Periodic solutions of second-order neutral functional differential systems with time-varying operator and delays
This paper aims to investigate the existence of periodic solutions for second-order neutral functional differential systems with time-varying operator and delays. The interesting issue is that the coefficient matrix of the neutral difference operator is not a constant matrix, which is different from...
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| Cite this: | Periodic solutions of second-order neutral functional differential systems with time-varying operator and delays / Zhengxin Wang, Shiping Lu, Jinde Cao // Нелінійні коливання. — 2014. — Т. 17, № 2. — С. 180-199. — Бібліогр.: 12 назв. — англ. |
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Zhengxin Wang Shiping Lu Jinde Cao 2021-02-02T08:05:01Z 2021-02-02T08:05:01Z 2014 Periodic solutions of second-order neutral functional differential systems with time-varying operator and delays / Zhengxin Wang, Shiping Lu, Jinde Cao // Нелінійні коливання. — 2014. — Т. 17, № 2. — С. 180-199. — Бібліогр.: 12 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/175629 517.9 This paper aims to investigate the existence of periodic solutions for second-order neutral functional differential systems with time-varying operator and delays. The interesting issue is that the coefficient matrix of the neutral difference operator is not a constant matrix, which is different from the other literature. Some properties of the time-varying neutral difference operator and the results on the existence of periodic solutions of second-order neutral functional differential systems are obtained. Moreover, a numerical simulation is given to illustrate the theoretical results. Дослiджується iснування перiодичних розв’язкiв функцiонально-диференцiальних систем другого порядку з залежним вiд часу оператором та запiзненнями. При цьому матриця коефiцiєнтiв нейтрального рiзницевого оператора не є сталою на вiдмiну вiд випадкiв, описаних у лiтературi. Отримано деякi властивостi нейтрального змiнного за часом рiзницевого оператора та результати щодо iснування перiодичних розв’язкiв нейтральних функцiональнодиференцiальних систем другого порядку. Наведено числовий приклад для iлюстрацiї теоретичних результатiв. This work was jointly supported by the National Natural Science Foundation of China under grants 61304169, 61272530, 11072059 and 11271197, the Natural Science Foundation of Jiangsu Province of China under grants BK 2012741 and BK 20130857, the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province of China under grant 13KJB110022, the Scientific Research Foundation of Nanjing University of Posts and Telecommunications under grant NY213052, the Scientific Research Foundation of Graduate School of Southeast University YBJJ1028, JSPS Innovation Program under grant CX10B_060Z. en Інститут математики НАН України Нелінійні коливання Periodic solutions of second-order neutral functional differential systems with time-varying operator and delays Періодичні розв'язки нейтральних функціонально-диференціальних систем другого порядку з залежним від часу оператором та запізненнями Периодические решения нейтральных функционально-дифференциальных систем второго порядка с зависимым от времени оператором и опозданиями Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Periodic solutions of second-order neutral functional differential systems with time-varying operator and delays |
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Periodic solutions of second-order neutral functional differential systems with time-varying operator and delays Zhengxin Wang Shiping Lu Jinde Cao |
| title_short |
Periodic solutions of second-order neutral functional differential systems with time-varying operator and delays |
| title_full |
Periodic solutions of second-order neutral functional differential systems with time-varying operator and delays |
| title_fullStr |
Periodic solutions of second-order neutral functional differential systems with time-varying operator and delays |
| title_full_unstemmed |
Periodic solutions of second-order neutral functional differential systems with time-varying operator and delays |
| title_sort |
periodic solutions of second-order neutral functional differential systems with time-varying operator and delays |
| author |
Zhengxin Wang Shiping Lu Jinde Cao |
| author_facet |
Zhengxin Wang Shiping Lu Jinde Cao |
| publishDate |
2014 |
| language |
English |
| container_title |
Нелінійні коливання |
| publisher |
Інститут математики НАН України |
| format |
Article |
| title_alt |
Періодичні розв'язки нейтральних функціонально-диференціальних систем другого порядку з залежним від часу оператором та запізненнями Периодические решения нейтральных функционально-дифференциальных систем второго порядка с зависимым от времени оператором и опозданиями |
| description |
This paper aims to investigate the existence of periodic solutions for second-order neutral functional differential systems with time-varying operator and delays. The interesting issue is that the coefficient matrix of the neutral difference operator is not a constant matrix, which is different from the other literature. Some properties of the time-varying neutral difference operator and the results on the existence of periodic solutions of second-order neutral functional differential systems are obtained. Moreover, a numerical simulation is given to illustrate the theoretical results.
