Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 242, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF POSITIVE PERIODIC SOLUTIONS FOR
NEUTRAL LIÉNARD DIFFERENTIAL EQUATIONS WITH A
SINGULARITY
FANCHAO KONG, SHIPING LU, ZAITAO LIANG
Abstract. By applying Mawhin’s continuation theorem, we study the existence of positive periodic solutions for a second-order neutral functional differential equation
((x(t) − cx(t − σ)))00 + f (x(t))x0 (t) + g(t, x(t − δ)) = e(t),
where g has a strong singularity at x = 0 and satisfies a small force condition at
x = ∞, which is different from the corresponding ones known in the literature.
1. Introduction
In recent years, the existence of periodic solutions for the second order differential
equations with a singularity have been studied in many literature. See [1]-[15] and
the references therein.
Wang [15] studied the Liénard equation with a singularity and a deviating argument
x00 (t) + f (x(t))x0 (t) + g(t, x(t − σ)) = 0,
(1.1)
where 0 ≤ σ < T is a constant, f : R → R, g : R × (0, +∞) → R is an L2 Carathéodory function, g(t, x) is a T -periodic function in the first argument and
can be singular at x = 0, i. e., g(t, x) can be unbounded as x → 0+ .
Let (1.1) be of repulsive type and set
Z
1 T
g(x) =
g(t, x)dt, x > 0.
T 0
Assume that
g(t, x)
ϕ(t) = lim sup
,
x→+∞
x
exists uniformly for a. e. t ∈ [0, T ], i.e., for any ε > 0, there is gε ∈ L2 (0, T ) such
that
g(t, x) ≤ (ϕ(t) + ε)x + gε ,
for all x > 0 and a. e. t ∈ [0, T ]. Assume that ϕ ∈ C(R, R) and ϕ(t + T ) = ϕ(t),
t ∈ R.
2010 Mathematics Subject Classification. 34B16, 34B20, 34B24.
Key words and phrases. Positive periodic solution; neutral differential equation;
deviating argument; singular; Mawhin’s continuation.
c
2015
Texas State University - San Marcos.
Submitted May 6, 2015. Published September 21, 2015.
1
2
F. KONG, S. LU, Z. LIANG
EJDE-2015/242
Wang established the following theorem.
Theorem 1.1. Assume that the following conditions are satisfied:
(H1) (Balance) There exist constants 0 < D1 < D2 such that if x is a positive
continuous T -periodic function satisfying
Z T
g(t, x(t))dt = 0,
0
then
D1 ≤ x(τ ) ≤ D2 , for some τ ∈ [0, T ].
(H2) (Degree) g(x) < 0 for all x ∈ (0, D1 ), and g(x) > 0 for all x > D2 .
(H3) (Decomposition) g(t, x) = g0 (x) + g1 (t, x), where g0 ∈ C((0, +∞), R) and
g1 : [0, T ] × [0, +∞) → R is an L2 -Carathéodory function, i. e., g1 is
measurable with respect to the first variable, continuous with respect to the
second one, and for any b > 0 there is hb ∈ L2 ((0, T ); [0, +∞)) such that
|g1 (t, x)| ≤ hb (t) for a.e. t ∈ [0, T ] and all x ∈ [0, b].
R1
(H4) (Strong force at x = 0) 0 g0 (x)dx = −∞.
(H5) (Small force at x = ∞)
√
π 2
) .
kϕk∞ < (
T
Then (1.1) has at least one positive T -periodic solution.
Meanwhile, the problem of the existence of periodic solutions to the neutral
functional differential equation was studied in many papers, see [16]-[11] and the
references therein. For example, in [7], Liu and Huang studied the following neutral
functional differential equation
(u(t) + Bu(t − τ ))0 = g1 (t, u(t)) + g2 (u(t − τ1 )) + p(t).
