R E S E A R C H
Open Access
Positive solutions of periodic boundary
value problems for the second-order
differential equation with a parameter
Ying Wang
*, Jing Li and Zengxia Cai
*Correspondence:
lywy1981@163.com School of Mathematics and Statistics, Linyi University, Linyi, Shandong 276000, People’s Republic of China
Abstract
In this paper, we investigate the existence of positive solutions for a class of singular second-order differential equations with periodic boundary conditions. By using the fixed point theory in cones, the explicit range for
λ
is derived such that for anyλ
lying in this interval, the existence of at least one positive solution to the boundary value problem is guaranteed.MSC: 34B16; 34B18
Keywords: positive solution; periodic boundary conditions; second-order differential equation; singularity
1 Introduction
Reaction-diffusion problems often arise in physics, chemistry, biology, economics, and various engineering fields. A class of reaction-diffusion equations
wt=wxx+H(w) ()
includes several known evolution equations. For equation (), if traveling wave satisfies w(x,t) =W(x–Ct) with speedC, then equation () can be converted to a second-order ordinary differential equation
W+CW+H(W) = . ()
With appropriate boundary value conditions, the existence of positive solution of equa-tion () is significant and helpful. The Liebau phenomenon, which is in honor of the physi-cian Liebauh’s pioneering work, is the occurrence of valveless pumping through the appli-cation of a periodic force at a place which lies asymmetric with respect to system config-uration. Propst [] made use of differential equations to model a periodically forced flow through different pipe-tank configurations. In one pipe-one tank configuration, ignore the singularity in the corresponding differential equation model, namely
⎧ ⎨ ⎩
u(t) +au(t) +
u(b(u(t))
–e(t)) +c= , t∈[,T],
u() =u(T), u() =u(T). ()
According to the physical meaning of the involved parameters, assumea≥,b> ,c> , andeis continuous andT-periodic on [, +∞). In [], Cid et al. applied the substitution u=xμ,μ=
b+, and then transformed the singular periodic boundary value problem ()
to the regular problem
⎧ ⎨ ⎩
x(t) +ax(t) +s(t)xβ–r(t)xα= , t∈[,T],
x() =x(T), x() =x(T), ()
wherer(t) = e(μt),s(t) = μc,α= – μ,β= –μ. Based on the lower and upper solution technique, the existence and asymptotic stability of positive solutions for () are obtained. In this paper, we discuss the positive solutions of the following periodic boundary value problem (PBVP):
⎧ ⎨ ⎩
x(t) +ax(t) +kx(t) =λf(t,x(t)), t∈(,T),
x() =x(T), x() =x(T), ()
wherea≥,k∈(–∞, +∞),λ> is a parameter,f: (,T)×(, +∞)→[, +∞) is a con-tinuous function andf(t,u) may be singular att= ,t=Tandu= .
In recent years, the existence of solutions for differential equations has been widely stud-ied by many scholars in the mathematical sense (see [–] and the references therein). In [, ], through the use of Guo-Krasnosel’skii’s fixed point theorem, the existence and multiplicity of positive solutions for the following periodic boundary value problem were established.
⎧ ⎨ ⎩
–x(t) +ρx(t) =f(t,x(t)), t∈[, π],ρ> ,
x() =x(π), x() =x(π), ()
⎧ ⎨ ⎩
x(t) +ρx(t) =f(t,x(t)), t∈[, π], <ρ< ,
x() =x(π), x() =x(π). ()
In [], the author researched PBVP () by using anLp-anti-maximum principle and ob-tained the existence results in order to overcome the difficulties of the symbol of Green’s functions for the corresponding linear periodic problem:
⎧ ⎨ ⎩
x(t) +a(t)x(t) =f(x(t)) +a(t)x(t), t∈[,T],
x() =x(T), x() =x(T). ()
Motivated by the above works, we consider PBVP (). In (), iff(t,x(t)) =mx(t)+s(t)xβ–
r(t)xα, then () is a special case of (). Compared with [, ], in which the existence and
multiplicity of positive solutions for () () are considered, we not only obtain the existence of positive solutions for (), but also increase the parameterλand get the explicit range of λby using the fixed point theory in cones. Therefore, our article contains, promotes, and
2 Preliminaries and lemmas
In this section, we present some notations and lemmas that will be used in the proof of our main results.
