A Global Description of the Positive Solutions of Sublinear Second Order Discrete Boundary Value Problems

doi:10.1155/2009/671625

Research Article

A Global Description of the Positive

Solutions of Sublinear Second-Order Discrete

Boundary Value Problems

Ruyun Ma, Youji Xu, and Chenghua Gao

Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, China

Correspondence should be addressed to Ruyun Ma,[email protected]

Received 12 February 2009; Accepted 20 August 2009

Recommended by Svatoslav Stanˇek

LetT∈Nbe an integer withT >1,T:{1, . . . , T},T:{0,1, . . . , T1}. We consider boundary value problems of nonlinear second-order difference equations of the formΔ2_{ut−1 λatfut 0,}

t ∈ T,u0 uT1 0, wherea : T → R,f ∈ C0,∞,0,∞ and,fs > 0 fors > 0, andf0f∞0,f0lims→0fs/s,f_∞lims_→∞fs/s. We investigate the global structure of positive solutions by using the Rabinowitz’s global bifurcation theorem.

Copyrightq2009 Ruyun Ma 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

LetT ∈Nbe an integer withT >1,T:{1, . . . , T},T :{0,1, . . . , T 1}. We study the global structure of positive solutions of the problem

Δ2_{ut−1 λatfut 0, t∈T,}

u0 uT1 0. 1.1

Hereλis a positive parameter,a:T → R andf :0,∞ → 0,∞are continuous. Denote f0lims→0fs/sandf_∞lim_s→∞fs/s.

There are many literature dealing with similar difference equations subject to various boundary value conditions. We refer to Agarwal and Henderson1 , Agarwal and O’Regan

of sign-changing solutions of some discrete boundary value problems in the case thatf0 ∈ 0,∞. However, relatively little result is known about the global structure of solutions in the case thatf00, and no global results were found in the available literature whenf0 0f∞.

The likely reason is that the Rabinowitz’s global bifurcation theorem 10 cannot be used directly in this case.

In the present work, we obtain a direct and complete description of the global structure of positive solutions of1.1under the assumptions:

A1a:T → 0,∞;

A2f :0,∞ → 0,∞is continuous andfs>0 fors >0;

A3f00, wheref0lims→0fs/s;

A4f∞0, wheref∞lims→∞fs/s.

LetYdenote the Banach space defined by

Y y|y: T−→R 1.2

equipped with the norm

y_Y max

t∈T yt. 1.3

LetEdenote the Banach space defined by

Eu:T−→R|u0 uT1 0 1.4

equipped with the norm

u₀max

t∈T |ut|. 1.5

Define an operatorL:E → Y by

Lut −Δ2_{ut−1, t∈T. 1.6}

To state our main results, we need the spectrum theory of the linear eigenvalue problem

Δ2_{ut−1 λatut 0, t∈T,}

u0 uT1 0. 1.7

Lemma 1.15,11 . Let (A1) hold. Then there exists a sequence{λn}nT1∈0,Rsatisfying that i{λn|n∈ {1,2, . . . , T}}is the set of eigenvalues of1.7;

iiifork∈ {1,2, . . . , T},kerL−λkIis one-dimensional subspace ofE;

ivfor each k ∈ {1,2, . . . , T}, if v ∈ kerL−λkI\ {0}, thenv has exactlyk−1 simple generalized zeros in0, T .

LetΣdenote the closure of set of positive solutions of 1.1in0,∞×E.

LetMbe a subset ofE. A component ofMis meant a maximal connected subset ofM, that is, a connected subset ofMwhich is not contained in any other connected subset ofM.

The main results of this paper are the following theorem.

Theorem 1.2. Let (A1)–(A4) hold. Then there exists a componentζin Σwhich joins ∞, θwith

∞,∞, and

Proj_Rζ ρ∗,∞ 1.8

for someρ∗>0. Moreover, there existsλ∗≥ρ∗>0such that1.1has at least two positive solutions forλ∈λ∗,∞.

We will develop a bifurcation approach to treat the casef00 directly. Crucial to this

approach is to construct a sequence of functions{fn} which is asymptotic linear at 0 and satisfies

fn _{−→f, f}n 0>0,

fn

0−→0. 1.9

By means of the corresponding auxiliary equations, we obtain a sequence of unbounded components{Cn}via Rabinnowitz’s global bifurcation theorem 10 , and this enables us to find an unbounded componentCsatisfying

C ⊂lim sup

n→ ∞ C n

. 1.10

2. Some Preliminaries

In this section, we give some notations and preliminary results which will be used in the proof of our main results.

