R E S E A R C H
Open Access
Positive periodic solution of
p
-Laplacian
Liénard type differential equation with
singularity and deviating argument
Yun Xin
1*and Zhibo Cheng
2*Correspondence:
1College of Computer Science and
Technology, Henan Polytechnic University, Jiaozuo, 454000, China Full list of author information is available at the end of the article
Abstract
In this paper, we consider the followingp-Laplacian Liénard type differential equation with singularity and deviating argument:
(ϕ
p(
x(t)))
+f(
x(t))
x(t) +g(
t,x(t–σ
))
=e(t).By applications of coincidence degree theory and some analysis techniques, sufficient conditions for the existence of positive periodic solutions are established.
MSC: 34C25; 34K13; 34K40
Keywords: positive solution;p-Laplacian; Liénard equation; singularity; deviating argument
1 Introduction
In this paper, we consider the followingp-Laplacian Liénard type differential equation with singularity and deviating argument:
ϕp
x(t)+fx(t)x(t) +gt,x(t–σ)=e(t), (.)
whereϕp:R→Ris given byϕp(s) =|s|p–s, herep> is a constant,f is continuous
func-tion;gis a continuous function defined onRand periodic intwithg(t,·) =g(t+T,·),g
has a singularity atx= ;σis a constant and ≤σ<T;e:R→Rare continuous periodic functions withe(t+T)≡e(t) andTe(t)dt= .
As is well known, the existence of periodic solutions for Liénard type differential equa-tions was extensively studied (see [–] and the references therein). In recent years, there also appeared some results on a Liénard type differential equation with singularity; see [, ]. In , using coincidence degree theory, Zhang considered the existence ofT -periodic solutions for the scalar Liénard equation
x(t) +fx(t)x(t) +gt,x(t)= ,
whengbecomes unbounded asx→+. The main emphasis was on the repulsive case,i.e.
wheng(t,x)→+∞, asx→+. Afterwards, Wang [] studied the existence of periodic
solutions of the Liénard equation with a singularity and a deviating argument,
x(t) +fx(t)x(t) +gt,x(t–σ)= ,
whereσ is a constant. Whenghas a strong singularity atx= and satisfies a new small force condition atx=∞, the author proved that the given equation has at least one positive
T-periodic solution.
However, the Liénard type differential equation (.), in which there is a p-Laplacian Liénard type differential equation, has not attracted much attention in the literature. There are not so many existence results for (.) even as regards thep-Laplacian Liénard type differential equation with singularity and deviating argument. In this paper, we try to fill this gap and establish the existence of a positive periodic solution of (.) using coincidence degree theory. Our new results generalize in several aspects some recent results contained in [, ].
2 Preparation
LetXandY be real Banach spaces andL:D(L)⊂X→Y be a Fredholm operator with index zero, hereD(L) denotes the domain ofL. This means thatImLis closed inY and
dim KerL=dim(Y/ImL) < +∞. Consider supplementary subspacesX,YofX,Y,
respec-tively, such thatX=KerL⊕X,Y=ImL⊕Y. LetP:X→KerLandQ:Y→Ydenote the
natural projections. Clearly,KerL∩(D(L)∩X) ={}and so the restrictionLP:=L|D(L)∩X
is invertible. LetKdenote the inverse ofLP.
Letbe an open bounded subset ofXwithD(L)∩=∅. A mapN:→Yis said to beL-compact inifQN() is bounded and the operatorK(I–Q)N:→Xis compact.
Lemma .(Gaines and Mawhin []) Suppose that X and Y are two Banach spaces,and L:D(L)⊂X→Y is a Fredholm operator with index zero.Let⊂X be an open bounded set and N:→Y be L-compact on.Assume that the following conditions hold:
() Lx=λNx,∀x∈∂∩D(L),λ∈(, ); () Nx∈/ImL,∀x∈∂∩KerL;
() deg{JQN,∩KerL, } = ,whereJ:ImQ→KerLis an isomorphism. Then the equation Lx=Nx has a solution in∩D(L).
For the sake of convenience, throughout this paper we will adopt the following notation:
|u|∞= max
t∈[,T]
u(t), |u|= min
t∈[,T] u(t),
|u|p=
T
|u|pdt
p
, h¯=
T T
h(t)dt.
