Unbounded Solutions of Second-Order Multipoint Boundary Value Problem on the Half-Line

Volume 2010, Article ID 236560,15pages doi:10.1155/2010/236560

Research Article

Unbounded Solutions of Second-Order Multipoint

Boundary Value Problem on the Half-Line

Lishan Liu,

_{Xinan Hao,}

_{and Yonghong Wu}

1_{School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China}

2_{Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia}

Correspondence should be addressed to Lishan Liu,[email protected] Received 14 May 2010; Revised 4 September 2010; Accepted 11 October 2010 Academic Editor: Vicentiu Radulescu

Copyrightq2010 Lishan Liu 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.

This paper investigates the second-order multipoint boundary value problem on the half-line

ut ft, ut,ut 0,t ∈ R,αu0−βu0−n_i1kiuξi a ≥ 0, limt→∞ut b > 0, whereα >0,β > 0,ki ≥0, 0≤ξi <∞ i1,2, . . . , n, andf :R×R×R → Ris continuous. We establish sufficient conditions to guarantee the existence of unbounded solution in a special function space by using nonlinear alternative of Leray-Schauder type. Under the condition thatf

is nonnegative, the existence and uniqueness of unbounded positive solution are obtained based upon the fixed point index theory and Banach contraction mapping principle. Examples are also given to illustrate the main results.

1. Introduction

In this paper, we consider the following second-order multipoint boundary value problem on the half-line

ut ft, ut, ut0, t∈R,

αu0−βu0−

n

i1

kiuξi a≥0, lim

t→∞u

_{t b >0,} 1.1

whereα >0, β >0, ki≥0, 0< ξ1< ξ2<· · ·< ξn<∞, andf:R×R×R → Ris continuous,

in whichR 0,∞, R −∞,∞.

Moiseev2,3 and Gupta4 . Since then, great efforts have been devoted to nonlinear multi-point BVPs due to their theoretical challenge and great application potential. Many results on the existence ofpositivesolutions for multi-point BVPs have been obtained, and for more details the reader is referred to5–10 and the references therein. The BVPs on the half-line arise naturally in the study of radial solutions of nonlinear elliptic equations and models of gas pressure in a semi-infinite porous medium11–13 and have been also widely studied

14–27 . Whenn1, β0, ab0, BVP1.1reduces to the following three-point BVP on the half-line:

ut ft, ut, ut0, t∈0,∞,

u0 αuη, lim

t→∞u

_{t 0,} 1.2

where_{α /}1, η ∈ 0,∞. Lian and Ge 16 only studied the solvability of BVP 1.2by the Leray-Schauder continuation theorem. Whenki 0, i 1,2, . . . , n, and nonlinearity f is

variable separable, BVP1.1reduces to the second order two-point BVP on the half-line

u Φtft, u, u0, t∈0,∞,

au0−bu0 u0≥0, lim

t→∞u

_{t k >0.} 1.3

Yan et al.17 established the results of existence and multiplicity of positive solutions to the BVP1.3by using lower and upper solutions technique.

Motivated by the above works, we will study the existence results of unbounded

positivesolution for second order multi-point BVP1.1. Our main features are as follows. Firstly, BVP 1.1 depends on derivative, and the boundary conditions are more general. Secondly, we will study multi-point BVP on infinite intervals. Thirdly, we will obtain the unboundedpositivesolution to BVP1.1. Obviously, with the boundary condition in1.1, if the solution exists, it is unbounded. Hence, we extend and generalize the results of16,17 to some degree. The main tools used in this paper are Leray-Schauder nonlinear alternative and the fixed point index theory.

The rest of the paper is organized as follows. InSection 2, we give some preliminaries and lemmas. InSection 3, the existence of unbounded solution is established. InSection 4, the existence and uniqueness of positive solution are obtained. Finally, we formulate two examples to illustrate the main results.

2. Preliminaries and Lemmas

Denotev0t t a/bδ/Δ, whereΔ α−

_n

i1ki/0, δβ

_n

i1kiξi. Let

EC1 ∞R,R

x∈C1_{R,R: lim}

t→∞

xt

1v0t

exists, lim

t→∞x

For anyx∈E, define

x_∞max

sup

t∈R

₁x_vt_0t , sup

t∈R

_xt

, 2.2

thenEC1∞R,Ris a Banach space with the norm · ∞see17 .