And in in [11], Lu and Ge studied the existence of periodic solutions for a kind of
second-order neutral functional differential equation of the form
n
X
d2 u(t)
−
c
u(t
−
r
)
j
j
dt2
j=1
= f (u(t))u0 (t) + α(t)g(u(t)) +
n
X
βj g(u(t − γj )) + p(t),
j=1
where f , g ∈ C(R; R), a(t), p(t), βj (t), γj (t) (j = 1, 2, . . . , n) are continuous periodic
functions defined on R with period T > 0, cj , rj ∈ R are constants with rj > 0
(j = 1, 2, . . . , n). By using the continuation theorem of coincidence degree theory
and some new analysis techniques, the authors obtained some new results on the
existence of periodic solution.
However, to the best of our knowledge, the studying of positive periodic solutions
for the neutral functional differential equation with a singularity is relatively infrequent. As we know, in order to establish the existence of positive periodic solutions,
a key condition is that the greatest lower bound must be estimated because of the
singularity. However, it is difficult to verify the greatest lower bound, especially for
the neutral functional differential equations with a singularity. Besides, because of
the singularity, the third condition of Mawhin’s continuation theorem is not easy
to verify.
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EXISTENCE OF POSITIVE PERIODIC SOLUTIONS
3
Inspired by the above facts, in this paper, we consider the following neutral
Liénard differential equation with a singularity and a deviating argument
((x(t) − cx(t − σ)))00 + f (x(t))x0 (t) + g(t, x(t − δ)) = e(t),
(1.2)
where c is a constant with |c| < 1, 0 ≤ σ, δ < T , f : R → R is continuous,
g : [0, T ] × (0, +∞) → R is a continuous function and can be singular at u = 0, i.
RT
e., g(t, u) can be unbounded as u → 0+ . e(t) is T -periodic with 0 e(t)dt = 0. And
we can easily see that when c = 0, the (1.2) transforms into (1.1). To sum up, our
results are essentially new.
The rest of this paper is organized as follows. In Section 2, we state some
necessary definitions and lemmas. In Section 3, we prove the main result. At last,
we will give an example of an application in section 4.
2. Preliminaries
To prove the announced result, we state the following necessary definitions and
lemmas. Denote the operator A by
A : CT → CT ,
(Ax)(t) = x(t) − cx(t − σ),
where CT = {ϕ ∈ C(R, R), ϕ(t + T ) = ϕ(t)}, with norm ||ϕ||0 = maxt∈[0,T ] |ϕ(t)|.
Clearly, CT is a Banach space. Define the operator
L : D(L) ⊂ X → Y,
Lx = (Ax)0 ,
where D(L) = {x ∈ C 1 (R, R), x(t) = x(t + T )}. Define
N : CT → CT ,
(N x)(t) = −f (x(t))x0 (t) − g(t, x(t − δ)) + e(t).
Then (1.2) can be rewritten by Lx = N x.
Lemma 2.1 ([10]). If |c| < 1 then A has continuous inverse on CT and
kxk0
for all x ∈ CT ;
(1) kA−1 xk ≤ |1−|c||
RT
RT
1
(2) 0 |(A−1 f )(t)|dt ≤ |1−|c||
|f (t)|dt for all f ∈ CT ;
0
R
R T −1 2
T
1
f 2 (t)dt for all f ∈ CT .
(3) 0 |A f | (t)dt ≤ (1−|c|)
2
0
From Hale’s terminology [4], a solution of the (1.2) is x ∈ C(R, R) such that
Ax ∈ C 1 (R, R) and (1.2) is satisfied on R. In general, x is not from C 1 (R, R).
Nevertheless, it is easy to see that (Au)0 = Au0 . Thus, a T -periodic solution x of
the (1.2) must be from C 1 (R, R). According to Lemma 2.1, we can easily obtain
RT
that ker L = R, Im L = {x : x ∈ X, 0 x(s)ds = 0}. Thus L is a Fredholm operator
with index zero.