Lemma .([, ]) Let h∈C(,T)∩L(,T),then the boundary value problem
⎧ ⎨ ⎩
x(t) +ax(t) +kx(t) =h(t), t∈[,T],
x() =x(T), x() =x(T)
()
has an integral representation
x(t) = T
G(t,s)h(s)ds,
where G(t,s)is the related Green’s function.
Lemma .([]) Assume that condition(H)holds,
(H) k> ,k< (πT)+ (a).
Then G(t,s)has the following properties:
(i) G(t,s) > ,(t,s)∈[,T]×[,T]; (ii) TG(t,s)ds=
k;
(iii) There exists a constantξ∈(, )such thatG(t,s)≥G(s,s)≥ξG(t,s),
(t,s)∈[,T]×[,T].
LetX=C[,T], thenXis a Banach space with the normx=maxt∈[,T]|x(t)|. Denote
K=x∈X:x(t)≥ξx,t∈[,T],
whereξis defined as Lemma .. It is easy to see thatKis a positive and normal cone inX. For any <r<R< +∞, letKr,R={x∈K:r≤ x ≤R}. In this paper, we always assume
that the following conditions hold.
(H) f : (,T)×(, +∞)→[, +∞)is a continuous function and
f(t,u)≤φ(t)g(u) +h(u), (t,u)∈(,T)×(, +∞),
whereφ: (,T)→[, +∞)is continuous and singular att= ,T,φ(t)≡on[, +∞),
g : (, +∞)→[, +∞)is continuous and nonincreasing,h: [, +∞)→[, +∞)is continuous.
(H)
T
G(s,s)φ(s)ds< +∞.
Under assumptions (H)-(H), for anyn∈N, Nis a natural number set, we define a
nonlinear integral operatorAn:K→Xby
(Anx)(t) =λ
T
G(t,s)fn s,x(s)
wherefn(t,u) =f(t, (u+n)). Obviously, the existence of solutions to () is equivalent to
the existence of solutions inKfor the operator equationAnx=xdefined by (). In this
paper, the proof of the main theorem is based on the fixed point theory in cones. We list the following lemmas which are needed in our study.
Lemma .([]) Let K be a positive cone in a real Banach space X.Denote Kr={x∈K:
x<r},Kr,R={x∈K:r≤ x ≤R}, <r<R< +∞.Let A:Kr,R→K be a completely
continuous operator.If the following conditions are satisfied:
() Ax ≤ x,∀x∈∂KR;
() There existsx∈∂Ksuch thatx=Ax+mx,∀x∈∂Kr,m> .
Then A has fixed points in Kr,R.
Remark . If () and () are satisfied for x ∈ ∂Kr and x∈ ∂KR, respectively, then
Lemma . is still true.
Lemma .([]) Let K be a positive cone in a Banach space E,andbe bounded
open sets in E,θ∈,⊂,A:K∩\→P be a completely continuous operator.
If the following conditions are satisfied:
Ax ≤ x, ∀x∈K∩∂, Ax ≥ x, ∀x∈K∩∂,
or
Ax ≥ x, ∀x∈K∩∂, Ax ≤ x, ∀x∈K∩∂,
then A has at least one fixed point in K∩(\).
3 Main results
Theorem . Assume that(H)-(H)hold,then An:K→K is a completely continuous
operator for any fixed n∈N.
Proof Letλ> andn∈Nbe fixed. For anyx∈Kandt∈[,T], by Lemma ., we have
λ T
G(s,s)fn s,x(s)
ds≤(Anx)(t) =λ
T
G(t,s)fn s,x(s)
ds
≤λ
ξ T
G(t,s)fn s,x(s)
ds.
This implies that (Anx)(t)≥ξAnx, thereforeAn(K)⊂K. By a standard argument, under
assumptions (H)-(H), we know thatAn:K→Kis well defined.
