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

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,[email protected]

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, whereαi ∈R,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

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, xt0, 0< t <∞,

x0

m−2

i1 αix

ηi

, lim

t→∞x

_{t 0,} 1.2

whereαi ∈ R,

_m−2

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,∞ {x∈C10,∞,limt→∞xtexists, limt→∞xtexists}

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

L1_{0,∞ {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, v∈R2_{, t→ft, u, vis measurable on0,∞,}

iifor almost everyt∈0,∞,u, v→ft, u, vis continuous onR2_,

iiifor eachr > 0, there existsϕrt ∈L10,∞withtϕrt ∈L10,∞, ϕrt > 0 on 0,∞such that max{|u|,|v|} ≤rimplies|ft, u, v| ≤ϕrt, for a.e.t∈0,∞.

Lemma 2.2. Supposem_i−₁2αi/1,if for anyvt ∈ L10,∞with tvt ∈ L10,∞, then the

BVP,

xt vt 0, 0< t <∞,

x0

m−2

i1 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.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.

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

lim

t→∞Gt, s Gs

_Λ1

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

s, s≤η1,

i

k1 αkηk

_m−2

ki1 αk Λ

s, ηi ≤s≤ηi1<∞, i1,2, . . . , m−3

m−2

i1

αiηi Λs, s≥ηm−2.

2.13

Lemma 2.5. For the functionx∈C10,∞,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 η∈η1, ηm−2satisfying

x0 αxη, 2.15

whereαmi−12αi.

Proof. Letαi i1,2, . . . , m−2are positive, and noteM∗ max{xt|t∈η1, ηm−2}, m∗

min{xt | t ∈ η1, ηm−2}, then for everyii 1,2, . . . , m−2, we havem∗ ≤ xηi ≤ M∗,

so m∗mi−12αi ≤

_m−2

i1 αixηi ≤ M∗

_m−2

i1 αi, that is,m∗ ≤

_m−2

i1 αixηi/

_m−2

i1 αix ≤ M∗.

Because xt is continuous on the interval η1, ηm−2, there exists η ∈ η1, ηm−2 satisfying x0 αxη, whereαm_i−₁2αi.

Theorem 2.6see5. LetM ⊂C∞0,∞ {x∈C0,∞,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|ft−f∞|< , for anyt > T, f ∈M.

3. Main Results

Consider the spaceX {x∈C1

∞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

Theorem 3.1. Letf :0,∞×R2 _{→Rbe an S-Carath´eodory function. Suppose further that there} exists functionspt, qtrt∈L1_{0,∞withtpt, tqttrt∈L}1_{0,∞such that}

_{ft, u, v≤pt|u|qt|v|rt} 3.2

for almost everyt∈0,∞and allu, v∈R2_{. 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. Letx∈X; then there existsr >0 such thatx ≤r. 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, s−Gt2, s|f

s, xs, xsds

≤λ _∞

0

|Gt1, s−Gt2, 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

Obviously,Tx, λ0 _im−₁2αiTx, ληi. Notice that

lim

t→∞Tx, λ

_{t lim}

t→∞

_∞

t

fs, xs, xsds0, 3.7

so we can getTx, λt∈X.

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.

Letxn → xasn → ∞inX. Next we prove that for eachλ∈0,1,Txn, λ → Tx, 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{maxn∈N\{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, λt−Txn, λ∞| ≤λ

_∞

0

Gt, s−Gsfs, xns, xnsds

≤

_∞

0

Gt, s−Gsϕr0sds

−→0, ast−→∞,

3.10

_{Txn, λt−Tx}

n, λ∞≤

_∞

t

_f

s, xns, xnsds

≤

_∞

t

ϕr0sds−→0, ast−→∞.

3.11

Similarly, we have

|Tx, λt−Tx, λ∞| −→0, ast−→∞,

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

|Txn, λt−Tx, λt| ≤

_∞

0

|Gt, s|fs, xns, xns

−fs, xs, xsds

−→0, asn−→∞, _{Txn, λt−Tx, λt≤}∞

t

_f

s, xns, xns

−fs, xs, xsds

−→0, asn−→∞.

3.13

Combining3.9–3.13, we can see thatT·, λis continuous. LetB⊂Xbe 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 thatx∈Xis a solution of the BVP1.2if and only ifxis a fixed point ofT·,1.Clearly,Tx,0 0 for eachx∈X.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 anyx∈ X, there existsη ∈η1, ηm−2satisfying x0 αxη. Hence, there are three cases as follow.

Case 1α <0. For anyx∈X, x0xη≤0 holds and, therefore, there exists at0∈0, ηsuch

thatxt0 0. Then, we have

|xt| t

t0

xsds≤tηx_∞≤tηm−2x_∞, t∈0,∞, 3.15

and so it holds that

_x

∞≤λft, x, xL1≤ft, x, x_L1

≤pt|xt|qt|xt|rt_L1

≤ηm−2PP1Qx_∞R,

3.16

therefore,

_x

∞≤ _1−η R

m−2P−P1−Q M

At the same time, we have

|xt| ≤λ _∞

0

Gt, sfs, xs, xsds

≤

_∞

0 _sf

s, xs, xsds

≤P1x∞Q1M1 R1, t∈0,∞,

3.18

and so

x_∞≤ Q1M

1R1

1−P1 M1.

3.19

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

Case 20≤α <1. For anyx∈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 allt∈0,∞.

In the same way as for Case1, we can get

_x

∞≤

1−αR

1−α1−P1−Q−αηm−2P M

2,

x_∞≤ Q1M

2R1

1−α−P1 M2.

3.21

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

Case 3α >1. Forx∈X, we have

|xt|xη t

η

xsds≤ 1

α|x0|t−ηx

∞, t∈0,∞, 3.22

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

Similarly, we obtain

_x ∞≤

α−1R

α−11−P1−Q−αηm−2P M

3,

x_∞≤ α

Q1M₃R1

αηm−2

QM₃ R α−1−αηm−2P M3.

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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