Let the projections P and Q be
Z
1 T
P : CT → ker L, P x =
x(s)ds;
T 0
Z
1 T
Q : CT → CT \ Im L, Qy =
x(s)ds.
T 0
Let Lp = L|D(L)∩ker P : CT ∩ ker P → Im L. Then LP has continuous inverse L−1
p
on Im L defined by
Z T
−1
−1
(Lp y)(t) = A
G(t, s)y(s)ds ,
0
4
F. KONG, S. LU, Z. LIANG
EJDE-2015/242
where
(
Gk (t) =
s−T
T
s
T,
, 0 ≤ t ≤ s;
s ≤ t ≤ T.
Lemma 2.2 ([2]). Let X and Y be two real Banach spaces, and Ω is an open and
bounded set of X, and L : D(L) ⊂ X → Y is a Fredholm operator of index zero
and the operator N : Ω̄ ⊂ X → Y is said to be L-compact in Ω̄. In addition, if the
following conditions hold:
(1) Lx 6= λN x for all (x, λ) ∈ ∂Ω × (0, 1);
(2) QN x 6= 0 for all x ∈ ker L ∩ ∂Ω;
(3) deg{JQN, Ω ∩ ker L, 0} =
6 0, where J : ImQ → ker L is a homeomorphism.
Then Lx = N x has at least one solution in D(L) ∩ Ω̄.
For the sake of convenience, we list the following assumptions:
(H1) There exist positive constants D1 and D2 with D1 < D2 such that
(1) for each positive continuous T -periodic function x(t) satisfying
RT
g(t, x(t))dt = 0, there exists a positive point τ ∈ [0, T ] such that
0
D1 ≤ x(τ ) ≤ D2 ;
(2) g(x) < 0 for all x ∈ (0, D1 ) and g(x) > 0 for all x > D2 , where
RT
g(x) = T1 0 g(t, x)dt, x > 0.
(H2) g(t, x) = g1 (t, x) + g0 (x), where g1 : [0, T ] × (0, +∞) → R is a continuous
function and
(1) there exist positive constants m0 and m1 such that
g(t, x) ≤ m0 x + m1 ,
R1
(2) 0 g0 (x)dx = −∞.
for all (t, x) ∈ [0, T ] × (0, +∞);
3. Main results
Theorem 3.1. Suppose that the conditions (H1)-(H2) hold, |c| < 1 and
|c|(1 + |c|) + m0 T 2
< 1,
(1 − |c|)2
then the (1.2) has at least one positive T -periodic solution.
Proof. Consider the operator equation
Lx = λN x,
λ ∈ (0, 1).
Let Ω1 = {x ∈ Ω, Lx = λN x, λ ∈ (0, 1)}. If x ∈ Ω1 , then x must satisfy
((Au)0 (t))0 + λf (u(t))u0 (t) + λg(t, u(t − δ)) = λe(t).
Integrating (3.1) on the interval [0, T ], we have
Z T
g(t, u(t − δ))dt = 0.
(3.1)
(3.2)
0
It follows from (H1)(1) that there exist positive constants D1 , D2 and τ ∈ [0, T ]
such that
D1 ≤ u(τ ) ≤ D2 .
(3.3)
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EXISTENCE OF POSITIVE PERIODIC SOLUTIONS
5
Then, we obtain
Z
kuk0 = max |u(t)| ≤ max |u(τ ) +
t∈[0,T ]
t∈[0,T ]
t
u0 (s)ds| ≤ D2 +
T
Z
τ
|u0 (s)|ds.
(3.4)
0
Multiplying the both sides of (3.1) by u(t) and integrating on the interval [0, T ],
we obtain
Z T
((Au)0 (t))0 u(t)dt
0
Z
T
e(t)u(t)dt (3.5)
0
0
T
T
Z
= −λ
T
Z
g(t, u(t − δ))u(t)dt + λ
0
Z
T
Z
f (u(t))u0 (t)u(t)dt − λ
= −λ
g(t, u(t − δ))u(t)dt + λ
e(t)u(t)dt.