Next, for any positive integersn,m∈N, we define an operatorAn,m:K→Xby
(An,mx)(t) =λ
T–m m
G(t,s)fn s,x(s)
ds, t∈[,T]. ()
In a similar discussion,An,m:K→Xis well defined andAn,m(K)⊆K. In what follows, we
An,m:K→K is continuous. Letxυ,x∈Ksatisfyxυ–x → asυ→+∞. Notice that
t∈[m,T–m],|fn(t,xυ(t)) –fn(t,x(t))| → asυ→+∞. Using the Lebesgue dominated
convergence theorem, we have
λ
T–m m
G(t,s)fn s,xυ(s)
ds–λ
T–m m
G(t,s)fn s,x(s)
ds
≤λ
ξ T–m
m
G(s,s)fn s,xυ(s)
–fn s,x(s)ds→, υ→+∞.
Therefore
An,mxυ–An,mx ≤
λ ξ
T–m m
G(s,s)fn s,xυ(s)
–fn s,x(s)ds→, υ→+∞.
So,An,m:K→C[,T] is continuous for any natural numbersn,m. ThenAn,m:K→Kis
continuous for any natural numbersn,m.
LetD⊂Kbe any bounded set, then for anyx∈D, we havex ≤r, and then <ξr≤ x(t)≤rfor anyt∈[,T]. By (H)-(H), for anyx∈D, we have
λ
T–m m
G(t,s)fn s,x(s)
ds
≤λ
ξ T–m
m
G(s,s)φ(s)
g
x(s) + n
+h
x(s) +
n
ds
≤λ
ξ T–m
m
G(s,s)φ(s)
g
ξr+ n
+h
x(s) +
n
ds
≤λ
ξ T–m
m
G(s,s)φ(s)
g(ξr) + max
y∈[ξr,r+]h(y)
ds
< +∞. ()
So,An,mDis bounded inK.
In order to show thatAn,m is a compact operator, we only need to show thatAn,mDis
equicontinuous. For anyε> , by the continuity ofG(t,s) on [,T]×[,T], there exists δ> such that for anyt,t∈[,T],s∈[m,T–m], and|t–t|<δ, we have
G(t,s) –G(t,s)<ε
λ
T–m m
φ(s)
g(ξr) + max
y∈[ξr,r+]h(y)
ds
–
.
Then, for anyx∈D, for anyt,t∈[,T],s∈[m,T–m], and|t–t|<δ, we have
(An,m)(t) – (An,m)(t)
≤λ T–m
m
G(t,s) –G(t,s)fn s,x(s)
ds
≤λ T–m
m
G(t,s) –G(t,s)φ(s)
g
x(s) +
n
+h
x(s) + n
≤λ T–m
m
G(t,s) –G(t,s)φ(s)
g(ξr) + max
y∈[ξr,r+]h(y)
ds
<ε,
which means thatAn,mDis equicontinuous. By the Arzela-Ascoli theorem,An,mDis a
rel-atively compact set and soAn,m:K→Kis a completely continuous operator.
Finally, we show thatAn:K→Kis a completely continuous operator. For anyt∈[,T]
andx∈S={x∈K,x ≤}, by (), (), we have
λ
m
G(t,s)fn s,x(s)
ds+λ
T T–m
G(t,s)fn s,x(s)
ds
≤λ
ξ
m
+ T
T–m
G(s,s)φ(s)
g
x(s) + n
+h
x(s) +
n
ds
≤λ
ξ
m
+ T
T–m
G(s,s)φ(s)
g
n
+ max
y∈[n,]
h(y)
ds
→, m→+∞.
Hence
An–An,m=sup x∈S
Anx–An,mx →, m→+∞.
Therefore, byAn,m:K→Kis a completely continuous operator, we get thatAn:K→K
is a completely continuous operator.
Theorem . Assume that(H)-(H)hold and f satisfies the following condition:
(H) There exists[a,b]⊂(,T)such that
lim
u→+∞tmin∈[a,b]
f(t,u) u = +∞.
Then there existsλ> such that PBVP()has at least one positive solution for anyλ∈ (,λ).
Proof Chooser> , let
λ=min
,T ξr
G(s,s)φ(s)(g(ξr) +maxy∈[ξr,r+]h(y))ds
.
LetKr={x∈K:x<r}. For anyx∈∂Kr,t∈[,T], by the definition of · , we have
x(t)≤ x ≤r, x(t)≥ξx ≥ξr.
