R E S E A R C H
Open Access
Existence of periodic solutions of a Liénard
equation with a singularity of repulsive type
Shiping Lu, Yajiao Wang
*and Yuanzhi Guo
*Correspondence:
College of Math & Statistics, Nanjing University of Information Science & Technology, Nanjing, 210044, China
Abstract
In this paper, the problem of positive periodic solutions is studied for the Liénard equation with a singularity of repulsive type,
x+f(x)x–
α
(t)xμ =h(t),
wheref: (0, +∞)→Ris continuous,
α
,hare continuous withT-periodic andα
(t)≥0 for allt∈R,μ
∈(0, +∞) is a constant. By means of a Manásevich-Mawhin’scontinuation theorem, a sufficient and necessary condition is obtained for the existence of positiveT-periodic solutions of the equation. The interesting point is that the weak singularity of restoring forceαx(μt)atx= 0 is allowed andfmay have a singularity atx= 0.
Keywords: Liénard equation; Manásevich-Mawhin’s continuation theorem; singularity; periodic solution
1 Introduction
In the past years, much attention from researchers in differential equations was paid to investigating the problem of periodic solutions for second order differential equations with singularities. This is due to the fact that the singularity has a significant background in applied sciences and physics (see [–] and the references therein). The first study on the periodic problem for second order singular differential equations seems to be the work of Nagumo [] in . After some work [–], the interest increased in this area with the pioneering paper of Lazer and Solimini []. They considered the existence of periodic solutions for the equation with a singularity of repulsive type,
x–
xα(t)=h(t), (.)
whereh:R→Ris continuous withT-periodic,α∈(, +∞) is a constant. Forα∈[, +∞) (called the strong force condition), by using topological degree methods, they found that the necessary and sufficient condition for the existence of positive periodic solutions for equation (.) is
h:=
T
T
h(s)ds< .
Forα∈(, ) (weak singularity condition), they produced some examples ofh(t) withh< and such that equation (.) does not have any positiveT-periodic solution. After that, the strong force conditionα≥ was regarded as crucial assumption in [–]. By using some fixed point theorems in cones, the existence of periodic solutions has been widely studied recently for the conservative equation of repulsive type,
x+a(t)x– b(t)
xα(t)=c(t), (.)
where a,b,c∈L[,T] witha(t)≥,b(t)≥ for a.e.t∈[,T] and being positive in a
set of positive measure [–]. Most of them focus on the case in which the singular nonlinearity was allowed to have a weak singularity (α∈(, )). Compared with the study of conservative equations with weak singularities, the corresponding ones of the Liénard equation with a weak singularity of repulsive type is considerably neglected. We find that the strong singularity is needed in the most recent papers associated to singular Liénard equation of repulsive type [–]. For example, Jebelean and Mawhin in [] considered the problem of the existence of positive periodic solutions for ap-Laplacian Liénard equa-tion like
xp–x+f(x)x– β
xμ=h(t), (.)
where p> ,β> ,μ> are constants,f : [, +∞)→Ris continuous,h:R→Ris a
T-periodic function withh∈L∞[,T]. Under the condition of strong singularityμ≥, they found that the necessary and sufficient condition for the existence of positive periodic solutions for equation (.) ish< . Wang in [] further studied the existence of positive periodic solutions for a delay Liénard equation with a strong singularity (μ∈[, +∞)) of repulsive type,
x+f(x)x+a(t)x(t–τ) – β
xμ(t–τ)=h(t). (.)
Hakl, Torres and Zamora in [] considered the periodic problem for the singular equation of repulsive type,
u(t) +fu(t)u(t) +ϕ(t)u(t)δ+gu(t)= , (.)
where δ ∈(, ] is a constant, ϕ is a T-periodic function with ϕ ∈ L([,T],R), f ∈ C((, +∞),R) may be singular at x= ,g∈C((, +∞),R) has a repulsive singularity at
x= ,i.e.,g(x)→–∞asx→+. By using Schauder’s fixed point theorem, some results
on the existence of positiveT-periodic solutions are obtained. However, a strong singu-larity,g(s)ds= –∞, is also required.
Now, the question is how to study the periodic problem of equation (.) under the condition of weak singularityg(s)ds> –∞. Motivated by this, the purpose of this paper is to investigate the existence of positiveT-periodic solutions for Liénard equation with a singularity of repulsive type
x+f(x)x–α(t)
where f : (, +∞)→Ris continuous, α, h are continuous T-periodic functions with
α(t)≥ for allt∈Randα(t)≡,μ∈(, +∞) is a constant. By using a continuation the-orem established by Mawhin and Manásevich [], some new results are obtained. The interesting point is that the weak singularity of restoring force term αx(t)μ atx= is allowed andf may have a singularity atx= . Furthermore, under the condition off(s)ds=∞, a sufficient and necessary condition is obtained for the existence of positiveT-periodic solutions of equation (.).