0
0
Furthermore,
Z T
Z
((Au)0 (t))0 u(t)dt = −
0
T
(Au)0 (t)u0 (t)dt
0
T
Z
(Au)0 (t)[u0 (t) − cu0 (t − σ) + cu0 (t − σ)]dt
=−
0
(3.6)
T
Z
0
=−
0
0
(Au) (t)[(Au )(t) + cu (t − σ)]dt
0
T
Z
|(Au0 )(t)|2 dt −
=−
0
Z
T
cu0 (t − σ)(Au)0 (t)dt.
0
Substituting (3.6) in (3.5), we obtain
Z T
|(Au0 )(t)|2 dt
0
T
Z
0
=−
T
Z
0
cu (t − σ)(Au) (t)dt + λ
0
g(t, u(t − δ))u(t)dt − λ
0
T
Z
|u0 (t − σ)||(Au)0 (t)|dt +
≤ |c|
Z
|g(t, u(t − δ))||u(t)|dt +
0
0
|u0 (t − σ)||(Au)0 (t)|dt + m0
0
Z
0
T
Z
|u(t)|dt +
+ m1
0
T
|e(t)||u(t)|dt.
0
T
Z
e(t)u(t)dt
0
T
Z
0
It follows from (H2)(1) that
Z T
Z
|(Au0 )(t)|2 dt ≤ |c|
T
Z
T
T
|u(t)|2 dt
(3.7)
|e(t)||u(t)|dt.
0
Moreover, by applying Hölder inequality and Minkowski inequality, we can have
Z T
|u0 (t − σ)||(Au)0 (t)|dt
0
≤
Z
T
1/2 Z
|(Au)0 (t)|2 dt
T
1/2 Z
0
2
|(Au) (t)| dt
0
=
Z
0
T
1/2
|u0 (t − σ)|2 dt
T
1/2
|u0 (t)|2 dt
0
0
6
F. KONG, S. LU, Z. LIANG
=
h Z
T
1/2 i Z
|u0 (t) − cu0 (t − σ)|2 dt
T
1/2 Z
0
2
|u (t)| dt
+
0
≤
h Z
≤
T
1/2
|u0 (t)|2 dt
0
0
h Z
EJDE-2015/242
T
1/2 i Z
|cu (t − σ)| dt
0
0
T
T
1/2 i Z
|u0 (t)|2 dt
0
Z
= (1 + |c|)
1/2
|u0 (t)|2 dt
0
1/2
Z
|u0 (t)|2 dt
+ |c|
0
T
2
T
1/2
|u0 (t)|2 dt
0
T
|u0 (t)|2 dt.
(3.8)
0
Substituting (3.8) into (3.7) and by (3.4), we can obtain
Z T
Z T
|(Au0 )(t)|2 dt ≤ |c|(1 + |c|)
|u0 (t)|2 dt + m0 T kuk20
0
0
+ (m1 + kek0 )T kuk0
Z T
Z
≤ |c|(1 + |c|)
|u0 (t)|2 dt + m0 T (D2 +
0
T
|u0 (s)|ds)2
(3.9)
0
Z
+ (m1 + kek0 )T (D2 +
T
|u0 (s)|ds).
0
By applying the third part of Lemma (2.1), we have
Z T
Z T
Z T
1
0
2
−1
0
2
|(Au0 )(t)|2 dt.
|u (t)| dt =
|(A A)u (t)| dt ≤
(1 − |c|)2 0
0
0
(3.10)
Substituting (3.10) into (3.9) and by applying Hölder inequality, we obtain
Z T
|(Au0 )(t)|2 dt
0
≤ [|c|(1 + |c|) + m0 T 2 ]
T
Z
|u0 (t)|2 dt
0
√ Z
+ [2m0 D2 + m1 + kek0 ]T T
T
1/2
|u0 (t)|2 dt
+ m0 T D22 + (m1 + kek0 )T D2
0
|c|(1 + |c|) + m0 T 2
≤
(1 − |c|)2
Z
T
|(Au0 )(t)|2 dt
√
Z
1/2
[2m0 D2 + m1 + kek0 ]T T T
|(Au0 )(t)|2 dt
+ m0 T D22
+
1 − |c|
0
+ (m1 + kek0 )T D2 .
