R E S E A R C H
Open Access
Existence and multiplicity of difference
φ
-Laplacian boundary value problems
Dingyong Bai
*and Xianliang Xu
*Correspondence:
[email protected] School of Mathematics and Information Science, and Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, 510006, P.R. China
Abstract
Concerned are the difference
φ
-Laplacian boundary value problems. The multiplicity result based on the lower and upper solutions method associated with Brouwer degree is applied to a differenceφ
-Laplacian eigenvalue problem. An existence result of at least three positive solutions is established for the eigenvalue problem with the parameter belonging to an explicit open interval. In addition, an example is given to illustrate the three solutions result.Keywords: difference
φ
-Laplacian boundary value problem; the lower and upper solution; positive solution; multiplicity; Brouwer degree1 Introduction
Recently, Kim [] studied a one-dimensional differentialp-Laplacian boundary value prob-lem with a positive parameter and established an existence result of three positive solu-tions by the lower and upper solusolu-tions method associated with Leray-Schauder degree theory. Kim and Shi [] studied the global continuum and multiple positive solutions of ap-Laplacian boundary value problem. Motivated by the methods in [, ], we consider differenceφ-Laplacian boundary value problems.
Fora,b∈Zwitha<b, let [a,b]Z={a,a+ ,a+ , . . . ,b– ,b}. First, by the upper and
lower solutions method associated with Brouwer degree theory, we establish the existence and multiplicity results for the following discreteφ-Laplacian boundary value problem:
(φ(u(k– ))) +f(k,u(k)) = , k∈[,T]Z,
u() =u(T+ ) = , ()
whereT> is a given positive integer,u(k) =u(k+ ) –u(k), and
(A) φ:R→Ris an odd and strictly increasing function; (A) f : [,T]Z×R→Ris continuous.
Then, we apply the multiplicity result of () to the following eigenvalue problem:
(φ(u(k– ))) +λp(k)g(u(k)) = , k∈[,T]Z,
u() =u(T+ ) = , ()
whereλis a positive parameter. Under some suitable assumptions imposed ong, we es-tablish the existence of three positive solutions of () withλbelonging to an explicit open interval.
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))+f(t,u) = , <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.
For discreteφ-Laplacian problems, there are fewer study results than for differentialφ -Laplacian problems. See Cabda [], Cabada and Espinar [] and Bondar []. To the best of our knowledge, there are no results on the existence and multiplicity of positive solutions for differenceφ-Laplacian problems.
The remaining part of this paper is organized as follows. In Section , we show the lower and upper solutions method and establish the existence and multiplicity of solutions of (). In Section , we establish the existence of three positive solutions of (). Finally, we give an example to illustrate our main results.
2 The upper and lower solutions method
In this section, we establish the existence and multiplicity results of solutions for problem () by lower and upper solutions method associated with Brouwer degree.
LetE={u: [,T+ ]Z→RT+}with the normu=maxt∈[,T+]Z|u(t)|.
Definition . Givenu,v,w∈E, we say that
() u≤vif for allk∈[,T+ ]Z,u(k)≤v(k). () u∈[v,w]ifv≤u≤w.
() u≺vif for allk∈[,T]Z,u(k) <v(k)andu()≤v(),u(T+ )≤v(T+ ).
Definition . α∈Eis called a lower solution of problem () if
(φ(α(k– ))) +f(k,α(k))≥, k∈[,T]Z,
α()≤, α(T+ )≤.
If the first inequality above is strict, thenαis called a strict lower solution of ().
In the same way, we define the upper solution and the strict upper solution of () by reversing the inequalities above.
Lemma . Let(A)hold.The problem
–(φ(u(k– ))) +u(k) = , k∈[,T]Z,
u() =u(T+ ) = ()
Proof It is clear that is a trivial solution of (). Suppose that () has a nontrivial solutionu. Let|u(m)|=maxk∈[,T]Z|u(k)|=u. Ifu(m) =maxk∈[,T]Zu(k), thenu(m) > andu(m)≤ ,u(m– )≥, which yields a contradiction:
u(m) =φu(m– )
=φu(m)–φu(m– )≤ <u(m).
Similarly, ifu(m) =mink∈[,T]Zu(k), thenu(m) < andu(m)≥,u(m– )≤, which
implies thatu(m) =(φ(u(m– )))≥ >u(m), which is a contradiction. The proof is
complete.
Theorem . Let(A)and(A)hold.
(i) Assume that there existαandβ,respectively lower and upper solutions of()such thatα≤β.Then problem()has at least one solutionuwithα≤u≤β.
(ii) Assume that problem()has two pairs of lower and upper solutions(α,β)and (α,β)withαandβbeing strict,satisfying that
α≤α≤β, α≤β≤β,
and that there existsk∈[,T+ ]Zsuch thatβ(k) <α(k).Then problem()has
at least three solutionsu,u,uwith
α≤u≺β, α≺u≤β, u∈[α,β]\
[α,β]∪[α,β].