The Arzela-Ascoli theorem fails to work in the Banach spaceEdue to the fact that the infinite interval0,∞is noncompact. The following compactness criterion will help us to resolve this problem.

Lemma 2.1see17 . LetM⊂EC1

∞R,R. Then,Mis relatively compact inEif the following

conditions hold:

aMis bounded inE;

bthe functions belonging to{y:yt xt/1v0t, x∈M}and{z:zt xt, x∈ M}are locally equicontinuous onR;

cthe functions from{y:yt xt/1v0t, x∈M}and{z:zt xt, x∈M} are equiconvergent, at∞.

Throughout the paper we assume the following.

H1Suppose that ft,0,0/≡0, t ∈ R, and there exist nonnegative functions pt, qt, rt∈L1_{0,∞withtpt, tqt, trt∈L}1_{0,∞such that}

_ft,1v0tu, v ≤pt|u|qt|v|rt, a.e.t, u, v∈R×R×R. 2.3

H2 Δ α−ni1ki>0. H3P1Q1<1, where

P1

_∞

0

ptdt, Q1

_∞

0

qtdt. 2.4

Denote

P2

_∞

0

1v0tptdt, Q2

_∞

0

1v0tqtdt,

R1

_∞

0

rtdt, R2

_∞

0

1v0trtdt.

2.5

Lemma 2.2. Supposing thatσt∈L1_{0,∞withtσt∈L}1_{0,∞, then BVP}

ut σt 0, t∈R,

αu0−βu0−

n

i1

kiuξi a≥0, lim

t→∞u

has a unique solution

ut

_∞

0

Gt, sσsdsa_Δbδbt, t∈R, 2.7

where

Gt, s

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

β Σj_i1kiξi Σn_ij1kis

Δ t,

t∈R, maxt, ξj

≤s≤ξj1, j 0,1,2, . . . , n,

β Σj_i1kiξi Σn_ij1kis

Δ s,

t∈R, ξj ≤s≤min

t, ξj1

, j 0,1,2, . . . , n,

2.8

in whichξ00, ξn1 ∞, and_im2_m1fi 0form2< m1.

Proof. Integrating the differential equation fromtto∞, one has

ut b

_∞

t

σsds, t∈R 2.9

Then, integrating the above integral equation from 0 tot, noticing thatσt∈L1_0,∞and tσt∈L1_{0,∞, we have}

ut u0 bt

_t

0

_∞

s

στdτ ds. 2.10

Sinceαu0−βu0−n_i1kiuξi a, it holds that

ut _Δ1

abδβ

_∞

0

σsds Σn_i1ki _ξi

0

_∞

s

στdτds

bt

_t

0

_∞

s

στdτds

_Δ1

β

_∞

0

σsds Σn_i1ki _ξi

0

sσsds Σn_i1ki

_∞

ξi

ξiσsds

t

0

sσsds

_∞

t

tσsdsbta_Δbδ.

2.11

Now, BVP1.1is equivalent to

ut

_∞

0

Gt, sfs, us, usdsa_Δbδ bt, t∈R. 2.12

Lettingvt ut−bt−abδ/Δ, t∈R,2.12becomes

vt

_∞

0

Gt, sf

s, vs a_Δbδbs, vs b

ds, t∈R. 2.13

Forv∈E, define operatorA:E → Eby

Avt

_∞

0

Gt, sf

s, vs a_Δbδbs, vs b

ds, t∈R. 2.14

Then,

Avt

_∞

t

f

s, vs a_Δbδbs, vs b

ds, t∈R. 2.15

Set

γt

⎧ ⎪ ⎨ ⎪ ⎩

t δ

Δ, t∈0,1 ,

1 δ

Δ, t∈1,∞.

2.16

Remark 2.3. Gt, sis the Green function for the following associated homogeneous BVP on the half-line:

ut ft, ut, ut0, t∈R,

αu0−βu0−

n

i1

kiuξi 0, lim

t→∞u

_{t 0.} 2.17

It is not difficult to testify that

Gt, s γt ≥

Gτ, s

1v0τ, ∀t, s, τ∈R _,

Gt, s≤Gs, s, Gt, s

1v0t ≤

1, ∀t, s∈R.

2.18

Let us first give the following result of completely continuous operator.

Proof. 1First, we show thatA:E → Eis well defined.