2 Preliminary lemmas
Let CT =x∈C(R,R) :x(t+T) =x(t) for all t∈Rwith the norm defined by |x|∞ =
maxt∈[,T]|x(t)|. For any T-periodic solution h(t) with h∈CT, h+(t) and h–(t) is
de-noted by max{(h(t), )} and –min{(h(t), )}, respectively, and h= T Th(s)ds. Clearly,
h(t) =h+(t) –h–(t) for allt∈R, andh=h+–h–. Furthermore, ϕ p:= (
T |ϕ(t)|
pdt)p,
p∈[, +∞),ϕ∈CT.
The following lemma is a corollary of Theorem . in [].
Lemma . Assume that there exist positive constants M,Mand Mwith <M<M, such that the following conditions hold:
. for eachλ∈(, ],each possible positiveT-periodic solutionxto the equation
u+λf(u)u–λα(t)
uμ =λh(t),
satisfies the inequalitiesM<x(t) <Mand|x(t)|<Mfor allt∈[,T];
. each possible solutionx∈(, +∞)to the equation
α
xμ +h=
satisfies the inequalityM<x<M;
. the inequality
α
Mμ +h
α
Mμ +h
<
holds.Then equation(.)has at least one positive T -periodic solution x(t)such that M< x(t) <Mfor all t∈[,T].
Lemma .([]) Let x(t)be a continuously differentiable T -periodic function.Then,for anyτ∈[,T],
T
x(t)dt
≤T
π
T
x(t)dt
+√Tx(τ).
In order to study the existence of positive periodic solutions to equation (.), we list the following assumptions:
(H) limx→+|
xf(s)ds|= +∞.
(H) (–αh)
μ >T[T
π h + (T
(–α
h)
Remark . Ifh< , then there are constantsDandDwith <D<Dsuch that
α
xμ +h> ∀x∈(,D)
and
α
xμ +h< ∀x∈(D,∞).
Now, we embed equation (.) into the following equations family with a parameter
λ∈(, ],
x+λf(x)x–λα(t)
xμ =λh(t), λ∈(, ]. (.)
3 Main results
Theorem . If(H)holds,then equation(.)has a positive T -periodic solution if and only if h< .
Proof Suppose that equation (.) has a positiveT-periodic solutiony(t), then
y+f(y)y–α(t)
yμ =h(t). (.)
Integrating (.) on the interval [,T], and by using
T
y(s)ds= T
fy(s)y(s)ds= ,
we have T
α(s)
yμ(s)ds= –Th,
which together with the assumption ofα(t)≥ andy(t) > for allt∈[,T] gives a neces-sary condition for the existence of a positiveT-periodic solution of equation (.):h< .
Below, we will show the proof of sufficiency. In order to do it, suppose thath< , and let
ube an arbitrary positiveT-periodic solution of (.). Then
u+λf(u)u–λα(t)
uμ =λh(t), λ∈(, ]. (.)
Integrating (.) over the interval [,T], we have
T
α(t)
uμ dt= – T
h(t)dt= –Th. (.)
Due to the fact thatα(t) is non-negative,
uμM T
α(t)dt≤
T
α(t) uμ(t)dt≤
uμm T
α(t)dt, where um,uM are the global minimum and maximum, respectively, ofu. Then there is a point
η∈[,T] such that
uμ(η) T
which results in
Tα= –uμ(η)Th,
and then
u(η) = –α h μ . (.)
Multiplying (.) withu(t), and integrating it over the interval [,T], we obtain
T
u(t)u(t)dt–λ
T
α(t)u(t)
uμ(t) dt=λ T
h(t)u(t)dt.
By usingTu(t)u(t)dt= –T|u(t)|dt, we have
T
u(t)dt+λ
T
α(t)u–μ(t)dt= –λ T
h(t)u(t)dt,
which together with the fact ofα(t)≥ for allt∈[,T] andα(t)≡ gives
T
u(t)dt< T
h(t)u(t)dt≤
T
h(t)d
T
u(t)dt
. (.)
By Lemma ., we have
T
u(t)dt
<T π T
u(t)dt
+√Tu(η).
It follows from (.) that
T
u(t)dt
<T π T
u(t)dt
+√T
–α h μ .
Substituting it into (.), we have
T
u(t)dt< T
h(t)dt
T
π
T
u(t)dt
+√T
–α h μ =T π T
h(t)dt
T
u(t)dt
+T
–α
h
μ T
h(t)dt
,
by the inequalityX–AX–B< , we can obtainX<A+B. LetX= (T
|u(t)|dt) ,
A=T
π( T
h
(t)dt),B=T(–α
h)
μ(T
h
(t)dt), we have
T
u(t)dt
<T
π h +T
–α h μ h
Combined (.) with (.), we have
u(t)≤u(η) +√Tu<
–α
h
μ
+Tρ:=M. (.)