It follows from
0
|c|(1+|c|)+m0 T 2
(1−|c|)2
< 1 that there exist a positive constant M such that
Z
T
|(Au0 )(t)|2 dt ≤ M,
0
which combining with (3.10) gives
Z T
|u0 (t)|2 dt ≤
0
M
.
(1 − |c|)2
(3.11)
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EXISTENCE OF POSITIVE PERIODIC SOLUTIONS
Then by (3.4), we obtain
7
√
kuk0 ≤ D2 +
TM
:= M1 .
1 − |c|
(3.12)
Since [Au](t) is T -periodic, there exists t0 ∈ [0, T ] such that [Au0 ](t0 ) = 0. Hence,
we have that, for t ∈ [0, T ],
Z t
0
0
|[Au ](t)| = |[Au ](t0 ) +
([Au0 ](s))0 ds|
t0
(3.13)
Z T
Z T
|f (u(t))||u0 (t)|dt + λ
≤λ
|g(t, u(t − δ))|dt.
0
0
Set FM1 = max|u|≤M1 |f (u)|, then by (3.11) we obtain
√
Z T
Z T
FM1 T M
|u0 (t)|dt ≤
.
|f (u(t))||u0 (t)|dt ≤ FM1
1 − |c|
0
0
(3.14)
Write
I+ = {t ∈ [0, T ] : g(t, u(t − δ)) ≥ 0};
I− = {t ∈ [0, T ] : g(t, u(t − δ)) ≤ 0}.
Then it follows from (3.2) and (H2)(1) that
Z T
Z
Z
|g(t, u(t − δ))|dt =
g(t, u(t − δ))dt −
g(t, u(t − δ))dt
0
I+
I−
Z
=2
g(t, u(t − δ))dt
I+
T
Z
≤ 2m0
Z
u(t − δ)dt + 2
0
(3.15)
T
m1 dt
0
≤ 2m0 T kuk0 + 2T m1 .
According to (3.14) and (3.15), we have
Z T
Z T
kAu0 k0 ≤ λ
|f (u(t))||u0 (t)|dt + λ
|g(t, u(t − δ))|dt
0
0
√
F
M1 T M
+ 2m0 T M1 + 2T m1 ,
≤λ
1 − |c|
which combining with the first part of Lemma 2.1, we see that
|u0 (t)| = |[A−1 Au0 ](t)| ≤
≤
i. e.,
ku0 k0 ≤
√
FM1 T M
1−|c|
√
F M1 T M
1−|c|
kAu0 k0
|1 − |c||
+ 2m0 T M1 + 2T m1
|1 − |c||
+ 2m0 T M1 + 2T m1
|1 − |c||
On the other hand, it follows from (3.1) and (H2) that
,
:= A3 .
((Au)0 (t + δ))0 = −λf (u(t + δ))u0 (t + δ) − λ[g1 (t + δ, u(t)) + g0 (u(t))]
+ λe(t + δ).
(3.16)
(3.17)
8
F. KONG, S. LU, Z. LIANG
EJDE-2015/242
Multiplying both sides of (3.17) by u0 (t), we have
((Au)0 (t + δ))0 u0 (t) = −λf (u(t + δ))u0 (t + δ)u0 (t)
− λ[g1 (t + δ, u(t)) + g0 (u(t))]u0 (t) + λe(t + δ)u0 (t).
