R E S E A R C H
Open Access
Multiplicity of solutions to a four-point
boundary value problem of a differential
system via variational approach
Weigao Ge
1and Zhihong Zhao
2**Correspondence: [email protected] 2Department of Mathematics,
University of Science and Technology Beijing, Beijing, 100083, P.R. China
Full list of author information is available at the end of the article
Abstract
By constructing an adequate real functional and choosing an appropriate admissible function space, the existence of multiple solutions to a four-point boundary value problem, which may be taken as an extension of Sturm-Liouville boundary value problems, is proved via a variational approach for a second-order differential system with a p-Laplacian.
MSC: 34B15
Keywords: four-point boundary value problem; functional; critical point; variational approach; differential system
1 Introduction
The variational approach, together with the critical point theory, is one of the important methods in the study of two-point boundary value problems of ordinary differential equa-tion [–], as well as impulsive differential equaequa-tions [–]. However, this approach is much more effective in the study of boundary value problems of differential systems [– ].
Mawhin and Willem [] studied the existence of periodic solutions of convex Hamilto-nian system in the form
Ju(t) +∇H(t,u(t)) = , u() –u(T) = ,
whereH: [,T]×Rn→Rand proved that the problem has at least one periodic solution
if
l(t),u≤H(t,u)≤α |u|
+γ(t),
T
H(t,u) dt→ ∞, as|u| → ∞,
withα∈(,Tπ) ([], Theorem .). Also, they proved the system
(M(t,u)u(t))–(∇u(M(t,u)u),u) +∇F(t,u) =f(t),
u() –u(T) =u() –u(T) = ,
has at least one periodic solution if|F(t,u)|+|∇F(t,u)| ≤h(t), (M(t,u)u,u)≥α|u|, and MisTi-periodic inui([], Theorem .).
Tian and Ge [] discussed the differential system with a p-Laplacian
d
dtϕp(u(t)) +∇F(t,u(t)) = ,
u() –u(T) =u() –u(T) = ,
whereϕp(x) =|x|p–xforx∈Rn, and they obtained an existence theorem of periodic
so-lutions under the condition
l(t),|u|p– u≤F(t,u)≤α
p|u|
p+γ(t),
T
F(t,u) dt→ ∞, as|u| → ∞.
The result extended that given by Mawhin and Willem ([], Theorem .).
Graefet al.studied in [] the existence of at least three classical solutions to the multi-point value system
(φp(u))+λF(t,u) +μG(t,u) = , <t< ,
u() =mj=aju(tj), u() =
m j=bju(tj),
whereφp(s) = (φp(s),φp(s), . . . ,φpn(sn)) withφpk(sk) =|sk|
pk–s
k,pk> ,aj,bj∈R,F,G:
[, ]×Rn→R,λ,μ> . By use of the existence theorem of three critical points given by
Ricceri [], they obtained sufficient conditions for the existence of three solutions to the discussed system, when the parameterλis defined in a certain interval [,δ].
In this paper, we are to study the existence of multiple solutions to the following four-point boundary value problem (BVP for short):
(P(t)x)=∇F(t,x), <t< ,
x() =αx(ξ), x() =βx(η), (.)
whereP: [, ]→Rn×nis a continuously symmetric matrix,i.e.,PT(t) =P(t) being
con-tinuous int;F: [, ]×Rn→Ris measurable intfor eachx∈Rnand continuously dif-ferentiable inxfor a.e.t∈[, ];α,β∈R, <ξ,η< .
Clearly, BVP (.) becomes a classic Sturm-Liouville BVP ifξ→ andη→.
Without loss of generality, we supposeξ≤η. Let{pi(t)}be the eigenvalue ofP(t).
As-sume
(H) <a≤min≤t≤min≤j≤npj(t)≤max≤t≤max≤j≤npj(t)≤b; (H) F(t, ) = ,F(t, –x) =F(t,x), and there arec,M> such that
F(t,x)≥c|x|–M.
Condition (H) implies thatP(t) is an invertible matrix for eacht∈[, ]. We are to show in this paper the following results via variational methods.
Theorem . Suppose assumptions(H)and(H)hold.BVP(.)has mn pairs of
non-trivial solutions if there are d,r> ,m∈N+,such that
∇F(t,x),x≤–d|x|< –bmπ|x|, (.)