Дослiджується iснування перiодичних розв’язкiв функцiонально-диференцiальних систем другого порядку з залежним вiд часу оператором та запiзненнями. При цьому матриця коефiцiєнтiв нейтрального рiзницевого оператора не є сталою на вiдмiну вiд випадкiв, описаних у лiтературi. Отримано деякi властивостi нейтрального змiнного за часом рiзницевого оператора та результати щодо iснування перiодичних розв’язкiв нейтральних функцiональнодиференцiальних систем другого порядку. Наведено числовий приклад для iлюстрацiї теоретичних результатiв.
|
| issn |
1562-3076 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/175629 |
| citation_txt |
Periodic solutions of second-order neutral functional differential systems with time-varying operator and delays / Zhengxin Wang, Shiping Lu, Jinde Cao // Нелінійні коливання. — 2014. — Т. 17, № 2. — С. 180-199. — Бібліогр.: 12 назв. — англ. |
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| first_indexed |
2025-11-26T03:43:59Z |
| last_indexed |
2025-11-26T03:43:59Z |
| _version_ |
1850610492023767040 |
| fulltext |
UDC 517.9
PERIODIC SOLUTIONS OF SECOND-ORDER NEUTRAL FUNCTIONAL
DIFFERENTIAL SYSTEMS WITH TIME-VARYING OPERATOR
AND DELAYS*
ПЕРIОДИЧНI РОЗВ’ЯЗКИ НЕЙТРАЛЬНИХ
ФУНКЦIОНАЛЬНО-ДИФЕРЕНЦIАЛЬНИХ СИСТЕМ
ДРУГОГО ПОРЯДКУ З ЗАЛЕЖНИМ ВIД ЧАСУ ОПЕРАТОРОМ
ТА ЗАПIЗНЕННЯМИ
Zhengxin Wang
College Sci., Nanjing Univ. Posts and Telecommunications
Nanjing 210003, China
and Research Center for Complex Systems and Network Sci. and Southeast Univ.
Nanjing 210096, China
e-mail: zwang@njupt.edu.cn
Jinde Cao
Research Center for Complex Systems and Network Sci. and Southeast Univ.
Nanjing 210096, China
and King Abdulaziz Univ.
Jeddah 21589, Saudi Arabia
e-mail: jdcao@seu.edu.cn
Shiping Lu
College Math. and Statistics, Nanjing Univ. Information Sci. and Technology
Nanjing 210044, China
and Anhui Normal Univ.
Wuhu 241000, China
e-mail: lushiping88@sohu.com
This paper aims to investigate the existence of periodic solutions for second-order neutral functional di-
fferential systems with time-varying operator and delays. The interesting issue is that the coefficient matrix
of the neutral difference operator is not a constant matrix, which is different from the other literature. Some
properties of the time-varying neutral difference operator and the results on the existence of periodic soluti-
ons of second-order neutral functional differential systems are obtained. Moreover, a numerical simulation
is given to illustrate the theoretical results.
∗ This work was jointly supported by the National Natural Science Foundation of China under grants 61304169,
61272530, 11072059 and 11271197, the Natural Science Foundation of Jiangsu Province of China under grants BK
2012741 and BK 20130857, the Natural Science Foundation of the Higher Education Institutions of Jiangsu Pro-
vince of China under grant 13KJB110022, the Scientific Research Foundation of Nanjing University of Posts and
Telecommunications under grant NY213052, the Scientific Research Foundation of Graduate School of Southeast
University YBJJ1028, JSPS Innovation Program under grant CX10B_060Z.
c© Zhengxin Wang, Jinde Cao, Shiping Lu, 2014
180 ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 2
PERIODIC SOLUTIONS OF SECOND-ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL SYSTEMS . . . 181
Дослiджується iснування перiодичних розв’язкiв функцiонально-диференцiальних систем
другого порядку з залежним вiд часу оператором та запiзненнями. При цьому матриця ко-
ефiцiєнтiв нейтрального рiзницевого оператора не є сталою на вiдмiну вiд випадкiв, описа-
них у лiтературi. Отримано деякi властивостi нейтрального змiнного за часом рiзницевого
оператора та результати щодо iснування перiодичних розв’язкiв нейтральних функцiонально-
диференцiальних систем другого порядку. Наведено числовий приклад для iлюстрацiї теоре-
тичних результатiв.