Definition 2.1see12 . LetX be a Banach space, and let{Cn | n 1,2, . . .}be a family of

subsets ofX. Then the superior limitDof{Cn}is defined by

D:lim sup

n→ ∞ Cn{x∈X| ∃{ni} ⊂Nand xni∈Cni, such that xni −→x}. 2.1

Definition 2.2see12 . Acomponentof a setMis meant a maximal connected subset ofM.

Using the above Whyburn’s lemma, Ma and An13 proved the following lemma.

Lemma 2.413, Lemma 2.2 . LetX be a Banach space, and let {Cn}be a family of connected subsets ofX. Assume that

ithere existzn∈Cn,n1,2, . . ., andz∗∈X, such thatzn → z∗; iilimn→ ∞rn∞, wherernsup{x |x∈Cn};

iiifor everyR >0,∪∞_n1Cn∩BRis a relatively compact set ofX, where

BR {x∈X| x ≤R}. 2.2

Then there exists an unbounded componentCinDandz∗∈ C.

Let

Gt, s 1 T1

⎧ ⎨ ⎩

T1−ts, 0≤s≤t≤T1,

tT1−s, 0≤t≤s≤T1. 2.3

It is easy to see that

Gt, s≥ 1

T1Gs, s, t, s∈T×T. 2.4

Denote the coneKinEby

K

x∈E|ut≥0 onT, and min

t∈T ut≥ 1 T1u

. 2.5

Now we define a mapAλ:K → Y by

Aλut λ

T

s1

Gt, sasfus, t∈T. 2.6

Define an operatorj:Y → Eby

jy1, . . . , yT0, y1, . . . , yT,0, ∀y1, . . . , yT∈Y. 2.7

Then the operatorTλ:j◦AλsatisfiesTλ:E → E.

Forr >0, let

Using the standard arguments, we may prove the following lemma.

3. Proof of the Main Results

Definefn :0,∞ → 0,∞by

To apply the global bifurcation theorem, we extendfto be an odd functiong:R → Rby

Now let us consider the auxiliary family of the equations

Δ2_{ut−1 λatg}n_{u 0, t∈T,}

u0 uT1 0. 3.5

Letξn ∈CRbe such that

gn_{u g}n 0uξ

n_{u nf}1

n

uξn_{u. 3.6}

Then

lim

|u| →0

ξn_u

u 0. 3.7

Let us consider

Lu−λatgn

0uλatξ

n_{u 3.8}

as a bifurcation problem from the trivial solutionu≡0.

Equation3.8can be converted to the equivalent equation

ut T

s1

Gt, sλasgn

0us λasξ

n_us

:λL−1a·gn

0u·

t λL−1_a·ξn_u·t.

3.9

Further we note thatL−1a·ξnu· ouforunearθinE.

The results of Rabinowitz10 for3.8can be stated as follows. For each integern≥1, ν∈ {,−}, there exists a continuumCνn of solutions of3.8joiningλ1/gn0, θto infinity

in0,∞×νK. Moreover,Cνn \ {λ1/gn0, θ} ⊂0,∞×νintK.

Lemma 3.1. Let (A1)–(A4) hold. Then, for each fixedn, Cn joins λ1/gn0, θ to∞,∞ in

0,∞×K.

Proof. We divide the proof into two steps.

Step 1. We show that sup{λ|λ, u∈Cn}∞.

Assume on the contrary that sup{λ |λ, u∈Cn }:c0 <∞. Let{ηk, yk} ⊂Cn be

such that

_ηkyk

Thenyk₀ → ∞. This together with the fact

min

t∈T ykt≥ 1

T1yk0 3.11

implies that

lim

k→ ∞ykt ∞, uniformly fort∈T. 3.12

Sinceηk, yk∈Cn, we have that

Δ2_y

kt−1 ηkatgnykt0, t∈T,

yk0 ykT1 0.

3.13

Setvkt ykt/yk0. Then

vk0 1. 3.14

Now, choosing a subsequence and relabelling if necessary, it follows that there existsη∗, v∗∈

0, c0 ×Ewith

v_∗01, 3.15

such that

lim

k→ ∞

ηk, vkη∗, v∗, inR×E. 3.16

Moreover, using3.13,3.12, and the assumptionf_∞0, it follows that

Δ2_v

∗t−1 η∗at·00, t∈T,

v_∗0 v_∗T1 0, 3.17

and consequently,v∗t≡0 fort∈T. This contradicts 3.15. Therefore

supλ|λ, y∈ C∞. 3.18

Step 2. We show that sup{||u||0 | λ, u∈Cn}∞.