Lett,tbe the maximum point and the minimum point ofu(t) on [,T], respectively,
then t
t
u(t)dt+λ
t
t
fu(t)u(t)dt–λ
t
t
α(t)
uμ(t)dt=λ t
t
h(t)dt,
which together withu(t) =u(t) = yields
Fu(t)
–Fu(t)
=
t
t
α(t)
uμ(t)dt+ t
t
h(t)dt,
whereF(x) =xf(s)ds, and then
Fu(t)≤F
u(t)+
T
α(t)
uμ(t)dt+ T
h(t)dt.
It follows from (.) and (.) that
Fu(t)≤ max (–α
h)
μ≤z≤M
F(z)–Th+T|h|
≤ max
(–α
h)
μ≤z≤M
F(z)+ Th–. (.)
It is easy to see from (H) that there is a constantγ> , such that
F(z)= z
f(s)ds> max
(–α
h)
μ≤z≤M
F(z)+ Th– forz∈(,γ].
By (.), we have
u(t) >γ,
and then
u(t) >γ for allt∈[,T]. (.)
Next, ifuattains its maximum over [,T] att∈[,T], thenu(t) = and we see from
(.) that
u(t) =λ
t
t
–f(u)u+α(t)
uμ +h(t)
Therefore,
u(t)≤λFu(t)–Fu(t)+λ
t+T
t
α(t)
uμ(t)dt+λ t+T
t
h+(s)ds
≤λ max γ≤u≤M
F(u)+λ
T
α(t)
uμ(t)dt+Th+. (.)
Substituting (.) into (.), we have
u(t)≤ max γ≤u≤M
F(u)–Th+Th+:=M∗ for allt∈[,T]
and then
u(t)≤M∗ for allt∈[,T]. (.)
From (.), (.), (.) and Remark ., we can chooseM:=min{γ,D}whereDis
de-termined in Remark .,M=MandM=M∗such that all the conditions of Lemma .
are satisfied. Thus, by using Lemma ., we see that equation (.) has at least one positive
T-periodic solution. The proof is complete.
Theorem . Suppose that h< and(H)holds.Then equation(.)has at least one positive T -periodic solution.
Proof Suppose thatuis an arbitrary positiveT-periodic solution of equation (.), then
u+λf(u)u–λα(t)
uμ =λh(t), λ∈(, ]. (.)
Similar to the proof of (.) and (.), we find that there is a pointξ∈[,T] such that
u(ξ) =
–α
h
μ
(.)
and
T
u(t)dt
<T
π h +T
–α
h
μ
h
:=ρ. (.)
In view of the inequality
u(t) =u(ξ) + t
ξ
u(s)ds≤u(ξ) + ξ+T
ξ
u(s)ds
=u(ξ) + T
u(s)ds, t∈[ξ,ξ+T],
and by using (.), together with Schwarz inequality, we have
max
t∈[,T]u(t)≤
–α
h
μ +T
T
u(s)ds
and
min
t∈[,T]u(t)≥u(ξ) –
ξtu(s)ds≥
–α
h
μ –T
T
u(s)ds
. (.)
Substituting (.) into (.) and (.), respectively, we have
max
t∈[,T]u(t)≤
–α
h
μ
+T T
π h +T
–α h μ h (.) and min
t∈[,T]u(t)≥
–α
h
μ
–T T
π h +T
–α h μ h . (.)
The rest of the proof works almost analogously to the corresponding ones of
Theo-rem ..
Example . Considering the following equation:
x(t) + x
(t)
x(t)–
sint
x
= – +cost. (.)
Corresponding to equation (.), we havef(x) =x,μ=,α(t) =sint,h(t) = –+cost, and
thenT= π,f(s)ds= +∞. This implies that assumption (H) holds. Sinceh= – < ,
by using Theorem ., we find that equation (.) has at least one positive π-periodic solution.
Example . Considering the following equation:
x(t) + x
(t)
x(t)
–sin
t
x
= –cost. (.)
Corresponding to equation (.),f can be regarded asf(x) =
x
,μ=,α(t) =sintand
h(t) = –cost. Since
f(s)ds= , it follows that assumption (H) does not hold. This
implies that Theorem . cannot be used to study the existence of periodic solutions to (.). But, by simple calculating, we can verify that
–h
α = , h
=
T
,
whereT=π
, and then
–α
h
μ
–T T
π h +
T
–α
h
μ
h
= – √ π + √ π > ,
which implies that assumption (H) holds. Thus, by using Theorem ., we find that (.)
has at least one positive π
Remark The above two examples can neither be studied by using the results in [, , ] and [], sincef(x) in (.) and in (.) are all singular atx= , nor be studied by using the results in [], since the restoring force terms ofsint
x
in (.) andsint
x
in (.)
have weak singularities atx= .
Acknowledgements
The authors thank the referees for valuable comments. This research is supported by the NSF of China (No. 11271197).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 5 April 2017 Accepted: 13 June 2017
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