R E S E A R C H
Open Access
A Lazer-Leach-type condition for singular
differential equations with a deviating
argument at resonance
Jin Li
**Correspondence: [email protected] School of Science, Jiujiang University, Jiujiang, 332005, China
Abstract
In this paper, we establish a Lazer-Leach-type condition depending on the delay for the existence of positive periodic solutions for singular differential equations with a deviating argument at resonance. The proof of the main result is based on the phase-plane analysis and topological degree methods.
MSC: 34C25; 34D40
Keywords: periodic solutions; deviating argument; singular; at resonance; Lazer-Leach-type condition
1 Introduction
In this paper, we deal with the following delay differential equations at resonance:
x+ n
x+gx(t),x(t–τ)=p(t), (.)
whereg: (,∞)×R→Ris continuous,τ≥ is a constant,p(t) is continuous and π -periodic, and the functiong has a singularity of repulsive type at the origin for its first variable, that is,lims→∞g(s,s) = –∞.
The periodic problems of singular differential equations had attracted the attentions of many researchers during more than the last two decades because of their background in applied science [–]. A landmark work on mathematical treatment of the differen-tial equations with singularities, as we all know, is done by Lazer and Solimini []. From then on, some classical mathematical tools were used successfully to study these singular equations, such as Mawhin’s continuation theorem in the coincidence degree theory [], the method of upper and lower solutions [], some fixed point theorems in cones [], the Poincaré-Birkhoff theorem [], the phase-plane analysis and topological degree methods [], and so on.
Wang and Ma [] first studied the resonant singular equation
x+ n
x+g(x) =p(t), (.)
whereghas a singularity and satisfies
lim
x→∞g(x) =g(+∞).
They obtained the existence of π-periodic solutions of (.) under the following so-called Lazer-Leach-type condition:
g(+∞) –
π
p(t)sin
θ+nt
dt= for allθ∈R. (.)
Note that they perfectly answered the open problem raised by del Pino and Manase-vich [].
After that, Wang [] discussed the periodic problem of the resonant Liénard equation with constant delay but without singularity and established some Lazer-Leach-type con-ditions depending on the delay. A natural and delicate idea is that the delay may affect the existence of periodic solutions not only for the equations in [], but also for many other kinds of equations. Based on this, in the present paper, we consider equation (.) and look for Lazer-Leach-type conditions. The main difficulty to overcome is the coexistence of the singularity and delay. A possible way for us is to use the phase-plane analysis and topolog-ical degree methods, also used in [], and give the following fundamental hypotheses.
(H) The variables ofg are separable, that is, there exist two functionsg and g such
thatg(x(t),x(t–τ)) =g(x(t)) +g(x(t–τ)). Moreover, g is bounded on [, +∞),
limx→+∞g(x) = , andlimx→+∞g(x) =g(+∞)is finite.
(H) For allθ∈R,
cosnτ
g(+∞)=
π
p(t)sin
n t+θ
dt. (.)
(H) PutG(x) =
x
g(s)ds. It satisfies
lim
x→+g(x) = –∞, xlim→+G(x) = +∞. (.)
Furthermore, there exist <ε≤, <l≤, and <L<∞such that
–g(x) G+l
(x)
>L for allx∈(,ε). (.)
We now state our main theorem.
Theorem . Let(H), (H),and(H)hold.Then(.)has at least one periodic solution.
Remark . (H) is a universal strong singularity condition. We give some examples to show that (H) can be satisfied.
• g(x) = –x. ThenG(x) = –lnx, and (.) holds. Letε=e–andl=L= . By using a
short calculation we can obtain that (.) holds.
• g(x) = –x. ThenG(x) =x –, and (.) holds. Letε=L= andl=. We can
• g,(x) +g,(x), where the functionsg,i(x)(i= , ) satisfy (.) and (.). In fact, it is easy to see that (.) holds forg,(x) +g,(x)andG,(x) +G,(x), where
G,i(x) =xg,i(s)ds(i= , ). On the other hand, we assume that there exist
<ε,i≤, <li≤, and <Li≤ ∞such that
–g,i(x) G+l
,i(x)
>Li for allx∈(,ε,i).