Assume on the contrary that sup{||u||0 | λ, u ∈ Cn} : M0 < ∞. Let{ηk, yk} ⊂

Cn be such that

Sinceηk, yk∈Cn, for anyt∈T, we have from2.6that exists a positive constantM, such that

supy₀|η, y∈Cn, η∈I≤M. 3.21

Now, choosing a subsequence and relabeling if necessary, it follows that there existsη∗, v∗∈

Moreover, from3.13,3.12, and the assumptionf∞0, it follows that

Lemma 3.3. Let (A1)–(A4) hold. Then there exitsρ_∗>0such that

Proof ofTheorem 1.2. Taker 1 . Let ρ∗ be as in Lemma 3.3, and letλ∗ be a fixed constant satisfyingλ∗≥ρ∗and

λ∗_mn 1

T

s1

G1, sas>1, 3.35

where

m₁n min

1/T1≤x≤1

gn_{x. 3.36}

It is easy to see that there existsn0 ∈N, such that

1 n0

< 1

T1. 3.37

This implies that

m₁n m1, ∀n > n0 3.38

see2.10for the definition ofm1, and accordingly, we may chooseλ∗which is independent

ofn > n0. FromLemma 2.6and3.35, it follows that forλ > λ∗,

Tλu0>u0, u∈∂Ω1. 3.39

This together with the compactness ofTλimplies that there exists∈0,1/2, such that

Cn ∩η, u|η≥λ∗_{; u∈K: 1−2≤ u}

0≤12

∅, ∀n > n0. 3.40

Notice that {Cn} satisfies all conditions in Lemma 2.4, and consequently, lim sup_{n→ ∞} Cn contains a component ζwhich is unbounded. However, we do not know whether ζ joins ∞, θ with ∞,∞ or not. To answer this question, we have to use the following truncation method.

Set

Γ: 0,∞×K\η, u|η≥λ∗; u∈K:u₀≤1. 3.41

Forn∈Nwithλ1/gn0 ≥λ∗, we defineζ n

0 a connected subset inC n

satisfying

1ζ₀n ⊂Cn \λ∗,∞×Ω1;

2ζ₀n joins{λ∗} ×Ω1with infinity inΓ.

Since

lim

n→ ∞

λ1

gn

0

lim

n→ ∞

λ1

nf1/n ∞, 3.42

we have from Lemmas3.1–3.3and3.40that forn > n0andλ1/gn0≥λ∗,

ζ₀n ∩{λ∗_{} ×Ω}

1−/∅. 3.43

Thus, there existsznj ∈ζ

nj

0 ∩{λ∗} ×Ω1−, such thatznj → z∗, and accordingly, conditioni

inLemma 2.4is satisfied. Obviously,

rnsupηy₀|η, y∈ζ₀n

∞, 3.44

that is, condition ii in Lemma 2.4 holds. Condition iii in Lemma 2.4 can be deduced directly from the Arzel`a -Ascoli theorem and the definition ofgn. Therefore, the superior limit of{ζ₀n}contains a componentζ0joining{λ∗} ×Ω1with infinity inΓ.

Similarly, for eachj ∈N, we may define a connected subset,ζ_jn , inCn satisfying

1ζ_jn ⊂Cn \λ∗j,∞×Ω1;

2ζ_jn joins{λ∗j} ×Ω1with infinity inΓ,

and the superior limit of{ζ_jn}contains a componentζjjoining{λ∗j}×Ω1with infinity inΓ.

It is easy to verify that

ζk⊆Σ, k0,1,2, . . . . 3.45

Now, for eachλ∗, v∈{λ∗} ×Ω1∩Σ, letEv ⊂Σbe a connected component containing λ∗_{, v. Let}

μv:sup{λ|λ, u∈ Ev, u∈Ω1}. 3.46

Set

Π:{λ∗, vλ∗, v ∈{λ∗} ×Ω1∩Σ, Ev is unbounded inΓ}, 3.47

thenΠ/∅since

ζj∩{λ∗} ×Ω1

⊆Π, j0,1,2, . . . . 3.48

FromLemma 2.4, it follows thatΠis closed in0,∞×E, and furthermore,Πis compact in

Let

Σ_:

λ∗_{, v∈Π}

Ev, 3.49

then

ζj⊆Σ, j0,1,2, . . . . 3.50

If for someλ∗, v∈Π,μv ∞, thenTheorem 1.2holds. Assume on the contrary thatμv<∞for allλ∗, v∈Π.