(3.18)
Let τ ∈ [0, T ] be as in (3.3). For any t ∈ [τ, T ], integrating (3.18) on the interval
[τ, T ], we have
Z u(t)
Z t
λ
g0 (u)du = λ
g0 (u(t))u0 (t)dt
u(τ )
τ
t
Z
0
t
Z
0 0
f (u(t + δ))u0 (t + δ)u0 (t)dt
((Au) (t + δ)) u (t)dt − λ
=−
τ
τ
Z
t
−λ
g1 (t + δ, u(t))u0 (t)dt + λ
Z
t
e(t + δ)u0 (t)dt.
τ
τ
Set GM1 = max|u|≤M1 |g1 (t, u)|, then from the inequality above, we obtain
Z u(t)
λ|
g0 (u)du|
u(τ )
t
Z
g0 (u(t))u0 (t)dt|
= λ|
τ
T
Z
0
≤
Z
0 0
T
|((Au) (t + δ)) u (t)|dt + λ
0
|f (u(t + δ))||u0 (t + δ)||u0 (t)|dt
0
T
Z
Z
0
|g1 (t + δ, u(t))||u (t)|dt + λ
+λ
0
|e(t + δ)||u0 (t)|dt
0
Z
≤ ku0 k0
T
T
|((Au)0 (t + δ))0 |dt + λFM1 ku0 k20 T + λGM1 ku0 k0 T + λkek0 ku0 k0 T,
0
i. e.,
Z
u(t)
λ|
g0 (u)du| ≤ ku0 k0
Z
u(τ )
T
|((Au)0 (t + δ))0 |dt + λFM1 ku0 k20 T
0
0
(3.19)
0
+ λGM1 ku k0 T + λkek0 ku k0 T.
Moreover,
Z T
Z
0
0
|((Au) (t + δ)) |dt =
0
T
|((Au)0 (t))0 |dt
0
≤λ
Z
0
T
|f (u(t))||u0 (t)|dt +
Z
T
|g(t, u(t − δ))|dt ,
0
which combining with (3.14) and (3.15) yields
Z T
Z T
Z T
|((Au)0 (t))0 |dt ≤ λ
|f (u(t))||u0 (t)|dt +
|g(t, u(t − δ))|dt
0
0
0
F √T M
M1
≤λ
+ 2m0 T M1 + 2T m1 .
1 − |c|
Substituting (3.20) into (3.19) and combining with (3.16), obtain
Z u(t)
F √T M
M1
|
g0 (u)du| ≤ A3
+ 2m0 T M1 + 2T m1 + FM1 A23 T
1 − |c|
u(τ )
(3.20)
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EXISTENCE OF POSITIVE PERIODIC SOLUTIONS
9
+ GM1 A3 T + kek0 A3 T < +∞.
According to (H2)(2), we can see that there exists a constant M2 > 0 such that,
for t ∈ [τ, T ],
u(t) ≥ M2 .
(3.21)
For the case t ∈ [0, τ ], we can handle similarly.
Let us define
0 < A1 = min{D1 , M2 },
A2 = max{D2 , M1 }.
Then by (3.3), (3.12) and (3.21), we obtain
A1 ≤ u(t) ≤ A2 .
(3.22)
Set
A1
< u(t) < A2 + 1, ||u0 ||0 < A3 + 1}.
2
Then condition (1) of Lemma 2.2 is satisfied.
RT
Assume that there exists x ∈ ∂Ω ∩ ker L such that QN x = T1 0 N x(s)ds = 0, i.
e.,
Z
1 T
[−f (u(t))u0 (t) − g(t, u(t − δ)) + e(t)]dt = 0,
(3.23)
T 0
then we have
Z
1 T
g(t, u(t − δ))dt = 0.