Theorem . Suppose assumptions (H) and(H) hold. Then BVP (.) has infinitely many pairs of nontrivial solutions if there are d,r> andσ∈(, ),such that
∇F(t,x),x≤–d|x|+σ, (.)
for|x| ≤r.
When condition (.) is replaced by a limitation condition, we have the following.
Theorem . Suppose assumptions (H) and(H) hold. Then BVP (.) has infinitely
many pairs of nontrivial solutions if
lim
|x|→min≤t≤
(∇F(t,x),x)
|x| = –∞. (.)
This paper is organized as follows. In Section , we discuss the relation of the critical point of functionaland the solution to BVP (.). In Section , we show thatsatisfies the (PS)-condition. Based on Sections and we prove in Section the theorems given above. Finally, an example is given in Section to illustrate our result.
To prove the above results we need the following.
Theorem A[] Suppose X is a Banach space and:X→Ra continuously differentiable
functional with() = andeven,bounded from below and satisfying(PS)-condition.If there is a set K⊂X such that K is homeomorphic to Sm–by an odd map,andsupK< . Thenpossesses at least m distinct pairs of critical points.
2 Critical point of functional and solution of BVP
SupposeXis a Banach space and:X→Ra differentiable functional with derivative given by
(u),v
withu,v∈X. LetY⊂Xbe a closed subspace. If there isu∈Xsuch that
(u),v=
holds for allv∈Y, thenuis called a critical point ofwith respect toY. Furthermore, uis called simply a critical point ofifY=X.
Obviously,uis a critical point ofwith respect toYif it is that of. LetX=H([, ],Rn). EquipXwith the norm · defined by
x=
x(t)dt+
x(t)dt
for eachx∈X. ThenXis a reflexive Banach space. Define
(x) =
P(t)x(t),x(t)+Ft,x(t)dt– δβ
P()x(η),x(η)
– δα
forx∈X, whereδis a constant.
It is easy to verify that() = and(–x) =(x). Furthermore, we have the following.
Lemma . If u is a critical point of(x),defined in(.)with respect to Y={x∈X:x() = –αδx(ξ),x() =βδx(η)},then u=u(t)is a solution to BVP(.).
Proof The properties ofFandPensureis continuously differentiable and the derivative ofis in the form
(x),y=
P(t)x(t),y(t)+∇Ft,x(t),y(t)dt
–P()βx(η),δβy(η)–P()αx(ξ),δαy(ξ), (.)
x∈X,y∈Y. Then the assumption thatuis a critical point ofrespect toY means that
(u),y= , y∈Y. (.)
LetZ={x∈C∞([, ],Rn) :x() =x(ξ) =x(η) =x() = }, thenZ⊂Y. Furthermore, let
Z=
x∈Z:x(t) = forξ≤t≤,
Z=x∈Z:x(t) = for ≤t≤ξorη≤t≤, Z=x∈Z:x(t) = for ≤t≤η,
andT= [,ξ],T= [ξ,η],T= [η, ]. Clearly equation (.) implies
(u),z= , z∈Zi, (.)
and then
=
Ti
P(t)u(t),z(t)+∇Ft,u(t),z(t)dt
=
Ti
–P(t)u(t),z(t)+∇Ft,u(t),z(t)dt
= –
Ti
P(t)u(t)–∇Ft,u(t),z(t)dt.
So one gets
P(t)u(t)=∇Ft,u(t), a.e.t∈Ti,
sincez∈Ziis arbitrary. Takei= , , , then we have
The equality (.) means (Pu)(t) is continuous on [, ] and as a critical point ofwith respect toY, we have
=
P(t)u(t),y(t)+∇Ft,u(t),y(t)dt –δβP()u(η),y(η)–δαP()u(ξ),y(ξ) =P(t)u(t),y(t)–
P(t)u(t)–∇Ft,u(t),y(t)dt –δβP()u(η),y(η)–δαP()u(ξ),y(ξ)
=P()u(),y()–P()u(),y()–P()βu(η),δy(η)–P()u(ξ),δαy(ξ),
fory∈Y.
