A global result for discrete ϕ Laplacian eigenvalue problems

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R E S E A R C H

Open Access

A global result for discrete

φ

-Laplacian

eigenvalue problems

Dingyong Bai

*

*_{Correspondence:}

[email protected] School of Mathematics and Information Science, Key Laboratory of Mathematics and

Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, 510006, P.R. China

Abstract

Concerned is the existence, nonexistence and multiplicity of positive solutions for discrete

φ

-Laplacian eigenvalue problems. By using lower and upper solutions method and ﬁxed point index theory, a global result with respect to parameter is established.

Keywords: discrete

φ

-Laplacian eigenvalue problem; positive solution; the lower and upper solutions method; ﬁxed point index theory

1 Introduction

Fora,b∈Zwitha<b, let [a,b]Z={a,a+ ,a+ , . . . ,b– ,b}. In this paper, we consider

the following discreteφ-Laplacian eigenvalue problem

(φ(u(k– ))) +λp(k)g(u(k)) = , k∈[,T]Z,

u() =u(T+ ) = , ()

whereT>  is a given positive integer,u(k) =u(k+ ) –u(k),λis a positive parameter,

p: [,T]Z→(,∞) andg:R+→(,∞) is continuous. We assume that

(A) φ:R→Ris an odd and strictly increasing homeomorphism; (A) lim_u_→∞_φg(u)_(u)=∞.

The functionφ(u) covers two important cases:φ(u) =uandφ(u) =|u|p–u(p> ). If

φ(u) =u, then problem () is the classical second order diﬀerence Dirichlet boundary value problem. For the case thatφ(u) =|u|p–_u_{, problem () is the well-known discrete}

p-Laplacian problem. The two cases have been widely studied. To name a few, see [–] and the references therein.

Problem () can be viewed as the discrete analogue of the following diﬀerential φ -Laplacian problem

(φ(u))+λp(t)g(u(t)) = ,  <t< ,

u() =u() = , ()

which rises from the study of radial solutions forp-Laplacian equations (φ(u) =|u|p–_u_{) on} an annular domain (see [], and references therein). Recently, the diﬀerentialφ-Laplacian problems have been widely studied in many diﬀerent papers. We refer the readers to [– ] and the references therein.

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For discrete φ-Laplacian problems, there are less study results than differential φ -Laplacian problems. See Cabada [], Cabada and Espinar [] and Bondar []. In [, ], some existence results were established by the upper and lower solutions method. In [], the existence and uniqueness of solutions were discussed for mixed and Dirichlet boundary value problems by the fixed point theory of contraction mapping. To the best of our knowledge, there are no results on the existence and multiplicity of positive solutions for differenceφ-Laplacian problems. Therefore, the purpose of this paper is to establish a global result of positive solutions of (). We state our main result as follows.

Theorem . Let(A)and(A)hold.Then there existsλ∗> such that problem()has at least two positive solutions forλ∈(,λ∗),at least one positive solution forλ=λ∗and no solution forλ>λ∗.

The result is motivated mainly by the ideas in [, ], in which some global results of positive solutions were established for boundary value problems ofp-Laplacian diﬀeren-tial systems andφ-Laplacian diﬀerential systems, respectively.

Generally, in order to make priori estimations on possible positive solutions of φ -Laplacian problems, the functionφ satisﬁes not only condition (A), but also other ad-ditional conditions. For example, Wang [] used the following condition.

(A∗) There exist two increasing homeomorphismsψ,ψ: (,∞)→(,∞)such that for

allxandy> ,

ψ(x)φ(y)≤φ(xy)≤ψ(x)φ(y).

In [], a more general condition was given.

(A∗∗) Forσ> , there exists a constantCσ> such that for alls∈R, φ_φ(σ_(s)s)<Cσ.

In our discussion for the single discrete problem (), we only assume thatφ satisﬁes condition (A). In addition, in the discussion of nonlinear diﬀerential systems in [, ], monotonicity conditions were imposed on nonlinear terms. In this papergdoes not have to satisfy monotonicity conditions.

The remaining part of this paper is organized as follows. In Section , we show some lemmas for the later use. In Section , we show the proof of Theorem .. Our proofs are mainly based on the upper and lower solutions technique arguments and the ﬁxed-point index theory for cones.

2 Some lemmas

First, we introduce an existence result of solutions based on lower and upper solutions method for discrete φ-Laplacian boundary value problems, which has been proved by Cabada [].

Consider the following boundary value problem

(φ(u(k– ))) +f(k,u(k)) = , k∈[,T]Z,

u() =u(T+ ) = , ()

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(B) φ:R→Ris an odd and strictly increasing function; (B) f: [,T]Z×R→Ris continuous.

LetE={u: [,T+ ]Z→RT+}with the normu=maxt∈[,T+]Z|u(t)|. Givenu,v∈E, we say thatu≤vifu(k)≤v(k) holds for allk∈[,T+ ]Z.

