we have

|Txn, λtTx, λt| ≤

0

|Gt, s|fs, xns, xns

fs, xs, xsds

−→0, asn−→∞, Txn, λtTx, λt

t

f

s, xns, xns

fs, xs, xsds

−→0, asn−→∞.

3.13

Combining3.9–3.13, we can see thatT·, λis continuous. LetBXbe 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 thatTBis equicontinuous and equiconvergent. Thus, byTheorem 2.6,T·, λ:X×0,1 → Xis completely continuous. The proof is completed.

Proof ofTheorem 3.1. In view ofLemma 2.2, it is clear thatxXis a solution of the BVP1.2if and only ifxis a fixed point ofT·,1.Clearly,Tx,0 0 for eachxX.If for eachλ∈0,1 the fixed pointsT·, λinX belong to a closed ball ofX independent ofλ, then the Leray-Schauder continuation theorem completes the proof. We have knownT·, λis completely continuous byLemma 3.2. Next we show that the fixed point ofT·, λhas a priori boundM

independently ofλ. AssumexTx, λand set

Q1

0

sqsds, R

0

rsds, R1

0

srsdt. 3.14

According toLemma 2.5, we know that for anyxX, there existsηη1, ηm−2satisfying x0 αxη. Hence, there are three cases as follow.

Case 1α <0. For anyxX, x0≤0 holds and, therefore, there exists at0∈0, ηsuch

thatxt0 0. Then, we have

|xt| t

t0

xsdstηxtηm−2x, t∈0,, 3.15

and so it holds that

x

∞≤λft, x, xL1≤ft, x, xL1

pt|xt|qt|xt|rtL1

ηm−2PP1QxR,

3.16

therefore,

x

∞≤ 1η R

m−2PP1−Q M

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At the same time, we have

|xt| ≤λ

0

Gt, sfs, xs, xsds

0 sf

s, xs, xsds

P1xQ1M1 R1, t∈0,,

3.18

and so

xQ1M

1R1

1−P1 M1.

3.19

SetMmax{M1, M1}, which is independent ofλ.

Case 20≤α <1. For anyxX, we have

|xt|αxη t

0

xsdsαxηtx, t∈0,, 3.20

which implies that|xt| ≤αη/1−α txαηm−2/1−α txfor allt∈0,∞.

In the same way as for Case1, we can get

x

∞≤

1−αR

1−α1−P1−Qαηm−2P M

2,

xQ1M

2R1

1−αP1 M2.

3.21

SetMmax{M2, M2}, which is independent ofλand is what we need.

Case 3α >1. ForxX, we have

|xt| t

η

xsds≤ 1

α|x0|tηx

, t∈0,, 3.22

and so|xt| ≤αη/α−1 txαηm−2−1 tx∞for allt∈0,∞.

Similarly, we obtain

x ∞≤

α−1R

α−11−P1−Qαηm−2P M

3,

xα

Q1M3R1

αηm−2

QM3 R α−1−αηm−2P M3.

3.23

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Acknowledgment

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

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