For anyv∈E, there existsd1>0 such thatv∞≤d1. Then,

|Avt|

1v0t ≤

_∞

0

Gt, s

1v0t

f

s, vs a_Δbδbs, vs b ds

≤

_∞

0 ps

_|

vs|

1v0sb

ds

_∞

0

qs vs bds

_∞

0

rsds

≤d1bP1Q1 R1, t∈R,

2.19

so

sup

t∈R

|Avt|

1v0t ≤d1bP1Q1 R1. 2.20

Similarly,

_Avt ∞

t

f

s, vs a_Δbδ bs, vs b

ds ≤ _∞ t

ps

_|

vs|

1v0sb

qs vs b rs

ds, t∈R,

2.21

sup

t∈R

_{Avt ≤}∞ 0

ps

_|

vs|

1v0sb

qs vs b rs

ds

≤d1bP1Q1 R1.

2.22

Further,

|Avt| ≤

_∞

0

Gt, s f

s, vs a_Δbδ bs, vs b ds

≤

_∞

0

1v0s

ps

_|

vs|

1v0sb

qs vs b rs

ds

≤d1bP2Q2 R2<∞, t∈R,

2.23

_{Avt ≤}∞ 0

f

s, vs a_Δbδ bs, vs b ds

≤

_∞

0

ps

_|

vs|

1v0sb

qs vs b rs

ds

≤d1bP1Q1 R1<∞.

On the other hand, for anyt1, t2∈Rands∈R, byRemark 2.3, we have

|Gt1, s−Gt2, s|

f

s, vs abδ

Δ bs, vs b

≤21v0s

ps

_|

vs|

1v0sb

qs vs b rs

≤21v0s

ps qsvb rs.

2.25

Hence, byH1, the Lebesgue dominated convergence theorem, and the continuity ofGt, s,

for anyt1, t2∈R, we have

|Avt1−Avt2| ≤

_∞

0

|Gt1, s−Gt2, s|

f

s, vs a_Δbδ bs, vs b ds

−→0, ast1−→t2,

_Avt

1−Avt2

_t2

t1

f

s, vs abδ

Δ bs, vs b

ds−→0, ast1−→t2.

2.26

So,Av∈C1_{R, Rfor anyv∈E.}

We can show thatAv∈E. In fact, by2.23and2.24, we obtain

lim

t→∞

|Avt|

1v0t

0, then lim

t→∞

Avt

1v0t

0,

lim

t→∞Av

_{t lim}

t→∞

_∞

t

f

s, vs a_Δbδ bs, vs b

ds0.

2.27

Hence,A:E → Eis well defined.

2We show thatAis continuous.

Suppose{vm} ⊆E, v∈E, and limm→ ∞vmv. Then,vmt → vt, vmt → vtas

m → ∞, t∈R, and there existsr0>0 such thatvm∞≤r0, m1,2, . . .,v∞≤r0. The

continuity offimplies that

f

t, vmt bta_Δbδ, vmt b

−f

t, vt bta_Δbδ, vt b −→0 2.28

asm → ∞, t∈R. Moreover, since

f

t, vmt bta_Δbδ, vm t b

−f

t, vt bta_Δbδ, vt b

≤2pt qtr0b rt

, t∈R,

we have from the Lebesgue dominated convergence theorem that

Avm−Av0_∞

max

sup

t∈R

|Avmt−Avt|

1v0t ,

sup

t∈R

_Avmt−Avt

≤

_∞

0

f

s, vms bsa_Δbδ, vm s b

−f

s, vs bsa_Δbδ, vs b ds

−→0 m−→ ∞.

2.30

Thus,A:E → Eis continuous.

3We show thatA:E → Eis relatively compact.

aLetB⊂Ebe a bounded subset. Then, there existsM >0 such thatv_∞≤Mfor all

v∈B. By the similar proof of2.20and2.22, ifv∈B, one has

Av_∞≤MbP1Q1 R1, 2.31

which implies thatABis uniformly bounded.

bFor anyT >0, ift1, t2∈0, T , v∈B, we have

Avt1

1v0t1−

Avt2

1v0t2

≤

_∞

0

Gt1, s

1v0t1−

Gt2, s

1v0t2

f

s, vs bsa_Δbδ, vs b ds

≤2

_∞

0

f

s, vs bs a_Δbδ, vs b ds

≤2MbP1Q1 R1 ,

_Avt

1−Avt2

≤

t2

t1

f

s, vs bs a_Δbδ, vs b ds

≤MbP1Q1 R1.