T 0
It follows from the (H1)(1) we can see that
Ω = {x = (u, v)> ∈ X :
A1
< D1 ≤ u(t) ≤ D2 < A2 + 1,
2
which contradicts the assumption x ∈ ∂Ω. So for all x ∈ ker L ∩ ∂Ω, we have
QN x 6= 0. Therefore, condition (2) of Lemma 2.2 is satisfied.
Finally, we prove that condition (3) of Lemma 2.1 is also satisfied. Let
z = Kx = x −
A1 + A2
,
2
then, we have
A1 + A2
.
2
Define J : ImQ → ker L is a linear isomorphism with J(u) = u, and define
x=z+
H(µ, x) = µKx + (1 − µ)JQN x, ∀(x, µ) ∈ Ω × [0, 1].
Then,
Z
µ(A1 + A2 ) 1 − µ T
+
g(t, x)dt.
(3.24)
2
T
0
Now we claim that H(µ, x) is a homotopic mapping. Assume, by way of contradiction, i. e., there exists µ0 ∈ [0, 1] and x0 ∈ ∂Ω such that H(µ0 , x0 ) = 0.
Substituting µ0 and x0 into (3.24), we have
H(µ, x) = µx −
H(µ0 , x0 ) = µ0 x0 −
µ0 (A1 + A2 )
+ (1 − µ0 )g(x0 ).
2
(3.25)
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F. KONG, S. LU, Z. LIANG
EJDE-2015/242
It follows H(µ0 , x0 ) = 0 that x0 = A1 or A2 . Furthermore, If x0 = A1 , it follows
from (H1)(2) that g(x0 ) < 0, then we have
A1 + A2
µ0 (A1 + A2 )
+ (1 − µ0 )g(x0 ) < µ0 (x0 −
) < 0.
2
2
If x0 = A2 , it follows from (H1)(2) that g(x0 ) > 0, then we have
µ0 x0 −
(3.26)
µ0 (A1 + A2 )
A1 + A2
+ (1 − µ0 )g(x0 ) > µ0 (x0 −
) > 0.
(3.27)
2
2
Combining with (3.26) and (3.27), we can see that H(µ0 , x0 ) 6= 0, which contradicts
the assumption. Therefore H(µ, x) is a homotopic mapping and x> H(µ, x) 6= 0,
for all (x, µ) ∈ (∂Ω ∩ ker L) × [0, 1], then
µ0 x0 −
deg(JQN, Ω ∩ ker L, 0) = deg(H(0, x), Ω ∩ ker L, 0)
= deg(H(1, x), Ω ∩ ker L, 0)
= deg(Kx, Ω ∩ ker L, 0)
X
=
sgn|K 0 (x)|
x∈K −1 (0)
= 1 6= 0.
Thus, condition (3) of Lemma 2.2 is also satisfied.
Therefore, by applying Lemma 2.1, we can conclude that (1.2) has at least one
positive T -periodic solution.
4. Example
In this section, we provide an example to illustrate results from the previous
sections.
Example 4.1. Consider the neutral Liénard differential equation with a singularity
and a deviating argument,
u2 (t)
+ 9 u0 (t)
((u(t) − 0.1u(t − π))0 ))0 +
3 + u(t)
(4.1)
1
1
1
+ (1 + sin 8t)u(t − δ) −
= sin 8t.
2
2
u(t − δ)
Corresponding to Theorem 3.1 and (1.2), we have
u2 (t)
+ 9, e(t) = sin 8t,
3 + u(t)
1
1
1
g(t, u(t − δ)) =
1 + sin 8t u(t − δ) −
.
2
2
u(t − δ)
Then, we choose
π
3
σ = π, c = 0.1, T = , m0 = , D1 = 2, D2 = 3.
4
4
Thus, |c| < 1 and the conditions (H1) and (H2) are satisfied. Meanwhile, we have
f (u(t)) =
|c|(1 + |c|) + m0 T 2
≈ 0.706 < 1.
(1 − |c|)2
Hence, by applying Theorem 3.1, we can see that (4.1) has at least one positive
π
4 -periodic solution.