1. Introduction. In recent years, the problem on the existence of periodic solutions to functional
differential systems was extensively studied, see [1, 2, 4, 5, 7 – 12] and their reference lists. In
[2], Các discussed the existence of 2π-periodic solutions of the systems
x′′ +
d
dt
[gradF (x(t))] + g(t, x(t)) = e(t),
under various asymptotic behaviors of g by applying a theorem in [12]. By using Mawhin’s
generalized continuation theorem, Ge [4] proved three theorems for the existence of harmonic
solutions to the systems
ẍ+
d
dt
gradF (x) + gradG(x) = p(t),
where F ∈ C2(Rn, R), G ∈ C1(Rn, R), p ∈ C(R,Rn) and p(t+ T ) ≡ p(t). Besides, Kiguradze
and Mukhigulashvili [7] studied the existence and uniqueness of an ω-periodic solution of two-
dimensional nonautonomous differential systems.
Recently, Agarwal and Chen [1] studied the periodic solutions of the first-order differential
system
x′ = G(t, x(t)), x(0) = x(2π).
By applying the inverse function theorem, they obtained the existence and uniqueness results
for periodic solutions. Mallet-Paret and Nussbaum [11] studied the stability of periodic soluti-
ons of state-dependent delay-differential equations of the form
ż(t) = g(z(t), z(t− r1), . . . , z(t− rn)),
where z : R → Rm, ri = ri(z(t)), i = 1, 2, . . . , n.
On the other hand, neutral functional differential systems were also studied. In virtue of
an extension of Mawhin’s continuation theorem which was established by the author, Lu [8]
studied the existence of periodic solutions to a second-order p-Laplacian neutral functional
differential systems in the form
d
dt
ϕp[(x(t) + Cx(t− τ))′] = f(t, x(t), x(t− µ(t)), x′(t)).
Lu and Ge [9] studied the following second-order neutral differential systems:
d2
dt2
(x(t) + Cx(t− r)) +
d
dt
gradF (x(t)) + gradG(x(t− τ(t))) = p(t),
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 2
182 ZHENGXIN WANG, JINDE CAO, SHIPING LU
which the coefficient matrix of the neutral difference operator being the constant matrix C.
Under this and other assumptions, they presented some results on the existence of periodic
solutions.
Very recently, Henrı́quez, Pierri, and Prokopczyk in [5] studied the existence of periodic
solutions of an abstract neutral functional differential equations with finite and infinite delays
of the form
d
dt
(x(t)−Bx(t)) = Ax(t) + L(xt) + f(t), t ∈ R,
where x(t) ∈ X and X is a Banach space. Lu, Xu and Xia [10] studied new properties of the
D-operator which is described as
D(xt) = x(t)−Bx(t− τ),
and the existence of periodic solutions for the neutral functional differential equation
dDxt
dt
= f(t, xt).
It is easy to see that the coefficient matrices C of the difference operator D, [Dx](t) =
= x(t)+Cx(t−r), are all constant matrices in above studies including the zero matrix. In some
cases, however, the coefficient matrix C is not a constant matrix which is related to the time-
variable t. The studies of the existence of periodic solutions to functional differential systems
with time-varying coefficient matrix C(t) are rare. Therefore, it is worthwhile to study how to
obtain the existence of periodic solution from the existing results. Since the difference operator
contains a time-varying matrix C(t), this paper is different from the literature.
Motivated by the above studies, this paper discusses the existence of periodic solutions to
the following second-order neutral functional differential systems with time-varying coefficient
matrix and deviating arguments:
d2
dt2
(x(t)− C(t)x(t− r)) +
d
dt
gradF (x(t)) + gradG(x(t− γ(t))) = e(t), (1.1)
where F ∈ C2(Rn, R), G ∈ C1(Rn, R), e ∈ C(R,Rn) with e(t+ T ) ≡ e(t) and
∫ T
0
e(t)dt = 0,
where 0 is the n-dimensional zero vector, γ ∈ C1(R,R) and γ(t + T ) ≡ γ(t) for a constant
T > 0, r is a constant. C ∈ C2(R,Rn×n) is a symmetric matrix for all t ∈ R. Because the
coefficient matrix in the difference operator is not a constant matrix, the methods of estimating
a priori bounds of periodic solutions in [2, 4, 8 – 10] can not be used directly. For this reason,
in this paper, we analyze some properties of the linear difference operator D : [Dx](t) =
= x(t)−C(t)x(t− r) firstly, and obtain some results on the existence of D−1 and on properties
of D−1. Then the existence results of periodic solutions to Eq. (1.1) are obtained by applying
Mawhin’s continuation theorem.