Deﬁnition . α∈Eis called a lower solution of problem () if

(φ(α(k– ))) +f(k,α(k))≥, k∈[,T]Z,

α()≤, α(T+ )≤.

In a same way, we deﬁne the upper solution of () by reversing the above inequalities.

Lemma .[] Let(B)and(B)hold.Assume that there existαandβ,respectively lower and upper solutions of()such thatα≤β.Then problem()has at least one solution u withα≤u≤β.

A functionuof integer variable is said to be concave if_u₍_k_{– )}_≤_{. This is the same} as saying that the ﬁrst diﬀerenceu(k– ) is non-increasing. Ifu(k– ) < , thenuis said to be strictly concave.

Lemma . Let u(k)be concave on[,  +T]Z,and u()≥,u(T+ )≥.Then:

(i) u(k)≥for allk∈[,T]Z,andu(k– )≥fork∈[,k∗]Z,u(k)≤for

k∈[k∗,T]Z,wherek∗∈[,  +T]Zsatisﬁesu(k∗) =maxk∈[,+T]Zu(k).

(ii) For eachk∈[,T+_ – ]Z(T+_ denotes the integer part of T+_ ),

u(k)≥ k

T+ u, u∈[k,  +T–k]Z. Specially,u(k)≥ 

T+u,u∈[,T]Z.

Proof (i) Ifk∗=  orT+ , the result is clear. Now we assume thatk∗∈[,T]Z. Since

uk∗– =uk∗–uk∗– ≥, uk∗=uk∗+ –uk∗≤.

We have by the monotonicity ofu(·) thatu(k– )≥ fork∈[,k∗]Z,u(k)≤ fork∈

[k∗,T]Z, which implies thatu(k)≥ holds for allk∈[,T]Zby the boundary conditions

u()≥,u(T+ )≥.

(ii) Fora,b∈Rwitha<b, let [a,b]Rand (a,b)Rdenote the closed and open interval on

R, respectively. For any givenk∈[,T+ ]Z, deﬁne

˜

u(t) =u(k) –u(k– )(t–k) +u(k), t∈[k– ,k]R.

Clearly,u˜(k) =u(k) fork∈[,T+ ]Zandu˜(t) is continuous concave on [,T + ]R. Let

· C[,T+]R denote the super norm in the spaceC[,T+ ]R. It is easy to see thatu=

uC[,T+]R. By the concavity ofu˜(t) on [,T+ ]R, we have that for anyδ∈(, T+

 )R, ˜

u(t)≥ δ

T+ uC[,T+]R, t∈[δ,T+  –δ]R.

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Remark Ifu(k) is strictly concave on [,  +T]Z, andu()≥,u(T+ )≥. Thenu(k) > 

fork∈[,T]Z, and there existsk∗∈[,  +T]Z such thatu(k∗) =u,u(k– ) >  for

k∈[,k∗– ]Z,u(k∗– )≥, andu(k) <  fork∈[k∗,T]Z.

Lemma . Let(B)hold and u satisfy the following diﬀerence inequality

–φu(k– )≥, k∈[,T]Z, ()

Lemmas . and . yield the following result.

Lemma . Let(B)hold.Then each solution u of()is strictly concave and u(k) > for all k∈[,T]Z.

Consider the following boundary value problem

(φ(u(k– ))) +h(k) = , k∈[,T]Z,

u() =u(T+ ) = , ()

whereh(k) >  for allk∈[,T]Zandφ satisﬁes (A). It is easy to check thatu∈E is a

solution of () if and only if

u(k) =

Sinceφ–_{is a homeomorphism from}_R_{onto itself, the solution}_C

of () is unique. Let

Then () can be rewritten as follows

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whereCsatisﬁesu() = ,i.e.,

By Lemmas . and ., the following result holds.

Lemma . Let(A)hold.Assume that u solves()andu=u(k∗).Then

whereCis the unique solution of the following equation

T+

Proof Suppose that the conclusion is not true. Then there exists a sequence{λn} ⊂ and

uncorresponding positive solutions of () atλnsuch thatun → ∞asn→ ∞. Letλ= holds fornsuﬃciently large. It follows that

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Letun=un(k∗). Ifk∗> , then by Lemma ., we have a contradiction that

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Let={λ∈(,∞) : the problem () has at least one positive solution atλ}.

Lemma . Let(A)and(A)hold.Thenis bounded.

Proof Suppose, on the contrary, that there exists a sequence{λn} ⊂andun correspond-ing positive solutions of () atλnsuch thatλn→ ∞asn→ ∞. By (A), there exists>  such that for allu≥,g(u)≥φ(u). Sinceλn→ ∞asn→ ∞, there existsδ>  satisfy-ingδp>  such thatλn>δholds fornsuﬃciently large. Letun=un(k∗). Similar to the arguments in the proof of Lemma ., ifk∗> , we have the following contradiction

un() >φ–δpφun()>un(),

and ifk∗= , we also have a contradiction that

un(T) >φ–δpφun(T)>un(T).

Lemma . Let(A)and(A)hold.Then there existsλ∗> such that= (,λ∗].