Remark We denote that the result (i) has been proved in [] by Brouwer fixed point theorem. Here, for the convenience of the proof of (ii), it is proven by Brouwer degree theory. The proof of (ii) is motivated by the idea in [].
Proof of Theorem.. (i) Defineγ: [,T]Z×R→Rby
γ(k,u) =
⎧ ⎪ ⎨ ⎪ ⎩
β(k), u>β(k),
u, α(k)≤u≤β(k),
α(k), u<α(k).
Consider the modified problem
(φ(u(k– ))) +f(k,γ(k,u(k))) – [u(k) –γ(k,u(k))] = , k∈[,T],
u() =u(T+ ) = . ()
Clearly, all solutionsuof () satisfyingα≤u≤βare solutions of (). Letube a solution of (). By the arguments in [], we know thatα≤u≤β. Now, we prove that problem () has at least one solution. Let E={u∈E:u() =u(T + ) = }and define operator
T:E→RTby
Obviously, each solutionuofTu= solves (). Define homotopic mapping: [, ]×
E→RTby
(λ,u)(k) = ( –λ)φu(k– )–u(k)+λTu(k)
=φu(k– )–u(k) +λfk,γk,u(k)+γk,u(k), k∈[,T]Z.
By the definition ofγ and the continuity off, there exists anR> , such that
max
k∈[,T]Z
max
u∈R
fk,γ(k,u)+γ(k,u)<R.
Let BR() ={u∈E: u <R}. We prove that if (λ,u)∈[, ]× E is a solution of
(λ,u) = , then u∈ BR(). Let |u(m)|=maxk∈[,T]Z|u(k)|=u. Then there are two
cases that u(m) = maxk∈[,T]Zu(k) and u(m) =mink∈[,T]Zu(k). For the first case, since
u(m+ ) –u(m)≤,u(m) –u(m– )≥ andφis odd, we have that
u(m) –λfm,γm,u(m)+γm,u(m)
=φu(m– )
=φu(m+ ) –u(m)–φu(m) –u(m– )
≤,
which implies that
u(m)≤λfm,γm,u(m)+γm,u(m)<R.
Similarly, for the second case, we have that
u(m)≥λfm,γm,u(m)+γm,u(m)> –R.
Therefore,u<R, anddeg((λ,·),BR(), ) is well defined. By the homotopy invariance of Brouwer degree, we get that
deg T,BR(), =deg(,·),BR(), =deg(,·),BR(), .
By Lemma ., the equation –(φ(u(k– ))) +u(k) = has the unique solutionu= in
E, thus we have
deg(,·),BR(), = .
Therefore,deg(T,BR(), ) = , which implies that problem () has at least one solution
u∈E.
In fact, if it is not true, then there exists an m∈[,T]Z such that α(m) =u(m). Since
u(m– )≤α(m– ),u(m)≥α(m), we have by the monotonicity ofφthat
φu(m– )–φα(m– )
=φu(m)–φα(m)+φα(m– )–φu(m– )
≥.
It yields a contradiction:
φu(m– )= –fm,γm,u(m)+u(m) –γm,u(m)
= –fm,α(m)<φα(m– ).
Thus α≺u. Similarly, one can check thatu≺β. By the excision property of Brouwer degree,
deg(T, αβ, ) =deg T,BR(),
= .
Now, consider the following modified problem:
(φ(u(k– ))) +f(k,γ∗(k,u(k))) – [u(k) –γ∗(k,u(k))] = , k∈[,T],
u() =u(T+ ) = , ()
whereγ∗: [,T]Z×R→Ris defined by
γ∗(k,u) =
⎧ ⎪ ⎨ ⎪ ⎩
β(k), u>β(k),
u, α(k)≤u≤β(k),
α(k), u<α(k).
It is easy to see that for sufficiently small> , (α–,β) and (α,β+) are two pairs of strict lower and upper solutions of (). Similarly to (), letT∗be the operator correspond-ing to problem (). For sufficiently largeR> , define
αβ=
u∈E,α–≺u≺β,u<R
,
αβ=
u∈E,α–≺u≺β+,u<R
,
and
αβ=
u∈E,α≺u≺β+,u<R
.
Then deg(T∗, αβ, ) = , deg(T∗, αβ, ) = and deg(T∗, αβ, ) = . Thus by the
additivity property of Brouwer degree, we havedeg(T∗, αβ\( αβ ∪ αβ, )) = –.
Therefore, problem () has three solutionsu,uanduwithu∈ αβ,u∈ αβ and u∈ αβ\ αβ∪ αβ. By the facts that all solutions of () satisfy [α,β] and are
3 Three positive solutions of eigenvalue problems
Lemma . Let(A)hold and u satisfy the following difference inequality:
–φu(k– )≥, k∈[,T]Z ()
with u()≥,u(T+ )≥.Then u(k)≥for all k∈[,T]Z,andu(k– )≥for k∈
[,k∗]Z,u(k)≤for k∈[k∗,T]Z,where k∗∈[, +T]Zsatisfies u(k∗) =maxk∈[,+T]Zu(k).