The main contributions of this paper include: (1) The difference operator contains a time-
varying matrix, which is rare and different from the constant matrix case. The paper obtains new
properties of time-varying difference operator. (2) The paper investigates periodic solutions
of the neutral functional differential systems under the time-varying difference operator. The
methods to estimate the priori bounds are different from those in the literature.
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 2
PERIODIC SOLUTIONS OF SECOND-ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL SYSTEMS . . . 183
The rest of the paper is organized as follows. Section 2 introduces preliminary results. The
point is that properties of time-varying difference operator are obtained. Section 3 investigates
existence of periodic solutions of neutral functional differential system (1.1). Section 4 gives an
example and two simulations to verify the theoretical results.
2. Preliminaries. In this section, we will make some preparations. At first, we recall Mawhin’s
continuation theorem [3].
LetX and Y be Banach spaces and L : D(L) ⊂ X → Y be a Fredholm operator with index
zero, here D(L) denotes the domain of L. This means that ImL is closed in Y and dim kerL =
= dim(Y/ImL) < +∞. Consider the supplementary subspaces X1, Y1 of X, Y respectively,
such that X = kerL
⊕
X1 and Y = ImL
⊕
Y1, and let P : X → kerL and Q : Y → Y1 be
natural projectors. Clearly, kerL ∩ (D(L) ∩X1) = {0}, thus the restriction LP := L|D(L)∩X1
is
invertible. Denote by KP the inverse of LP .
Now, let Ω be an open bounded subset of X with D(L) ∩ Ω 6= ∅. A map N : Ω → Y is
said to be L-compact in Ω, if QN(Ω) is bounded and the operator KP (I − Q)N : Ω → X is
compact.
Lemma 2.1 [3]. Suppose that X and Y are two Banach spaces, and L : D(L) ⊂ X → Y is a
Fredholm operator with index zero. Furthermore, Ω ⊂ X is an open bounded set and N : Ω →
→ Y is L-compact on Ω. If:
(1) Lx 6= λNx, for all x ∈ ∂Ω ∩D(L), λ ∈ (0, 1);
(2) QNx 6= 0, for all x ∈ ∂Ω ∩ kerL;
(3) deg (JQN,Ω ∩ kerL, 0) 6= 0, where J : ImQ → kerL is an isomorphism.
Then the equation Lx = Nx has a solution in Ω ∩D(L).
We denote by (·, ·) the inner product in Rn, moreover, |x| =
√
(x, x) is the Euclidean norm
for x ∈ Rn. In order to apply Lemma 2.1, we take
X = {x ∈ C1(R,Rn) : x(t+ T ) ≡ x(t)},
with the norm ‖x‖ = maxt∈[0,T ]
√
|x(t)|2 + |x′(t)|2, for all x ∈ X, and
Y = {x ∈ C(R,Rn) : x(t+ T ) ≡ x(t)},
with the norm ‖y‖0 = maxt∈[0,T ] |y(t)|, for all y ∈ Y. Then X and Y are both Banach spaces.
Denote
In = {1, 2, . . . , n}, [x, y] =
T∫
0
(x(t), y(t))dt, for all x, y ∈ Y.
A vector-valued function x(t) = (x1(t), x2(t), . . . , xn(t))> is continuous if xi(t) are conti-
nuous functions for all t ∈ R, i ∈ In. Similarly, a matrix-valued function A(t) = (aij(t))n×n is
continuous if aij(t) are continuous for all t ∈ R, i, j ∈ In, and the definitions of their derivati-
ves and integrals are similar. Let
x′(t) = (x′1(t), x
′
2(t), . . . , x
′
n(t))>, A′(t) = (a′ij(t))n×n,
T∫
0
x(t)dt =
T∫
0
x1(t)dt,
T∫
0
x2(t)dt, . . . ,
T∫
0
xn(t)dt
> ,
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 2
184 ZHENGXIN WANG, JINDE CAO, SHIPING LU
and
T∫
0
A(t)dt =
T∫
0
aij(t)dt
n×n
.
Let Z = {A ∈ C(R,Rn×n) : A(t + T ) ≡ A(t)} represent the set of all continuous matrix-
valued functions, with the norm ‖A‖m = maxt∈[0,T ] ‖A(t)‖F , where
‖A(t)‖F =
√√√√ n∑
i=1
n∑
j=1
(aij(t))2.