For anyλ∈(,λ), we have
(Anx)(t)=
λ
T
G(t,s)fn s,x(s)
ds
≤ λ
ξ T
G(s,s)φ(s)
g
x(s) + n
+h
x(s) +
n
≤ λ
ξ T
G(s,s)φ(s)
g
ξr+ n
+h
x(s) +
n
ds
≤ λ
ξ T
G(s,s)φ(s)
g(ξr) + max
y∈[ξr,r+]
h(y)
ds
<r.
Thus,
Anx ≤ x for anyx∈∂Kr. ()
On the other hand, by the inequality in (H), choosel> such thatλlξr
b
aG(s,s)ds> ,
then there existsN∗> such that
f(t,u)≥lu, u≥N∗, t∈[a,b].
Letr>max{r,N ∗
ξ },Kr={x∈K:x<r}. Takeq≡∈∂K={x∈K:x= }. For any
x∈∂Kr,μ> ,n∈N, we will show
x=Anx+μq. ()
Otherwise, there existx∈∂Kr andμ> such thatx=Anx+μq. Fromx∈∂Kr,
we know thatx=r. Then, fort∈[a,b], we have
x(t)≥ξx ≥ξr≥N∗.
Hence, we conclude that
x(t) =λ
T
G(t,s)fn s,x(s)
ds+μ
≥λ b
a
G(s,s)fn s,x(s)
ds+μ
≥λ b
a
G(s,s)lξrds+μ
≥r+μ>r.
This implies thatr>r, which is a contradiction. This yields that () holds.
It follows from the above discussion, (), (), Lemma . and Theorem . that, for any n∈N,λ∈(,λ),Anhas a fixed pointxn∈Kr\Kr.
Let{xn}∞n=be the sequence of solutions of PBVP (). It is easy to see that they are
uni-formly bounded. Fromxn∈Kr\Kr, we know that
r≥ xn ≥xn(t)≥ξxn ≥ξr, t∈[,T].
For anyε> , by the continuity ofG(t,s) on [,T]×[,T], there existsδ> such that for
anyt,t,s∈[,T],|t–t|<δ, we have
G(t,s) –G(t,s)<ε
λ
T
φ(s)
g(ξr) + max
y∈[ξr,r+]
h(y)
ds –
Then, for anyt,t,s∈[,T],|t–t|<δ, we obtain
xn(t) –xn(t)
≤λ T
G(t,s) –G(t,s)fn s,xn(s)
ds
≤λ T
G(t,s) –G(t,s)φ(s)
g
xn(s) +
n
+h
xn(s) +
n
ds
≤λ T
G(t,s) –G(t,s)φ(s)
g(ξr) + max
y∈[ξr,r+]h(y)
ds
<ε. ()
Similarly to (), together with (), by the Ascoli-Arzela theorem, the sequence{xn}∞n=
has a subsequence being uniformly convergent on [,T]. Without loss of generality, we still assume that{xn}∞n=itself uniformly converges toxon [,T]. Since{xn}∞n=∈Kr\Kr⊂K,
we havexn≥. Besides, we have
xn(t) =xn
+xn
t–
–a t
xn(s) –xn
ds
–k t
s
(xn(ς) –λfn ς,xn(ς)
dςds, t∈(,T). ()
Since{xn()}∞n=is bounded, without loss of generality, we may assumexn()→casn→
+∞. Then, by () and the Lebesgue dominated convergence theorem, we have
x(t) =x
+c
t–
–a t
x(s) –x
ds
–k t
s
(x(ς) –λf ς,x(ς)dςds, t∈(,T). ()
By (), the direct computation shows that
x(t) +ax(t) +kx(t) =λf t,x(t), t∈(,T).
On the other hand, letn→+∞in the following boundary conditions:
xn() =xn(T), xn() =xn(T).
Therefore, we deduce thatxis a solution of PBVP (). The proof is completed.
Theorem . Assume that(H)-(H)hold and f satisfies the following condition:
(H) There exists[c,d]⊂(,T)such that
lim inf
u→+∞tmin∈[c,d]f(t,u) >
d
c G(s,s)ds
, lim
u→+∞
Then there existsλ> such that PBVP()has at least one positive solution for anyλ∈ (λ, +∞).