Especially, wheny∈Y=y∈Y:y() =y(ξ) = , one gets
=P()u(),y()–P()βu(η),δβy(η) =P()u() –βu(η),y(),
and then
P()u() –βu(η),y()= . u() =βu(η),
sincey()∈Rnis arbitrary andP() is invertible. At the same time the casey∈Y=y∈
Y:y(η) =y() = implies
u() =αu(ξ).
Sou=u(t) is a solution to BVP (.).
Therefore our task is to discuss the existence of critical points ofinX.
Lemma . For each x∈X,
x(t)≤x. (.)
Proof From
xi(t) –x¯i≤
xi(t)dt,
one has
x(t) –x¯≤
wherex¯i=
xi(t) dtandx¯=
x(t) dt. Then
x(t)≤ |¯x|+
x(t)dt
≤
x(t)dt+
x(t)dt
≤
x(t)dt
+
x(t)dt
≤
x(t)
+x(t)dt
= x.
3 A lemma on the (PS)-condition
We show at first a lemma which will be applied in the proof of our main results.
Lemma . The functional,defined in(.),satisfies the(PS)-condition if assumptions
(H)-(H)hold.
Proof Suppose{uk} ⊂X is a sequence such that{uk} is bounded and(uk)→ as
k→ ∞. We are to show that there is in{uk}a subsequence which converges inX.
To this end, letθ=min{a,c}> and choose
δ∈
, θ
b(α+β)
in the functional (.). Then
(uk) =
P(t)uk(t),uk(t)+Ft,uk(t)
dt
– δβ
P()u
k(η),uk(η)
– δα
P()u
k(ξ),uk(ξ)
≥θ
uk(t)+uk(t)
dt–M
– δ
βP()uk(η),uk(η)
+αP()uk(ξ),uk(ξ)
.
Notice that
uk(η)≤ |¯uk|+
uk(t)dt
≤
uk(t)dt+
u k(t)dt
≤
uk(t)
dt +
uk(t)dt
and then
uk(η)≤
uk(t)+uk(t)
dt= uk. (.)
Similarly,
uk(ξ)
≤
uk(t)+uk(t)
dt= uk. (.)
Therefore, we have
(uk)≥θuk–
δb
α+βuk–M
= θuk
–M,
which implies{uk}is bounded inX. Going, if necessary, to a subsequence, we assume that
ukuinXanduk→uinC([, ],Rn). Then
(uk) –(u),uk–u
→ ask→ ∞. (.)
Using (.) and assumptions (H)-(H), we have
(uk) –(u),uk–u
=
P(t)uk(t) –uk(t)
,uk(t) –uk(t)
+∇Ft,uk(t)
–∇Ft,u(t),uk(t) –u(t)
dt
–δP()βuk(η) –u(η)
,βuk(η) –u(η)
+P()αuk(ξ) –u(ξ)
,αuk(ξ) –u(ξ)
≤a
uk(t) –u(t)dt+
∇Ft,uk(t)
–∇Ft,u(t),uk(t) –u(t)
dt
–δbβuk(η) –u(η)+αuk(ξ) –u(ξ)
. (.)
The fact thatuk→uinC([, ],Rn) implies
∇Ft,uk(t)
–∇Ft,u(t),uk(t) –u(t)
dt→,
uk(η) –u(η)→, uk(ξ) –u(ξ)→,
uk(t) –u(t)
dt→ ∞,
ask→ ∞. Then from (.) and (.) we get
Therefore, we have
uk→u inX.
Thensatisfies the (PS)-condition.
4 Proof of theorems
Proof of Theorem. First we show that(x) is bounded from below. From the definition ofin (.), one has
(x)≥θ
x(t)+x(t)dt–M– δb
βx(η)+αx(ξ)
=θx– δb
βx(η)+αx(ξ)–M ≥θx–
δb
β+αx–M ≥
θx –M,
which implies that(x) is bounded from below.
Second, we prove the existence of a setK⊂Xsuch thatKis homeomorphic toSmn–by an odd map, andsupK< .
To this end we choose the linear spaceXmnin the following way.