Proof By Lemmas .-.,is a nonempty bounded interval open at the left with the left endpoint  /∈. We only need to show that it is closed at the right. Letsup=λ∗. Then there exists a sequence{λn} ⊂satisfyingλn<λn+such thatλn→λ∗asn→ ∞. Since {λn}is bounded, Lemma . implies that there exists a constantℵsuch that for allnand all possible positive solutionsunof () atλn,un ≤ ℵ. It follows that{un}has a convergent subsequence, say again{un}, which converges tou∗. Since (λn,un) solves problem (), we know that

(φ(un(k– ))) +λnp(k)g(un(k)) = , k∈[,T]Z,

un() =un(T+ ) = .

Letn→ ∞, we have by the continuity ofgas well asφthat

(φ(u∗(k– ))) +λ∗p(k)g(u∗(k)) = , k∈[,T]Z,

u∗() =u∗(T+ ) = ,

which implies that (λ∗,u∗) solves problem (). By Lemma .,u∗is positive on [,T]Zand

λ∗∈. The proof is completed.

In the succeeding arguments, we need the following well-known ﬁxed point index the-orem on cones. For proof and details, see Guo [].

Lemma . Let E be a Banach space,let P be a cone in E,and letbe a bounded open set in E with∈.Let T:P∩ ¯→P be a complete continuous operator satisfying that

Tx=μx, x∈P∩∂⇒μ< .

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3 The proof of Theorem 1.1

By Lemma ., there existsλ∗>  such that problem () has at least one positive solution forλ∈(,λ∗], and no solution forλ>λ∗. So we only need to show the existence of the

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and () implies that

φu(k– )

= –λp(k)g∗u(k)≥–λp(k)gu∗(k)

>φu∗(k– )

,

which contradicts (). Thus, ifuis a solution of (), thenu∈. DeﬁneT_λ∗the same asTλ

replacinggbyg∗. ThenTλ∗:K→Kis continuous. Sinceg∗is bounded, there existsR>  such thatTλ∗u<Rfor allu∈K. Thus, we can chooseRsuﬃciently large such thatR>R and⊂BR. HereBR={u∈E:u<R}. Then by Lemma ., we havei(Tλ∗,BR∩K,K) = . Since all positive ﬁxed points ofTλ∗are contained in, andTλu=uis equivalent toTλ∗u=u

on∩K, one has

i(Tλ,∩K,K) =i

Tλ∗,∩K,K

=iTλ∗,BR∩K,K

= . ()

By Lemma ., there existsλ¯ >λ∗ such that problem () has no positive solution atλ¯. Thus, for any open setUinE,

i(Tλ¯,U∩K,K) = . ()

Take=_λ_p, wherep=mink∈[,T]Zp(k). By (A), there existsL>  such thatg(u) >φ(u)

foru>L. Letbe a compact set of (,∞) containingλandλ¯. Then by Lemma ., we can chooseb>  suﬃciently large such that all possible solutionsuof () at anyλ˜∈

satisfyu<band thatb> (T+ )Land⊂Bb. HereBb={u∈E:u<b}. Deﬁne h: [, ]×(B¯b∩K)→Kby

h(t,u) =Tτλ¯+(–τ)λ(u).

It is easy to see thathis continuous on [, ]×K, h(,u) =Tλuandh(,u) =T¯λu. We

need to prove thath(τ,u)=ufor all (τ,u)∈[, ]×(∂B¯b∩K). Assume that there exists

(τ,u)∈[, ]×(∂B¯b∩K) such that

u(k) = k

s=

φ–

C_(h)+τλ¯+ ( –τ)λ

T

l=s

p(l)gu(l)

, k∈[,T+ ]Z,

whereC_(h)satisﬁesu(T+ ) = . Then by Lemma .,u(k)≥_T+ u>Lfork∈[,T]Z. Let

u=u(k∗). Similar to the proof of Lemma ., ifk∗> , then one can get a contradiction that

u() >φ–τλ¯+ ( –τ)λpφu()>u().

Ifk∗= , one can also have a contradiction. Therefore,h(τ,u)=ufor all (τ,u)∈[, ]× (∂B¯b ∩K) andi(h(τ,·),Bb∩K,K) is well deﬁned. Thus, by the property of homotopy

invariance and (),

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Then by the additive property, we have from () and (),

iTλ, (Bb\ ¯)∩K,K

= –.

Therefore, problem () has one positive solution in∩K and another in (Bb\ ¯)∩K.

The proof is completed.

Competing interests

The author declares that he has no competing interests.

Author’s contributions

The author worked on the results independently.

Acknowledgements

The author is very grateful to the referees for their helpful comments. This research is supported partially by the Research Funds for the Doctoral Program of Higher Education of China (No. 20104410120001, 20114410110002), PCSIRT of China (No. IRT1226) and the Natural Science Fund of China (No. 11171078).

Received: 11 June 2013 Accepted: 15 August 2013 Published: 29 August 2013 References

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