EJDE-2015/242
EXISTENCE OF POSITIVE PERIODIC SOLUTIONS
11
Remark 4.2. Since only a few papers consider positive periodic solutions for the
neutral Liénard equation. One can easily see that all the results in [1]-[4] and
the references therein are not applicable to (4.1) for obtaining positive periodic
solutions with period π4 . This implies that the results in this paper are essentially
new.
Acknowledgments. The research was supported by the the National Natural
Science Foundation of China (Grant No.11271197). The authors would like to
express their gratitude to Professor Giovanni Monica Bisci for his valuable guidance,
and also to the reviewers for their valuable comments and constructive suggestions,
which help us improve this article.
References
[1] A. Fonda, R. Manásevich, F. Zanolin; Subharmonic solutions for some second order differential equations with singularities, SIAM J. Math. Anal. 24 (1993) 1294–1311.
[2] R. Gaines, J. Mawhin; Coincidence Degree and Nonlinear Differential Equations, Springer
Verlag, Berlin, 1977.
[3] R. Hakl, P. Torres, M. Zamora; Periodic solutions to singular second order differential equations: the repulsive case, Nonlinear Anal. TMA. 74 (2011) 7078–7093.
[4] J. Hale; Theory of Functional Differential Equations, Springer, New York, 1977.
[5] I. Kiguradze, B. Shekhter; Singular boundary value problems for ordinary second-order differential equations, J. Sov. Math. 43(2) (1988) 2340–2417.
[6] A. Lazer, S. Solimini; On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc. 88 (1987) 109–114.
[7] B. Liu, L. Huang; Existence and uniqueness of periodic solutions for a kind of first order
neutral functional differential equation, J. Math. Anal. Appl. 322 (2006) 121–132.
[8] X. Li, Z. Zhang; Periodic solutions for second order differential equations with a singular
nonlinearity, Nonlinear Anal. TMA. 69 (2008) 3866–3876.
[9] S. Lu, W. Ge; On the existence of periodic solutions for neutral functional differential equation, Nonlinear Anal. TMA. 54 (2003) 1285–1306.
[10] S. Lu, W. Ge; Periodic solutions for a kind of second order differential equations with multiple
deviating arguments, Appl. Math. Comput. 146 (1) (2003) 195–209.
[11] S. Lu, W. Ge; Existence of periodic solutions for a kind of second-order neutral functional
differential equation, Appl. Math. Comput. 157 (2004) 433–448.
[12] P. Torres; Existence of one-signed periodic solutions of some second order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations. 190 (2003) 643–662.
[13] Z. Wang; Periodic solutions of the second order differential equations with singularities,
Nonlinear Anal. TMA. 58 (2004) 319–331.
[14] Z. Wang, T. Ma; Existence and multiplicity of periodic solutions of semilinear resonant
Duffing equations with singularities, Nonlinearity. 25 (2012) 279–307.
[15] Z. Wang; Periodic solutions of Liénard equation with a singularity and a deviating argument,
Nonlinear Anal. Real World Appl. 16 (2014) 227–234.
[16] M. Zhang; Periodic solutions to linear and quasilinear neutral functional differential equation, J. Math. Anal. Appl. 189 (1995) 378–392.
[17] M. Zhang; Periodic solutions of Liénard equations with singular forces of repulsive type, J.
Math. Anal. Appl. 203 (1996) 254–269.
Fanchao Kong
College of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000,
Anhui, China
E-mail address: fanchaokong88@yahoo.com
Shiping Lu
College of Mathematics and Statistics, Nanjing University of Information Science and
Technology, Nanjing 210044, China
E-mail address: lushiping26@sohu.com
12
F. KONG, S. LU, Z. LIANG
EJDE-2015/242
Zaitao Liang
Department of Mathematics, College of Science, Hohai University, Nanjing 210098,
China
E-mail address: liangzaitao@sina.cn