2.32

then

Avt1

1v0t1−

Avt2

1v0t2

< ε,

_Avt

1−Avt2 < ε.

2.33

Since T is arbitrary, then {ABt/1 v0t} and {ABt} are locally

equi-continuous onR.

cForv∈B, from2.27, we have

lim

t→∞

₁Av_v0t_t− lim

s→∞

Avs

1v0s

lim

t→∞

₁Av_v0t_t 0,

lim

t→∞

Avt− lim

s→∞Av

_{s lim}

t→∞ Av

_{t 0,}

2.34

which means that {ABt/1 v0t} and {ABt} are equiconvergent at ∞. By Lemma 2.1,A:E → Eis relatively compact.

Therefore,A:E → Eis completely continuous. The proof is complete.

Lemma 2.5see28,29 . LetEbe Banach space,Ωbe a bounded open subset of E, θ ∈ Ω, and A : Ω → Ebe a completely continuous operator. Then either there existx ∈ ∂Ω, λ > 1such that Fx λx, or there exists a fixed pointx∗∈Ω.

Lemma 2.6see28,29 . LetΩbe a bounded open set in real Banach spaceE, letP be a cone of

E, θ∈Ω, and letA:Ω∩P → Pbe completely continuous. Suppose that

λAx /x, ∀x∈∂Ω∩P, λ∈0,1 . 2.35

Then,

iA,Ω∩P, P 1. 2.36

3. Existence Result

In this section, we present the existence of an unbounded solution for BVP1.1by using the Leray-Schauder nonlinear alternative.

Proof. Sinceft,0,0/≡0, byH1, we havert ≥ |ft,0,0|, a.e.t ∈ R, which implies that R1 >0. Set

R bP1Q1 R1

1−P1−Q1 , ΩR{v∈E

:v_∞< R}. 3.1

From Lemmas2.2and2.4, BVP1.1has a solutionvvtif and only ifvis a fixed point of

AinE. So, we only need to seek a fixed point ofAinE. Supposev∈∂ΩR, λ >1 such thatAvλv. Then

λRλv_∞Av_∞max

sup

t∈R

|Avt|

1v0t,

sup

t∈R

_Avt

≤

_∞

0

f

s, vs bsa_Δbδ, vs b ds

≤P1Q1v_∞ P1Q1bR1

P1Q1R P1Q1bR1.

3.2

Therefore,

λ≤P1Q1 P1Q1bR1

R 1, 3.3

which contradictsλ >1. ByLemma 2.5,Ahas a fixed pointv∗ ∈ΩR. Lettingu∗t v∗t

bt abδ/Δ, t ∈ R, boundary conditions imply thatu∗is an unbounded solution of BVP1.1.

4. Existence and Uniqueness of Positive Solution

In this section, we restrict the nonlinearityf ≥0 and discuss the existence and uniqueness of positive solution for BVP1.1.

Define the coneP ⊂Eas follows:

P

u∈E:ut≥γtsup

s∈R

₁u_vs_0s , t∈R, u0

1v00 ≥ β

δ Δ a/bsups∈R _us

.

4.1

Lemma 4.1. Suppose thatH1andH2hold. Then,A:P → Pis completely continuous.

we have

Avt

_∞

0

Gt, sf

s, vs a_Δbδbs, vs b

ds

≥γt

_∞

0

Gτ, s

1v0τf

s, vs a_Δbδ bs, vs b

ds

γt

_∞

0 Gτ, sfs, vs abδ/Δ bs, vs bds

1v0τ

γt Avτ

1v0τ, ∀t, τ∈R _.

4.2

Then,

Avt≥γtsup

τ∈R

Avτ

1v0τ, t∈R _,

Av0 1v00

_∞

0 G0, sfs, vs abδ/Δ bs, vs bds

1v00

≥

β/Δ ₀∞fs, vs abδ/Δ bs, vs bds

1 a/bδ/Δ

≥ β

δ Δ a/bsupt∈R

_{Avt .}

4.3

Therefore,AP⊂P.

Theorem 4.2. Suppose that conditionsH2andH3hold and the following condition holds: H₁suppose thatft,0,0, tft,0,0∈L1_{0,∞, ft,0,0/≡0and there exist nonnegative}

functionspt, qt∈L1_{0,∞withtpt, tqt∈L}1_{0,∞such that}

_ft,1v0tu1, v1−ft,1v0tu2, v2

≤pt|u1−u2|qt|v1−v2|, a.e.t, ui, vi∈R×R×R, i1,2.

