R E S E A R C H
Open Access
Sign-changing solution for a third-order
boundary-value problem in ordered Banach
space with lattice structure
Xiuli Lin
*and Zengqin Zhao
*Correspondence:
School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, P.R. China
Abstract
In this paper, the sign-changing solution of a third-order two-point boundary-value problem is considered. By calculating the eigenvalues and the algebraic multiplicity of the linear problem and using a new fixed point theorem in an ordered Banach space with lattice structure, we give some conditions to guarantee the existence for a sign-changing solution.
Keywords: third-order boundary-value problem; sign-changing solution; Green’s function; fixed point; lattice
1 Introduction
In this paper, we consider the following nonlinear third-order two-point boundary-value problem
–u(t) =f(t,u(t)), t∈[, ],
u() =u() =u() = , (.)
wheref ∈C([, ]×R,R).
The study on the existence of the sign-changing solutions for the boundary-value prob-lem is very useful and interesting both in theory and in application. Recently, there has been much attention focused on the problem, especially to the two-point or multi-point boundary-value problem. For the second-order two-point or multi-point boundary-value problem, many beautiful results have been given on the existence and multiplicity of the sign-changing solutions (see [–] and the references therein). For example, Xu and Sun [] obtained an existence result of the sign-changing solutions for the second-order three-point boundary-value problem
–u(t) =f(u), t∈[, ], u() = , αu(η) =u(),
where <α< , <η< ,f ∈C(R,R). Xu [] considered the sign-changing solutions for the second-order multi-point boundary-value problem
–u(t) =f(u), t∈[, ],
u() = , u() =mi=–αiu(ηi),
where αi> ,i= , , . . . ,m– , <η <· · ·<ηm–< , f ∈C(R,R). In [], Zhang and
Sun obtained the existence and multiplicity of the sign-changing solutions for the inte-gral boundary-value problem
–u(t) =f(u), t∈[, ],
u() = , u() =a(s)u(s)ds, (.)
wheref ∈C(R,R),a∈L(, ) is nonnegative witha(s)ds< . For the integral
boundary-value problem (.), Li and Liu [] also obtained the existence and multiplicity of the sign-changing solutions in ordered Banach space with the lattice structure.
For the third-order boundary-value problem, the existence and multiplicity of solutions have also been discussed in many papers (see [–] and the references therein). However, the research on the sign-changing solutions has been proceeded slowly. For the problem (.), Yao and Feng [, ] established several existence results for the solutions includ-ing the positive solutions usinclud-ing the lower and upper solutions and a maximum princi-ple, respectively. To our knowledge, however, there are fewer papers considered the sign-changing solutions of the problem (.). Motivated by the work mentioned above, using the eigenvalues of linear operator, we give an existence result for the sign-changing solu-tions of the problem (.).
The main contribution of this paper are as follows: (a) for the sign-changing solutions of the problem (.), to our knowledge, there is no result using the eigenvalues of the linear operator until now; (b) we obtain the eigenvalues and the algebraic multiplicity of the linear problem corresponding the problem (.), which is one of the key points that we can use to prove our main result; (c) some conditions are given to guarantee the existence for a sign-changing solution of the problem (.).
2 Notations and preliminaries
The following results will be used throughout the paper.
LetE=C[, ],u=maxt∈[,]|u(t)|. ThenEis a real Banach space with the norm · .
LetP={u∈E:u(t)≥,t∈[, ]}, andPis a normal solid cone ofE,{u∈E:u(t) > ,t∈ (, ]} ⊂P◦={x∈P|xis an interior point ofP}.
Let the operatorsK,F,Abe defined by
(Ku)(t) =
G(t,s)u(s)ds, (Fu)(t) =ft,u(t), fort∈[, ],u∈E, (.)
andA=KF, respectively, where
G(t,s) =
ts–s–ts, ≤s≤t≤,
t( –s), ≤t≤s≤. (.)
Remark ()A,K:E→Eare completely continuous. ()G(t,s)≥,t,s∈[, ]. In fact, since it is obvious in the other case, we only need to prove the case ≤s≤t≤. Now we suppose that ≤s≤t≤. Then
G(t,s) =
ts–s–ts= s
Definition .[] We callEa lattice under the partial ordering≤, ifsup{x,y}andinf{x,y} exist for arbitraryx,y∈E.