Without loss of generality, we assume also thatG,i(x) > andg,i(x) < –for all
x∈(,ε,i). Setε=min{ε,,ε,}. For allx∈(,ε), ifG,(x) >G,(x), then
–g,(x) –g,(x) [G,(x) +G,(x)]+l >
–g,(x) [G,(x)]+l>
L +l;
otherwise,
–g,(x) –g,(x) [G,(x) +G,(x)]+l >
–g,(x) [G,(x)]+l >
L +l.
Hence (.) holds.
Remark . Whenτ= , condition (.) degenerates to condition (.). Therefore The-orem . generalizes the result in []. Moreover, the delayτ may affect the existence of periodic solutions.
This paper is structured into three sections. Section is devoted to the proof of a useful lemma. In Section , we state some lemmas to prove the main theorem.
2 Preliminary lemma
To use the phase-plane analysis and topological degree methods, we embed (.) into a family of equations with one parameterλ∈[, ],
x+ n
x+ ( –λ)
– – x
+λgx(t),x(t–τ)=λp(t). (.)
Now, we give the following fundamental lemma.
Lemma . Suppose that there exist three positive constants M,M,and Msuch that, for anyπ-periodic solution x(t)of (.),
M<x(t) <M for all t∈R
and
x ∞ max
t∈[,π]
x(t)<M.
Then Eq. (.)has at least oneπ-periodic solution.
Remark . In fact, Lemma . is also valid if we embed (.) into the following family of equations with one parameterλ∈[, ]:
x+ n
x+ ( –λ)
– x
+λgx(t),x(t–τ)=λp(t). (.)
3 The proof of the main theorem Condition (H) reduces to
(H) for allθ∈R,
cosnτ
g(+∞) <
π
p(t)sin
n t+θ
dt
or
(H) for allθ∈R,
cosnτ
g(+∞) >
π
p(t)sin
n t+θ
dt.
In this section, we always assume that (H) holds. The argument for (H) is similar. We first suppose that the sequence{(xk,yk)}∞k=satisfies
xk=yk, yk= – n
xk–gxk,xk(t–τ),λ k
(.)
with xk ∞+ yk ∞→ ∞ask→ ∞, where
gxk,xk(t–τ),λk
= ( –λk)
– – xk
+λkg(xk) +λkgxk(t–τ)–λkp(t).
In this position, we only considerλk→λ∈[, ]. Even ifλkhas no limit, we can consider its convergent subsequence{λki}and the corresponding solution sequence{(xki,yki)}for (.) because the sequence{λk}is bounded. For simplicity, we omit these discussions. It is easy to see that xk ∞+ yk ∞→ ∞is equivalent to xk ∞→ ∞and yk ∞→ ∞as
k→ ∞. Define
g(xk,λk) = –( –λk)
xk+λkg(xk)
and
gxk(t–τ),λk
= –( –λk) +λkgxk(t–τ).
Obviously,g(xk,λk) satisfies condition (H). Take the transformation
Then system (.) is equivalent to the following system:
⎧ ⎨ ⎩
dθk dt = –
n –
nrkg( +rkcosθk, +rk(t–τ)cosθk(t–τ),λk)cosθk+ n rkcosθk, drk
dt = – n sinθk–
ng( +rkcosθk, +rk(t–τ)cosθk(t–τ),λk)sinθk.
(.)
Lemma . Assume that(H)and(H)hold.For k large enough,we have
θk(t) < ∀t∈R.
Proof By (H) and the definition ofg(xk,xk(t–τ),λk) we have that there exists <δ< such that
gxk,xk(t–τ),λk
–n
< .
Noticing thatcosθk< for <xk≤δ< , we get, for <xk≤δ,
dθk dt ≤–
n
. (.)
Since xk ∞→ ∞and yk ∞→ ∞ask→ ∞, we have that ifxk>δ, then
rk→ ∞ ask→ ∞.
Meanwhile, ifxk>δ, by (H), then we get that there existsM> such that
gxk,xk(t–τ),λk<M.
Hence, forklarge enough, ifxk>δ, then
–
nrk
g +rkcosθk, +rk(t–τ)cosθk(t–τ),λk
cosθk+ n rk
cosθk
<n
,
and then
dθk dt ≤–
n +
n = –
n
. (.)