Proof By the first inequality of (H), we have that there existsN∗> such that for any
t∈[c,d],u>N∗, we have
f(t,u) >d
c G(s,s)ds
. ()
Selectλ=max{,Nξ∗}. In the following proof, we supposeλ>λ, chooseR=λ,KR={x∈
K:x<R}. For anyx∈∂KR,t∈[c,d], we have
x(t)≥ξx ≥ξR>N∗.
Then, by (), we have
(Txn)(t)=λ
T
G(t,s)fn s,x(s)
ds≥λ
d c
G(s,s)fn s,x(s)
ds
≥λ d
c
G(s,s)d
c G(s,s)
ds≥R.
Therefore, we have
Anx ≥ x for anyx∈∂KR. ()
Based on the second inequality in (H) and the continuity ofh(u) on [, +∞), for
c=max
,
λ ξ
T
G(s,s)φ(s)ds –
,
there existsN∗> such that whenx≥N∗, for any ≤z≤x, we haveh(z)≤cx. Select
R≥
,R,N∗,
λ ξ
T
G(s,s)φ(s)g(ξR)ds
.
Then, for anyx∈∂KR,t∈[, +∞), we have
x(t)≤ x ≤R, x(t)≥ξx ≥ξR.
Hence, we gain
(Anx)(t)=
λ
T
G(t,s)fn s,x(s)
ds
≤ λ
ξ T
G(s,s)φ(s)
g
x(s) + n
+h
x(s) +
n
ds
≤ λ
ξ T
G(s,s)φ(s)
g
ξR+
n
+h
x(s) +
n
ds
≤ λ
ξ T
G(s,s)φ(s) g(ξR) +c(R+ )
Thus,
Anx ≤ x for anyx∈∂KR. ()
It follows from the above discussion, (), (), Lemma . and Theorem . that, for n∈N,λ∈(λ, +∞),Anhas a fixed pointxn∈KR\KRsatisfyingR≤ xn ≤R. The rest
of the proof is similar to Theorem .. That is the proof of Theorem ..
Corollary . The conclusion of Theorem.is valid if(H)is replaced by the following:
(H∗) There exists[c,d]⊂(,T)such that
lim inf
u→+∞tmin∈[c,d]f(t,u) = +∞, u→lim+∞
h(u) u = .
Remark . From the proof of Theorems . and ., we can obtain the main results under the condition that the function f(t,u) not only has singularity ont but also has singularity onu, and we use the approximation method to overcome the difficulty caused by singularity.
Remark . In this paper, we can get the positive solution of PBVP () when the param-eterλis sufficiently large and small; concretely, we can chooseλ∈(, ) andλ∈(, +∞). What is more, the solutionxin PBVP () satisfiesx(t) > for anyt∈[,T].
4 Examples Consider the PBVP
⎧ ⎨ ⎩
x(t) +x(t) +x(t) =λf(t,x(t)), t∈[, ],
x() =x(), x() =x(), ()
wherea= ,k= , T= . Obviously, (H) holds. Takef(t,u) = √t(– t)(√u+u), we can
supposeφ(t) =√
t(–t),g(u) =
√
u,h(u) =u
. Since the continuous functionG(t,s) is positive
for allt,s∈[,T], there exist constantsC> ,C> such that <C<G(t,s) <Cfor all
t,s∈[,T]. Together withG(t,s)≥G(s,s)≥ξG(t,s), (t,s)∈[,T]×[,T] in Lemma ., we have <ξC<G(s,s) <C, so we can get
T
G(s,s)φ(s)ds=
G(s,s)φ(s)ds<C
√
s( –s)ds
=C
– (s–
)
ds=C
– ((s– ))
ds
=C
–
√
–udu
u=
s–
=Clim
u→arcsinu=π< +∞,
lim
u→+∞tmin∈[a,b]
So all the conditions of Theorem . are satisfied. By Theorem ., PBVP () has at least one positive solution providedλis small enough.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors typed, read, and approved the final manuscript.
Acknowledgements
The authors were supported financially by the National Natural Science Foundation of China (11626125), Natural Science Foundation of Shandong Province of China (ZR2016AP04), the Project of Shandong Province Higher Educational Science and Technology Program (J16LI03), the Science Research Foundation for Doctoral Authorities of Linyi University (LYDX2016BS080), and the Applied Mathematics Enhancement Program of Linyi University.
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Received: 8 October 2016 Accepted: 21 March 2017 References
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