Remark E=C[, ] is a lattice under the partial ordering≤that is deduced by the cone P={u∈E:u(t)≥,t∈[, ]}ofE.
Definition . [] LetE be a Banach space with a coneP,A:E−→Ebe a nonlinear operator. We call thatAis a unilaterally asymptotically linear operator alongPw={x∈E:
x≥w,w∈E}, if there exists a bounded linear operatorLsuch that
lim x∈Pw,x−→∞
Ax–Lx
x = .
Lis said to be the derived operator ofAalongPwand will be denoted byAPw. Similarly, we
can also define a unilaterally asymptotically linear operator alongPw={x∈E:x≤w,w∈
E}. Specially, ifw=θ, We call thatAis a unilaterally asymptotically linear operator along Pand –P.
Definition .[] LetD⊆EandA:D−→Ebe a nonlinear operator.Ais said to be quasi-additive on lattice, if there existsv∗∈Esuch that
Ax=Ax++Ax–+v∗, ∀x∈D,
wherex+=x+=sup{x,θ},x–=x–= –sup{–x,θ}.
Remark It is easy to see that the operatorsFandA=KFdefined by (.) are both quasi-additive on the latticeE=C[, ].
Let us list some conditions and preliminary lemmas to be used in this paper.
(H) f ∈C([, ]×R,R)is strictly increasing inu, andf(t,u)u> for all t∈[, ],u∈
R\{}.
(H) lim|u|→+∞f(tu,u) =β uniformly on [, ]. There exists a positive integer n such
that
λn
<β<λ
n+,
where <λ<λ<· · ·<λn<· · · are the positive solutions of the equation
– e
–λ=cos
√
λ+
π
. (.)
(H) limu→f(tu,u) =βuniformly on[, ], and <β<λ.
Lemma . For any f ∈C[, ],u∈C[, ]is a solution of the following problem:
if and only if u(t)is a solution of the integral equation
u(t) =
G(t,s)f(s)ds,
where G(t,s)is defined by(.).
Proof On the one hand, integrating the equation
–u(t) =f(t), t∈[, ]
over [,t] for three times, we have
u(t) = –
t
(t–s)f(s)ds+ u
()t+u()t+u().
Then
u(t) = –t
t
f(s)ds+
t
sf(s)ds+u()t+u().
Combining them with boundary conditionu() =u() =u() = , we conclude that
u() =
( –s)f(s)ds.
Therefore,
u(t) = –
t
(t–s)f(s)ds+ t
( –s)f(s)ds
= –
t
(t–s)–t( –s)f(s)ds+ t
t
( –s)f(s)ds
=
G(t,s)f(s)ds.
On the other hand, since
u(t) =
G(t,s)f(s)ds=
t
ts–s–tsf(s)ds+
t
t( –s)f(s)ds,
therefore,
u(t) =
t
(s–ts)f(s)ds+
t
t( –s)f(s)ds,
u(t) = –
t
sf(s)ds+
t
( –s)f(s)ds,
and
u(t) = –tf(t) – ( –t)f(t) = –f(t).
Remark Considering Lemma ., we find thatuis a solution of the problem (.) if and only ifuis a fixed point of the operatorA=KF.
From the following lemma, we can obtain the eigenvalues and the algebraic multiplicity of the linear operatorK.
Lemma . The eigenvalues of the linear operator K are
λ >
λ >· · ·>
λ
n
>· · ·
and the algebraic multiplicity of each positive eigenvalue λ
n is equal to,where <λ<
λ<· · ·<λn<· · · are the positive solutions of(.).
Proof Letλbe a positive eigenvalue of the linear operatorK, andy∈E\ {θ}be an eigen-function corresponding to eigenvalueλ. By Lemma ., we have
–y(t) =λy(t), t∈(, ),
y() =y() =y() = . (.)
The auxiliary equation of the differential equation (.) has roots –μ,μ( +√i), μ( –
√
i), whereμ=
√
λ. Thus the general solution of (.) is
y(t) =Ce–μt+Ce
μtcos
√
μt +Ce μtsin
√
μt . Then
y(t) =C
–μe–μt+Cμe
μtcos
√
μt+
π
+Cμe μtsin
√
μt+
π
.