From (.) and (.) we obtain the conclusion of Lemma ..
From Lemma . we conclude that, forklarge enough, the solution (xk(t),xk(t)) of (.) makes clockwise rotations around the point (, ). Without loss of generality, we take the initial point (xk(tm,k ),yk(tm,k)) of themth rotation that satisfies
xk
tm,k= , yk
tm,k=xktm,k > , and
θk
tm,k=π
– (m– )π,
Let (xk(t),yk(t)) take exactly one rotation fromtm,k . Then there are two points where the curve (xk(t),yk(t)) meets the linexk= and two points where the curve (xk(t),yk(t)) meets thexk-axis. We denote them by (xk(tm,k ), ), (,yk(tm,k)), (xk(tm,k ), ), and (,yk(tm,k)),
Therefore we haveR
k=rk(t) +Gk(t).
Proof Without loss of generality, we assume that there exist a positive integerm∗ such that
=
tm∗,k tm∗,k
λk n yk(s)g
xk(s–τ)ds =orktm∗,k.
It follows from the relationR
k=rk(t) +Gk(t) that
rktm∗+,k=rktm∗,k–Gk
tm∗+–Gk
tm∗,k=rktm∗,k+orktm∗,k. Hence,rk(tm
∗+,k
)→ ∞. By using the second equality of (.) again we obtainrk(tm
∗+,k
)→ ∞. Equivalently,limk→∞xk(tm
∗+,k
) = +∞.
Denote byτkmthe required time for the solution (xk(t),yk(t)) to complete themth rota-tion around the point (, ). Then,
τkm=tm,k –tm,k +tm,k –tm,k.
Lemma . Assume that(H)and(H)hold.Then,for k large enough,
τkm=π n +o().
Proof We first computetm,k–tm,k. Sinceg(xk,xk(t–τ),λk) is bounded on the interval [tm,k ,tm,k], we get from the first equality of (.) that
dt dθk
= – n+o(),
which impliestm,k–tm,k =nπ +o().
Next, we estimate (tm,k–tm,k ). By (.) we have, fort∈[tm,k ,tm,k],
dt dxk =
yk = –
n
R
k– (xk(t) – )–Gk(t) .
Furthermore,
– ndxk=
R k–
xk(t) – –Gk(t)dt. (.) By (H) and Remark . we have that there exist <ε≤, <l≤, and <L<∞such that
–g(xk,λk) G+l
(xk,λk)
>L for allx∈(,ε).
For abovel, we can chooset–m,k∈[tm,k ,tm,k] such that
Gktm,k– =R
+l
k ,
R +l
k ≤Gk(t)≤Rk fort∈
and
On the other hand, we get
tm–,k
Thus we have
Passing to limits and using the Lebesgue dominated convergence theorem, we get
Since xk(t)→ ask→ ∞ fort∈[tm,k – ,t
m,k
], we obtain, fork large enough, [xk(t m,k ), xk(t–m,k)]⊂(,ε). Then, fort∈[tm,k– ,tm,k],
–g(xk,λk)<
L
G+l(xk,λk),
which yields that, fort∈[tm,k – ,t
m,k ],
dt dyk ≤
L
G+l(xk,λk)+o
G+l (xk,λk)
= L
G+l
k (t) +o
G+l
k (t)
≤ L
R (+l)
+l
k +o
R (+l)
+l
k
.
Integrating over [tm,k
– ,tm,k ] and noticing thatyk(t)≤nRk, we have
tm,k–tm,k– ≤ n
LR
(+l) +l
k – +o
R (+l)
+l
k –
,
which implies that
tm,k–tm,k– =o
Rk
. (.)
It follows from (.) and (.) that
tm,k–tm,k = nRk
+o
Rk
. (.)
Using a similar argument, we can prove thattm,k–tm,k =nRk +o(
Rk). It follows that
tm,k–tm,k = nRk
+o
Rk
. (.)
Thentm,k–tm,k =o(). This completes the proof.
Since (xk,yk) is π-periodic, by Lemma . we know that, forksufficiently large, (xk,yk) makes exactlynclockwise revolutions around the point (, ) astvaries from to π.