For everyλ∗, v∈Π, letEvbe the component inEv∩λ∗,∞×Ω1which contains λ∗_{, v. Using the standard method, we can find a bounded open setUvinλ}∗_,∞×Ω

1, such

that

E_{v⊂Uv, ∂Uv∩Σ∅, 3.51}

supλ|λ, u∈Uv <∞, 3.52

where∂UvandUvare the boundary and closure ofUvinλ∗,∞×Ω1, respectively.

Evidently, the following family of the open sets of{λ∗} ×Ω1:

{Uv∩{λ∗_{} ×Ω}

1|λ∗, v∈Π} 3.53

is an open covering ofΠ. SinceΠis compact set in{λ∗} ×Ω1, there existv1, . . . , vmsuch that λ∗_{, vi∈Π, i1, . . . , m, and the family of open sets in{λ}∗_{} ×Ω}

1:

{Uvi∩{λ∗} ×Ω1|i1, . . . , m} 3.54

is a finite open covering ofΠ. There is

Π⊆ {Uvi∩{λ∗} ×Ω1|i1, . . . , m}. 3.55

Let

U1 m

i1

Uvi. 3.56

ThenU1is a bounded open set inλ∗,∞×Ω1,

and by3.52, we have

supλ|λ, u∈U1

<∞, 3.58

where∂U1andU1are the boundary and closure ofU1inλ∗,∞×Ω1, respectively.

Equation3.58together with3.55and3.57implies that

supλ|λ, u∈Σ, u∈Ω1

for some constanta >0, then

a

Thus

{ηn}is bounded. This is a contradiction.

Step 3. We show that limλ,u∈ζ∩Γ, λ→∞u0 ∞.

Suppose on the contrary that there exists{ηn, yn} ⊂ζ∩Γwith

ηn −→∞, yn₀≤M 3.66

for some constantM >0, then

1

that{ηn}is bounded. This is a contradiction.

To sum up, there exits a componentζwhich joins∞, θand∞,∞.

Acknowledgments

References

1 R. P. Agarwal and J. Henderson, “Positive solutions and nonlinear eigenvalue problems for third-order difference equations,”Computers & Mathematics with Applications, vol. 36, no. 10–12, pp. 347–355, 1998.

2 R. P. Agarwal and D. O’Regan, “Boundary value problems for discrete equations,” Applied Mathematics Letters, vol. 10, no. 4, pp. 83–89, 1997.

3 R. P. Agarwal and F.-H. Wong, “Existence of positive solutions for nonpositive difference equations,”

Mathematical and Computer Modelling, vol. 26, no. 7, pp. 77–85, 1997.

4 I. Rachunkova and C. C. Tisdell, “Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions,”Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 4, pp. 1236–1245, 2007.

5 J. Rodriguez, “Nonlinear discrete Sturm-Liouville problems,” Journal of Mathematical Analysis and Applications, vol. 308, no. 1, pp. 380–391, 2005.

6 S. S. Cheng and H.-T. Yen, “On a discrete nonlinear boundary value problem,”Linear Algebra and Its Applications, vol. 313, no. 1–3, pp. 193–201, 2000.

7 G. Zhang and W. Feng, “On the number of positive solutions of a nonlinear algebraic system,”Linear Algebra and Its Applications, vol. 422, no. 2-3, pp. 404–421, 2007.

8 R. Ma and H. Ma, “Existence of sign-changing periodic solutions of second order difference equations,”Applied Mathematics and Computation, vol. 203, no. 2, pp. 463–470, 2008.

9 R. Ma, “Nonlinear discrete Sturm-Liouville problems at resonance,” Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 11, pp. 3050–3057, 2007.

10 P. H. Rabinowitz, “Some global results for nonlinear eigenvalue problems,” Journal of Functional Analysis, vol. 7, pp. 487–513, 1971.

11 A. Jirari, “Second-order Sturm-Liouville difference equations and orthogonal polynomials,”Memoirs of the American Mathematical Society, vol. 113, no. 542, p. 138, 1995.

12 G. T. Whyburn,Topological Analysis, Princeton University Press, Princeton, NJ, USA, 1958.