Lemma .([]) Ifω∈C(R,R)andω() =ω(T) = ,then
T
ω(t)pdt≤
T
πp
p T
ω(t)pdt,
where≤p<∞,πp=
(p–)/p
ds
(–psp–)/p =
π(p–)/p
Lemma . If x∈C(R,R)with x(t+T) =x(t),and t
∈[,T]such that|x(t)|<d,then
T
x(t)pdt
p
≤
T
πp
T
x(t)pdt
p
+dTp.
Proof Letω(t) =x(t+t) –x(t), and thenω() =ω(T) = . By Lemma . and Minkowski’s
inequality, we have
T
x(t)pdt
p
=
T
ω(t) +x(t)
p
dt
p
≤
T
ω(t)pdt
p
+
T
x(t)pdt
p
≤
T
πp
T
ω(t)pdt
p
+dTp
=
T
πp
T
x(t)pdt
p
+dTp.
This completes the proof of Lemma ..
In order to apply the topological degree theorem to study the existence of a positive periodic solution for (.), we rewrite (.) in the form
⎧ ⎨ ⎩
x(t) =ϕq(x(t)),
x(t) = –f(x(t))x(t) –g(t,x(t–σ)) +e(t),
(.)
where p +q = . Clearly, ifx(t) = (x(t),x(t)) is anT-periodic solution to (.), then x(t) must be anT-periodic solution to (.). Thus, the problem of finding anT-periodic
solution for (.) reduces to finding one for (.).
Now, setX=Y ={x= (x(t),x(t))∈C(R,R) :x(t+T)≡x(t)} with the normx= max{|x|∞,|x|∞}. Clearly,XandYare both Banach spaces. Meanwhile, define
L:D(L) =x∈CR,R:x(t+T) =x(t),t∈R⊂X→Y
by
(Lx)(t) =
x(t)
x(t)
andN:X→Yby
(Nx)(t) =
ϕq(x(t))
–f(x(t))x(t) –g(t,x(t–σ)) +e(t)
Then (.) can be converted to the abstract equationLx=Nx. From the definition ofL, one can easily see that
KerL∼=R, ImL=
y∈Y:
T
y(s) y(s)
ds=
.
SoLis a Fredholm operator with index zero. LetP:X→KerLandQ:Y→ImQ⊂Rbe
defined by
Px=
(Ax)() x()
; Qy=
T T
y(s) y(s)
ds,
thenImP=KerL,KerQ=ImL. LetK denote the inverse ofL|Kerp∩D(L). It is easy to see
thatKerL=ImQ=Rand
[Ky](t) =
T
G(t,s)y(s)ds,
where
G(t,s) =
⎧ ⎨ ⎩
s
T, ≤s<t≤T; s–t
T, ≤t≤s≤T.
(.)
From (.) and (.), it is clear thatQNandK(I–Q)Nare continuous,QN() is bounded and then K(I–Q)N() is compact for any open bounded⊂X, which means N is
L-compact on.
3 Main results Assume that
ψ(t) = lim
x→+∞sup g(t,x)
xp– (.)
exists uniformly a.e.t∈[,T],i.e., for anyε> there isgε∈L(,T) such that
g(t,x)≤ψ(t) +εx+gε(t), (.)
for allx> and a.e.t∈[,T]. Moreover,ψ∈C(R,R) andψ(t+T) =ψ(t).
For the sake of convenience, we list the following assumptions which will be used re-peatedly in the sequel:
(H) (Balance condition) There exist constants <D<D such that ifxis a positive
continuousT-periodic function satisfying
T
gt,x(t)dt= ,
then
D≤x(τ)≤D,
(H) (Degree condition)g¯(x) < for allx∈(,D), andg¯(x) > for allx>D.
(H) (Decomposition condition)g(t,x) =g(x) +g(t,x), where g∈C((,∞);R) and g: [,T]×[,∞)→Ris anL-Carathéodory function,i.e.it is measurable in the first
variable and continuous in the second variable, and for anyb> there ishb∈L(,T;R+)
such that
g(t,x)≤hb(t), a.e.t∈[,T],∀≤x≤b.
(H) (Strong force condition atx= )
g(x)dx= –∞.
Theorem . Assume that conditions(H)-(H)hold.Suppose the following condition is satisfied:
(H) (πTp)
p|ψ|
∞< .
Then(.)has at least one positive T -periodic solution.
Proof Consider the equation
Lx=λNx, λ∈(, ).
Set={x:Lx=λNx,λ∈(, )}. Ifx(t) = (x(t),x(t))∈, then
⎧ ⎨ ⎩
x(t) =λϕq(x(t)),
x(t) = –λf(x(t))x(t) –λg(t,x(t–σ)) +λe(t).