4.4

Then, BVP1.1has a unique unbounded positive solution.

Proof. We first show thatH₁impliesH1. By4.4, we have

_ft,1v0tu, v ≤pt|u|qt|v| ft,0,0 , a.e.t, u, v∈R×R×R. 4.5

ByLemma 4.1,A:P → Pis completely continuous. LetR₀∞ft,0,0dt. Then,R >0. Set

R >bP1Q1 R

1−P1−Q1 , Ω {v∈E

For anyv∈P∩∂Ω, by4.5, we have

|Avt|

1v0t

∞

0

Gt, s

1v0tf

s, vs bsa_Δbδ, vs b

ds

≤RbP1Q1 R < R, t∈R,

_Avt ∞

t

f

s, vs bsa_Δbδ, vs b

ds

≤

_∞

0

f

s, vs bs a_Δbδ, vs b ds

≤RbP1Q1 R < R, t∈R.

4.7

Therefore,Av_∞<v_∞,for allv∈P∩∂Ω, that is,_{λAv /}vfor anyλ∈0,1 , v∈P∩∂Ω. Then,Lemma 2.6yieldsiA, P∩Ω, P 1, which implies thatAhas a fixed pointv∗∈P∩Ω. Letu∗t v∗t bt abδ/Δ, t ∈R. Then,u∗is an unbounded positive solution of BVP1.1.

Next, we show the uniqueness of positive solution for BVP1.1. We will show thatA

is a contraction. In fact, by4.4, we have

Av1−Av2∞

max

sup

t∈R

|Av1t−Av2t|

1v0t , supt∈R

_Av1t−Av 2t

≤

_∞

0

f

s, v1s bsa_Δbδ, v1s b

−f

s, v2s bsa_Δbδ, v2s b

ds

≤

_∞

0

ps|v1s−v2s|

1v0s qs

_v

1s−v2s

ds

≤P1Q1v1−v2_∞.

4.8

So, A is indeed a contraction. The Banach contraction mapping principle yields the uniqueness of positive solution to BVP1.1.

5. Examples

Example 5.1. Consider the following BVP:

ut 2e−4t u2t

1u2_t2e

−3t ut

3

1 ut4 −

arctant

1t3 0, t∈R _,

89u0−3u0−

7

i1

iu

i3 4

2, lim

t→∞u

_{t 1,}

We have

Δ 89−

7

i1

i61, δ3

7

i1

ii3

4 59, v0t t

259

61 t1,

ft, x, y2e−4t x 2

1x2 2e −3t y3

1y4 −

arctant

1t3 ∈CR

_×R×R,R.

_f

t,2tx, y ≤2te−4t|x|e−3t y arctant

1t3 .

5.2

Let

pt 2te−4t, qt e−3t, rt arctant

1t3 . 5.3

Then,pt, qt, rt∈ L1_{0,∞, tpt, tqt, trt ∈L}1_{0,∞, and it is easy to prove thatH} 1

is satisfied. By direct calculations, we can obtain thatP1 9/16, Q1 1/3, P1Q1 < 1. By Theorem 3.1, BVP5.1has an unbounded solution.

Example 5.2. Consider the following BVP:

ut 1 2t2e

−4t u2t

1u2_t

1 4e

−3t |ut|

3

1 ut4

arctant

1t3 0, t∈R _,

89u0−3u0−

7

i1

iu

i3 4

2, lim

t→∞u

_{t 1.}

5.4

In this case, we have

Δ 61, δ59, v0t t1,

ft, x, y 1 2t2e

−4t x2

1x2

1 4e

−3t y

3

1y4

arctant

1t3 ∈CR

_×R×R,R,

_f

t,2tx1, y1

−ft,2tx2, y2 ≤2e−4t|x1−x2|

3 4e

−3t _y

1−y2 .

5.5

Let

pt 2e−4t_{, qt} 3

4e

−3t_{. 5.6}

Then,P1 1/2, Q1 1/4, P1Q1 <1. ByTheorem 4.2, BVP5.4has a unique unbounded

Acknowledgments

The authors are grateful to the referees for valuable suggestions and comments. The first and second authors were supported financially by the National Natural Science Foundation of China11071141, 10771117 and the Natural Science Foundation of Shandong Province of ChinaY2007A23, Y2008A24. The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.

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