Applying the conditiony() =y() = , we obtainC= –C,C= √
C, whereC= .
Applying the second conditiony() = , that is,
e–μ+ eμcos
√
μ+
π
= . (.)
Considering (.), we see thatμis one ofλ,λ, . . . ,λn, . . . , that is,
λ
>
λ
>· · ·>
λ
n
>· · ·
are eigenvalues of the linear operatorKand the eigenfunction corresponding to the eigen-value λ
n is
yn(t) =C
e–λnt–eλntcos
√
λnt +
√
eλntsin
√
λnt
, (.)
Next we prove that the algebraic multiplicity of the eigenvalue λ
n is . From (.), any
two eigenfunctions corresponding to the same eigenvalueλ
n are merely nonzero constant
multiples of each other, that is,
dim ker
λ
n
I–K =dim kerI–λnK= .
Now we show that
kerI–λnK=kerI–λnK.
Obviously, we only need to show that
kerI–λnK⊂kerI–λnK.
In fact, for anyy∈ker(I–λnK), (I–λ
nK)yis an eigenfunction of linear operatorK
cor-responding to the eigenvalueλ
nif (I–λ
nK)y=θ. Considering (.), there exists a nonzero
constantσsuch that
I–λnKy(t) =σ
e–λnt–eλntcos
√
λnt +
√
eλntsin
√
λnt
.
By direct computation, we have
⎧ ⎪ ⎨ ⎪ ⎩
y(t) +λ
ny(t)
= –σ λ
n(e–λnt–e
λntcos(
√
λnt) +
√
eλntsin( √
λnt)), t∈[, ],
y() =y() =y() = .
(.)
It is easy to see that the solution for the corresponding homogeneous equation of (.) is of the form
y(t) =Ce–λnt+Ce
λntcos
√
λnt +Ce λntsin
√
λnt .
Then, by an ordinary differential equation method, we see that the general solution of (.) is of the form
y(t) =y(t) +y∗(t) +y∗(t), t∈[, ],
where
y∗(t) = – σ λnte
–λnt
is the special solution of the equation
and
y∗(t) = σ λnte
λnt
cos
√
λnt +
√
sin
√
λnt
= σ λnte
λntcos
π
–
√
λnt is the special solution of the equation
y(t) +λny(t) =σ λneλnt
cos
√
λnt –
√
sin
√
λnt
.
Then
y∗(t) = – σ λne
–λnt+
σ λ
nte
–λnt,
y∗(t) = σ λne
λntcos
π
–
√
λnt +
σ λ
nte
λntcos
π
–
√
λnt +
√
σ λ
nte
λntsin
π
–
√
λnt .
Applying the conditiony() =y() = , we obtainC= –C,C= √
C. From the
condi-tiony() = , we obtain
(y)() +
y∗() +y∗() = . (.)
From (.), we have (y)() = . Thus it follows from (.) that
–e–λn+λ
ne–λn+ e
λncos
π
–
√
λn +λne λncos
π
–
√
λn +√λne
λnsin
π
–
√
λn = , which implies that
e–λn+eλncos
π
–
√
λn +
√
eλnsin
π
–
√
λn = . That is
– e
–λn=cos √
λn, which is a contradiction of
– e
–λn=cos
√
λn+
π
.
Therefore, the algebraic multiplicity of the eigenvalue λ
Lemma . Suppose that(H)holds and y∈P\{θ}is a solution of the(.),then y∈ ◦
P. Similarly,if y∈(–P)\{θ}is a solution of the(.),then y∈–P.◦
Proof The proof is obvious.
Lemma . Suppose that(H)-(H)hold.Then the operator A is Fréchet differentiable at
θ and∞,and Aθ=βK,A∞=βK.
Proof Since (H):limu→f(ut,u) =βuniformly on [, ]. That is, for anyε> , there exists
δ> such that
f(t,u) –βu≤ε|u|, ∀t∈[, ], <|u|<δ.