Proof Since[φ(u(k– ))] =φ(u(k)) –φ(u(k– ))≤,k∈[,T]Z, we have by the
monotonicity ofφthatu(k)≤u(k– ),k∈[,T]Z. 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+ )≥.
Remark If inequality () is strict, thenu(k) > fork∈[,T]Z, and there existsk∗∈[, + T]Zsuch thatu(k∗) =u, andu(k– ) > fork∈[,k∗– ]Z,u(k∗– )≥, andu(k) <
fork∈[k∗,T]Z.
Consider the following problem:
(φ(u(k– ))) +h(k) = , k∈[,T]Z,
u() =u(T+ ) = , ()
whereh: [,T]Z→(,∞).
In the following arguments, we assume that
(B) φ:R→Ris an odd and strictly increasing homeomorphism.
Lemma . Let(B)hold and u solve().If h is symmetric on[,T]Z,i.e.,h(k) =h(T+–k), k∈[,T]Z,then u(k)is symmetric on[,T]Z.Moreover,
(i)if T+ (T ≥)is odd,thenu=u(T) =u(T + ),and the solution u of()can be expressed as
u(k) =
⎧ ⎨ ⎩
k s=φ–(
T
l=sh(l)), k≤ T
,
T s=kφ–(
s
l=T+h(l)), k≥
T
+ .
(ii)if T+ (T≥)is even,thenu=u(T+
),and the solution u of()can be expressed
as
u(k) =
⎧ ⎨ ⎩
k s=φ–(
T+ –
l=s h(l) +h(
T+
)), k≤
T+ ,
T s=kφ–(
s
l=T++h(l) +
h(
T+
)), k≥
Proof It is easy to see that
Since φ– is a homeomorphism from R onto itself, the solution C of the equation
and fork≥T+,
Now, we state the existence result of at least three positive solutions of (). Throughout the following arguments, we suppose thatT≥. Letvbe the unique positive solution of the following boundary value problem:
(φ(u(k– ))) +p(k) = , k∈[,T]Z, u() =u(T+ ) = ,
andp=mink∈[,T]Zp(k).
We make the following assumptions.
(B) There exists an increasing homeomorphismψ: (,∞)→(,∞)such that for all
We denote that condition (B) implies thatlimu→∞ g
∗(u)
ψ(uv) = (see [], Lemma .). Clearly,g∗is nondecreasing on [,∞).
Assumption (B) is satisfied by two important casesφ(x) =xandφ(x) =|x|p–x(p> ). We also provide the following two functions as examples:
Theorem . Let(B)-(B)hold.Then forλ∈(λ,λ),problem()has at least three posi-tive solutions.Here
λ=
Thusβis a strict upper solution of (). Now, letαsolve the following problem:
Thusb<α(k) <Mfork∈[,T]Z. Therefore,
= φα(k– )+λ∗pg(b) ≤φα(k– )
+λ∗pg
α(k)
< φα(k– )+λp(k)gα(k), k∈[,T]Z,
which implies thatαis a strict lower solution of (). It is easy to see that
α(k) >b>a≥β(k), k∈[,T]Z.
Bylimu→∞ g
∗(u)
ψ(uv)= , one can choose a sufficiently large positive numberCλ, such that
g∗(λCλ) ψ(λCλ
v)
<
λ,
and
λCλ
v > , α≤β,
whereβ=λCλvv. Then by (B) and the monotonicity ofg∗,βis a strict upper solution of (). In fact, fork∈[,T]Z,
φβ(k– )=φβ(k)–φβ(k– )
=φ
λCλ
vv(k)
–φ
λCλ
vv(k– )
≤ψ
λCλ
v
φv(k)–φv(k– )
= –ψ
λCλ
v
p(k)
< –λg∗(λCλ)p(k)
≤–λg∗β(k)
p(k)
≤–λgβ(k)p(k).
Thus by Theorem ., problem () has three positive solutions forλ∈(λ,λ).
Remark Ifgis nondecreasing on [,∞), then we takeM=∞andλ=
ψ(av)
g(a) .
4 An example
Takingφ(u) =u+u,g(u) = (u+ ),p(k)≡,T= , consider
((u(k– )) + (u(k– ))
) +λ(u(k) + ) = , k∈[,T]Z,
Letψ(u) =u. It is easy to see that (B)-(B) hold. Choosea= ,b= , then (B) is satis-fied. In fact, after some simple calculations, we get thatv ≈. and that
λ= φ(b)
pg(b) =
+ ≈.,
λ=
ψ(av)
g(a) = √
v ≈..
Thus by Theorem ., problem () has at least three positive solutions forλ: . <λ< ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors jointly worked on the results, and they read and approved the final manuscript.
Acknowledgements
The authors are 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: 5 June 2013 Accepted: 22 August 2013 Published: 8 September 2013 References
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doi:10.1186/1687-1847-2013-267
Cite this article as:Bai and Xu:Existence and multiplicity of differenceφ-Laplacian boundary value problems.