Let{ei}be the orthogonal basis ofRn. As Banach spaceHis a subspace ofL([, ],Rn),
its element can be expressed in the form
x(t) =a+
∞
k=
(coskπt·ak+sinkπt·bk), <t< ,
wherea,ak,bk∈Rn. In this case, let
x() =x+, x() =x–,
and
x() = lim t→+
x(t) –x() t =tlim→+
t
x(t) –x+,
x() = lim t→–
x() –x(t) –t =tlim→–
–t
x––x(t).
LetXmn={x(t) =
m
k=sinkπt·bk,bk∈Rn}. ThendimXmn=mn. For a functionx(t) =
m
k=sinkπt·bk, one gets
x(t) =
m
k=
and
x(t)dt=
m
k= |bk|,
x(t)dt=
m
k=
(kπ)|bk|≤mπ
x(t)dt.
(.)
It follows that
x=
m
k=
+ kπ|bk|≤
+ mπ
m
k= |bk|
and
+ mπx
≤
x(t)dt≤ + πx
. (.)
Now chooseK={x∈Xmn,
m
k=|bk|=mr}. Obviously,Kis closed inXwithdimK=
mn– . Furthermore, for eachx∈K,
(x)≤
b x
(t)
+Ft,x(t)dt,
x(t)≤
m
k= |bk| ≤
√ m
m
k= |bk|
=r. (.)
At the same time we have, from (.),
Ft,x(t)=Ft,x(t)–F(t, ) =
∇Ft,sx(t),x(t)ds
= s
∇Ft,sx(t),sx(t)ds
< –bmπ
sx(t)ds = –bmπx(t), which yields
Ft,x(t)dt< –bmπ
x(t)dt
and
(x) <b
x(t)dt– bmπ
x(t)dt
≤bmπ
x(t)dt– bmπ
x(t)dt
Then we have
sup K
< .
Finally, define the odd mappingG:K→Smn–in the following way. For a functionx∈K with the expression
x(t) =
m
k=
sinkπt·bk,
let x(t)→G(x) = (
ρb,
ρb, . . . ,
ρbm), whereρ= (
m k=|bk|)
. ThenGis a homeomor-phism betweenKandSm–. It is clear thatGis an odd mapping.
Then Theorem A gives the conclusion of Theorem ..
Proof of Theorem. It suffices to show that for anym∈N, condition (.) implies that there isˆr> such that the condition holds forr∈(,ˆr).
In fact, fromlim|x|→+|x|+σ/|x|= +∞, we know that there isrˆ∈(,r) such that
d|x|+σ > bmπx(t)
, <|x| ≤ ˆr.
In this case, we have
∇F(t,x),x≤–d|x|+σ < –bmπx(t), <|x| ≤ ˆr,
which implies, by Theorem ., that BVP(.) has at leastmnpairs of distinct nontrivial
solutions.
Proof of Theorem. Condition (.) implies that for anym∈Nthere isr> such that
∇F(t,x),x≤–b(m+ )πx(t)< –bmπ|x|, <|x| ≤r.
Then the conclusion comes from Theorem ..
5 Example
Example . Suppose x,x: (, )→R.Then the BVP
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
x+ x= [–( +sint)(x+ x)–
+ ( +t)(x +x)
]x, x+ x= [–( +sint)(x
+ x)–
+ ( +t)(x +x)
]x, x() = x(), x() = x(
), x() = x(), x() = x(),
(.)
has infinitely many solutions.
Proof LetM(t) = ,α= ,β= ,ξ = ,η=,x=xx,F(t,x) = –( +sint)(x + x)+ ( +t)(x
+x)
Obviously, the eigenvalues ofMare and , which means
<a= , b= .
On the other hand, we have
lim
|x|→∞F(t,x) = +∞, F(t, ) = , F(t, –x) =F(t,x),
lim
|x|→∞
(∇F(t,x),x) |x| = –∞,
and
Ft,x(t)≥–
x+x+x +x
≥–
+x+x+ x+x– =
x+x– .
Letc= andM=. Then Theorem . gives the conclusion.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, P.R. China.2Department of Mathematics,
University of Science and Technology Beijing, Beijing, 100083, P.R. China.
Acknowledgements
This work is supported by Beijing Higher Education Young Elite Teacher Project Grant No. YETP0388, and Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP), Grant No. 20120006120007. The authors would like to thank the referee for their valuable comments and suggestions on the original manuscript.
Received: 2 December 2015 Accepted: 14 February 2016 References
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