Then Z is a Banach space. Suppose that C ∈ C2(R,Rn×n) and C(t) ≡ C(t+ T ). We denote
Ĉ = max
t∈[0,T ]
√√√√ n∑
i=1
n∑
j=1
(aij(t))2, Ĉ1 = max
t∈[0,T ]
√√√√ n∑
i=1
n∑
j=1
(a′ij(t))
2, Ĉ2 = max
t∈[0,T ]
√√√√ n∑
i=1
n∑
j=1
(a′′ij(t))
2.
Then Ĉ, Ĉ1, Ĉ2 are all finite constants.
The following lemma holds by calculation, moreover, it also can be found in books on matrix
analysis such as [6].
Lemma 2.2. Let a, b ∈ Y and A,B ∈ Z. Then the following relations hold:
(1) ‖A+B‖F ≤ ‖A‖F + ‖B‖F ,
(2) ‖AB‖F ≤ ‖A‖F ‖B‖F ,
(3) |Aa| ≤ ‖A‖F |a|,
(4) ‖εA‖F ≤ |ε|‖A‖F , where ε ∈ R is a constant,
(5) |a+ b| ≤ |a|+ |b|.
These results, as well as conclusion (3), which shows compatibility of the Frobenius norm
and the Euclidean norm, are very crucial to obtain a priori bounds for periodic solutions to
Eq. (1.1).
Lemma 2.3 [9]. Let 0 ≤ α ≤ T be a constant, s ∈ C(R,R) with s(t + T ) ≡ s(t) and
maxt∈[0,T ] |s(t)| ≤ α, then for all x ∈ X we have
T∫
0
|x(t)− x(t− s(t))|2dt ≤ 2α2
T∫
0
|x′(t)|2dt.
Defining an operator D as follows:
D : Y → Y, [Dx](t) = x(t)− C(t)x(t− r). (2.1)
It is easy to see that D is a continuous linear operator with ‖D‖ ≤ 1 + Ĉ.
Lemma 2.4. If Ĉ < 1, then D has its continuous inverse D−1 with the following properties:
(1) ‖D−1‖ ≤ 1
1− Ĉ
,
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 2
PERIODIC SOLUTIONS OF SECOND-ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL SYSTEMS . . . 185
(2) [D−1f ](t) = f(t) +
∑∞
j=1
∏j
i=1C(t− (i− 1)r)f(t− jr), for all f ∈ Y,
(3) |[D−1f ](t)| ≤ ‖f‖0
1− Ĉ
, for all f ∈ Y,
(4)
∫ T
0
|[D−1f ](t)|dt ≤ 1
1− Ĉ
∫ T
0
|f(t)|dt, for all f ∈ Y,
(5) [Df ′](t) = [Df ]′(t) + C ′(t)f(t− r), for all f ∈ X.
Proof. Define a linear operator
E : Y → Y, [Ex](t) = C(t)x(t− r).
Then |[Ex](t)| ≤ ‖C(t)‖F |x(t− r)| ≤ Ĉ‖x‖0, so ‖E‖ ≤ Ĉ. Moreover, for all f ∈ Y,
[Ejf ](t) =
j∏
i=1
C(t− (i− 1)r)f(t− jr),
which yields ∣∣∣∣∣∣
∞∑
j=1
[Ejf ](t)
∣∣∣∣∣∣ ≤
∣∣∣∣∣∣
∞∑
j=1
j∏
i=1
C(t− (i− 1)r)f(t− jr)
∣∣∣∣∣∣ ≤
≤
∞∑
j=1
j∏
i=1
‖C(t− (i− 1)r)‖F |f(t− jr)| ≤
≤
∞∑
j=1
Ĉj‖f‖0 ≤
Ĉ
1− Ĉ
‖f‖0.
Since D = id− E and ‖E‖ ≤ Ĉ < 1 it follows that
D−1 : Y → Y, D−1 = (id− E)−1 = id+
∞∑
j=1
Ej ,
and ‖D−1‖ ≤ 1
1− ‖E‖
≤ 1
1− Ĉ
. For all f ∈ Y,
[D−1f ](t) = f(t) +
∞∑
j=1
[Ejf ](t) = f(t) +
∞∑
j=1
j∏
i=1
C(t− (i− 1)r)f(t− jr),
furthermore,
|[D−1f ](t)| =
∣∣∣∣∣∣f(t) +
∞∑
j=1
[Ejf ](t)
∣∣∣∣∣∣ ≤ ‖f‖01− Ĉ
.