From (H), it is easy to see that∀t∈[, ],f(t, ) = . Then, for anyu∈Ewithu<δ, we
have
(Au–Aθ–βKu)(t)=K(Fu–βu)(t)
≤ Kmax
s∈[,]
fs,u(s)–βu(s)≤ Kuε.
Then
Au–Aθ–βKu ≤ Kuε, u<δ.
Thus,
lim
u→
Au–Aθ–βKu
u = ,
which meansAθ=βK.
Since (H):lim|u|→+∞f(tu,u) =β uniformly on [, ]. That is, for anyε> , there exists
R> such that
f(t,u) –βu≤ε|u|, ∀t∈[, ],|u|>R.
LetMR=max≤|u|≤R|f(t,u)|. Then
(Fu)(t) –βu(t)=f
t,u(t)–βu(t)≤MR+βR+εu, t∈[, ],u∈E.
Thus,
(Au–βKu)(t)=K(Fu–βu)(t)≤ K
MR+βR+εu
.
Then
lim
u→+∞
Au–βKu
u ≤εK.
Remark Suppose (H) holds. Similar to Lemma ., we have
AP=A–P=A∞=βK.
Lemma .[] Suppose that E is an ordered Banach space with a lattice structure,P is a normal solid cone in E,and the nonlinear operator A is quasi-additive on the lattice. Assume that
(i) Ais strongly increasing onPand–P;
(ii) bothAPandA–Pexist withr(AP) > andr(A–P) > ;is not an eigenvalue ofAPor A–Pcorresponding a positive eigenvector;
(iii) Aθ=θ;the Fréchet derivativeAθofAatθ is strongly positive,andr(Aθ) < ; (iv) the Fréchet derivativeA∞ofAat∞exists;is not an eigenvalue ofA∞;the sumβ
of the algebraic multiplicities for all eigenvalues ofA∞lying in the interval(,∞)is an even number.
Then A has at least three nontrivial fixed points containing one sign-changing fixed point.
3 Main result
We state the main result of this paper.
Theorem . Suppose that(H)-(H)hold.Then the problem(.)has at least three
solu-tions including a sign-changing solution.
We need only to prove thatA=KFsatisfies the four conditions of Lemma ..
Proof Noticing
Aθ=βK, AP=A–P=A∞=βK.
(i) Ais strongly increasing onPand–P. In fact, from (H) andK(P\ {θ})⊆ ◦
P, we see thatAis strongly increasing onP. Similarly,Ais strongly increasing on–P. (ii) FromAP=A–P=βK,λn<β<λ
n+and Lemma ., we find thatis not an
eigenvalue ofAPorA–Pandr(AP) =r(A–P) = β
λ > .
(iii) FromAθ=βK, <β<λ,K(P\ {θ})⊆ ◦
P, Lemma . and (H), we haveAθis
strongly positive,Aθ=θ andr(Aθ) =β
λ < .
(iv) SinceA∞=βK,λn<β<λ
n+and Lemma ., the condition (iv) of Lemma .
is satisfied.
Therefore, from Lemma ., we see thatAhas at least three nontrivial fixed points in-cluding one sign-changing fixed point. Then, the problem (.) has at least three solutions,
including one sign-changing solution.
Example . Consider the following third-order boundary-value problem
–u(t) =f(t,u(t)), t∈[, ],
where
f(t,u) =
⎧ ⎪ ⎨ ⎪ ⎩
(u– ) + ( +t)√u+ , t∈[, ],u∈(, +∞), ( +t)u + u, t∈[, ],u∈[–, ], (u+ ) + ( +t)√u– , t∈[, ],u∈(–∞, –).
By simple calculations, we haveλ≈.,λ≈.,λ≈.,β= ,β=
. Then it is easy to see thatf(t,u) satisfies the conditions (H)-(H). Therefore, the
boundary-value problem (.) has at least three solutions, including one sign-changing solution.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
The authors are highly grateful for the referees’ careful reading and comments on this paper. The research is supported by Program for Scientific research innovation team in Colleges and universities of Shandong Province, the Doctoral Program Foundation of Education Ministry of China (20133705110003), the Natural Science Foundation of Shandong Province of China (ZR2011AM008, ZR2012AM010, ZR2012AQ024).
Received: 11 December 2013 Accepted: 7 April 2014 Published:22 May 2014
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