(.)
Substitutingx(t) = λp–ϕp(x(t)) into the second equation of (.)
ϕp
x(t)+λpfx(t)
x(t) +λpgt,x(t–σ)
=λpe(t). (.)
Integrating both sides of (.) over [,T], we have
T
gt,x(t–σ)
dt= . (.)
From (H), there exist positive constantsD,D, andξ∈[,T] such that
D≤x(ξ)≤D. (.)
Then we have
x(t)= x(ξ) +
t
ξ
x(s)ds≤D+ t
ξ
x(s)ds, t∈[ξ,ξ+T],
and
x(t)=x(t–T)= x(ξ) –
ξ
t–T
x(s)ds≤D+ ξ
t–T
Then we get from (.) and (.)
By the second equations of (.) and (.), we obtain
T
On the other hand, it follows from (.) that
Letτ∈[,T], for anyτ≤t≤T, we integrate (.) on [τ,t] and get
From these inequalities we can derive from (.) that
condition (H), we know that there exists a constantM> such that
x(t)≥M, ∀t∈[τ,T]. (.)
From (.), (.), and (.), we let
=x= (x,x):E≤ |x|∞≤E,|x|∞≤E,∀t∈[,T]
,
where <E<min(M,D),E>max(M,D),E>M. ={x:x∈∂∩KerL}then ∀x∈∂∩KerL
QNx=
T T
ϕq(x(t))
–f(x(t))x(t) –g(t,x(t–σ)) +e(t)
dt.
IfQNx= , thenx(t) = ,x=Eor –E. But ifx(t) =E, we know
=
T
g(t,E) –e(t)
dt.
From assumption (H), we havex(t)≤D≤E, which yields a contradiction. Similarly if x= –E. We also haveQNx= ,i.e.,∀x∈∂∩KerL,x∈/ImL, so conditions () and () of
Lemma . are both satisfied. Define the isomorphismJ:ImQ→KerLas follows:
J(x,x)= (x, –x).
LetH(μ,x) = –μx+ ( –μ)JQNx, (μ,x)∈[, ]×, then∀(μ,x)∈(, )×(∂∩KerL),
H(μ,x) =
–μx––Tμ T
[g(t,x) –e(t)]dt
–μx– ( –μ)|x|p–x
.
We haveTe(t)dt= . So, we can get
H(μ,x) =
–μx––Tμ T
g(t,x)dt
–μx– ( –μ)|x|p–x
,
∀(μ,x)∈(, )×(∂∩KerL).
From (H), it is obvious thatxH(μ,x) < ,∀(μ,x)∈(, )×(∂∩KerL). Hence
deg{JQN,∩KerL, }=degH(,x),∩KerL,
=degH(,x),∩KerL,
=deg{I,∩KerL, } = .
So condition () of Lemma . is satisfied. By applying Lemma ., we conclude that the equationLx=Nxhas a solutionx= (x,x) on¯ ∩D(L),i.e., (.) has anT-periodic
solutionx(t).
Example . Consider thep-Laplacian Liénard type differential equation with singularity and deviating argument:
ϕp
x(t)+fx(t)x(t) +
(cost+ )x
(t–σ) –
xκ(t–σ)=sint, (.)
whereκ≥ andp= ,f is a continuous function,σis a constant, and ≤σ<T. It is clear thatT=π,g(t,x) =(cost+ )x(t–σ) –
xκ(t–σ),ψ(t) =(cost+ ). It is obvious that (H)-(H) hold. Now we consider the assumption (H). Since|ψ|∞≤, we
have
T
πp
p
|ψ|∞=
T π(p–)/p
psin(π/p) p
|ψ|∞≤
π
π(–)/ sinπ/
×
= < .
So by Theorem ., we know (.) has at least one positiveπ-periodic solution.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
YX and ZBC worked together in the derivation of the mathematical results. Both authors read and approved the final manuscript.
Author details
1College of Computer Science and Technology, Henan Polytechnic University, Jiaozuo, 454000, China.2School of
Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, China.
Acknowledgements
YX and ZBC would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by NSFC Project (No. 11501170), Fundamental Research Funds for the Universities of Henan Province (NSFRF140142), Henan Polytechnic University Outstanding Youth Fund (J2015-02) and Henan Polytechnic University Doctor Fund (B2013-055).
Received: 22 May 2015 Accepted: 8 December 2015
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