We use the following lemma to estimate the maxima.
Lemma . Assume that(H), (H),and(H)hold.There exist two positive constants M
and Msuch that,for anyπ-periodic solution x(t)of Eq. (.),
Proof Assume by contradiction that there exists a sequence {(xk,yk)}∞k= satisfying the system (.) with xk ∞ + yk ∞ → ∞ as k→ ∞. It follows that xk ∞ → ∞ and
yk ∞→ ∞.
Using the transformation
xk= +rkcosθk, yk=n rksinθk,
we have system (.).
Without loss of generality, we take the initial point (xk(tm,k),yk(tm,k)) of themth rotation satisfying
xktm,k= , yktm,k=xktm,k > ,
and
θk
tm,k
=π
– (m– )π
form= , , . . . ,n.
Let (xk(t),yk(t)) take exactly one rotation fromtm,k . Denote byτkthe required time for the solution (xk(t),yk(t)) to completenrotations around the point (, ). Then,
τk= n
m=
τkm= n
m=
tm,k –tm,k= n
m=
tm,k–tm,k+ n
m=
tm,k–tm,k .
We further estimatenm=(tm,k–tm,k ). By the second equality of (.) we get, fort∈ (tm,k ,tm,k),
rk(t) =rktm+O().
Hence, for allt∈[tm,k,tm,k ], ≤m≤n,
rk(t) =Rk+o(Rk). (.)
Furthermore, r k(t)=
Rk +o(
Rk) fort∈[t m,k ,tm,k].
By the first equality of (.), we obtain, fort∈[tm,k ,tm,k],
dt dθk
= – n
·
+nR
kg( +rkcosθk, +rk(t–τ)cosθk(t–τ),λk)cosθk+
Rkcosθk+o( Rk)
= – n+
nR
k
g +rkcosθk, +rk(t–τ)cosθk(t–τ),λk
cosθk
+ nRk
cosθk+o
Rk
Integrating over [–π – (m– )π,π – (m– )π], we get
Therefore, by (.) we have, forklarge enough,
τk=
which is a contradiction. Consequently, (.) holds.
Passing to limits, we get
Therefore we have
mπ
Consequently, we get
n
Consequently, forklarge enough,
n
which also contradicts the π-periodicity of (xk(t),yk(t)). This completes the proof.
Remark . Similarly, if (H), (H), and (H) hold, then the result in Lemma . is valid. Under these conditions, to estimate the maxima, we can we embed (.) into (.).
Lemma . Assume(H), (H),and(H)hold.Then there exists a positive constant M such that,for anyπ-periodic solution x(t)of Eq. (.),
min
Proof LetRkbe defined by (.). We first prove that there existsc> such that, for all k∈N+,
xk
t,k>c. (.)
Assume by contradiction thatxk(t,k )→ (ask→ ∞). By Lemma . and Remark . there existM> andM> such that
xk ∞<M, xk ∞<M.
ThenRk(t)≡c< +∞. However, by (H),
rkt,k
–rkt,k
= n
t,k
t,k
ykgxk,xk(t–τ),λk
dt
=λk n
t,k t,k
yk
xkdt– ( –λk) n
t,k t,k
ykg(xk)dt+O()
=λk n ·
x
k(t ,k )
– ( –λk) n
x(t,k)
g(xk)dxk+O()
→+∞,
which is impossible. Therefore (.) holds. Similarly, we can obtain
xk
tm,k>cm
form= , , . . . ,n. Consequently, we get the conclusion of Lemma ..
Proof of Theorem . The result is obtained directly by Lemma ., Lemma .,
Re-mark ., and Lemma ..
Acknowledgements
Research was supported by the National Natural Science Foundation of China (11401274, 11661046), Science and Technology Landing Project of Colleges and Universities in Jiangxi Province (KJLD14092), Science and Technology Project of Education Department of Jiangxi Province (GJJ151094), and Scientific Research Project of Jiujiang University (2013KJ18, 2014KJYB022).
Competing interests
The author declares that they have no competing interests.
Author’s contributions
All 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: 8 April 2017 Accepted: 13 June 2017
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