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186 ZHENGXIN WANG, JINDE CAO, SHIPING LU
On the other hand, for all f ∈ Y,
T∫
0
|[D−1f ](t)|dt =
T∫
0
∣∣∣∣∣∣f(t) +
∞∑
j=1
j∏
i=1
C(t− (i− 1)r)f(t− jr)
∣∣∣∣∣∣ dt ≤
≤
T∫
0
|f(t)|dt+
T∫
0
∞∑
j=1
∥∥∥∥∥
j∏
i=1
C(t− (i− 1)r)
∥∥∥∥∥
F
|f(t− jr)|dt ≤
≤
T∫
0
|f(t)|dt+
∞∑
j=1
Ĉj
T∫
0
|f(t)|dt ≤ 1
1− Ĉ
T∫
0
|f(t)|dt.
Therefore, the conclusions (1) to (4) hold, and the conclusion (5) can be obtained by a direct
calculation.
Lemma 2.4 is proved.
Now, we define
L : D(L) ⊂ X → Y, Lx =
d2(Dx)
dt2
, (2.2)
where D(L) = {x : x ∈ C2(R,Rn), x(t+ T ) ≡ x(t)} and
N : X → Y, [Nx](t) = − d
dt
gradF (x(t))− gradG(x(t− γ(t))) + e(t). (2.3)
Let x ∈ kerL, one has
d2(Dx)
dt2
= 0, that is, x(t)−C(t)x(t− r) = ω1t+ω2, where ω1, ω2 ∈ Rn.
Since x(t) − C(t)x(t − r) is T -periodic, we have ω1 = 0. Let ψi(t) be a solution to x(t) −
−C(t)x(t−r) = ei,where ei = (0, 0, . . . , 1︸ ︷︷ ︸
i
, . . . , 0)> ∈ Rn. Let Ψ(t) = (ψ1(t), ψ2(t), . . . , ψn(t)) ∈
∈ Rn×n, K = (k1, . . . , kn)> ∈ Rn. Then for all ω =
∑n
i=1 kiei ∈ Rn, Ψ(t)K is a solution to
x(t) − C(t)x(t − r) = ω. Therefore, kerL = {Ψ(t)K : K ∈ Rn}. Furthermore, kerL =
= codim ImL = Rn, ImL =
{
y ∈ Y :
∫ T
0
y(t) dt
}
. Hence, L is a Fredholm operator with
index zero. Define operators P, Q as follows, respectively,
P : X → kerL, Px = Ψ(t)
∫ T
0
Ψ(t)x(t)dt∫ T
0
‖Ψ(t)‖2Fdt
, Q : Y → ImQ, Qy =
1
T
T∫
0
y(t)dt.
Then ImP = kerL, kerQ = ImL. Let KP represent the inverse of LkerP∩D(L), so KP :
ImL → D(L) ∩ kerP. Since ImL ⊂ Y and D(L) ∩ kerP ⊂ X, KP is an embedding operator.
Furthermore, KP is a completely operator in ImL, which together with (2.3), makes it is easy
to see that N is L-compact on Ω, where Ω is an arbitrary open bounded subset of X.
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PERIODIC SOLUTIONS OF SECOND-ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL SYSTEMS . . . 187
For the sake of convenience, we list the following assumptions.
(H1) There is a positive constant η such that (H(v)x, x) ≥ η|x|2, where H(x) =
∂2F (x)
∂x2
,
for all v, x ∈ Rn.
(H2) There is a positive constant M such that
xi
∂G
∂xi
> 0 or xi
∂G
∂xi
< 0, for all i ∈ In with |xi| > M.
(H3) There is a positive constant L such that |gradG(x) − gradG(y)| ≤ L|x − y|, for all
x, y ∈ Rn.
(H4) There is an integer m ≥ 0 such that |γ(t)−mT | ≤ α and γ′(t) < 1 for all t ∈ [0, T ].
Since the matrices of H(x) and C(t) are symetric, it follows that C ′(t), H(x)C(t)C(t)H(x)
and H(x)C ′(t)C ′(t)H(x) are also symmetric for all x ∈ Rn, t ∈ R. Therefore the matrices
C(t), C ′(t), H(x)C(t)C(t)H(x) and H(x)C ′(t)C ′(t)H(x) have n real eigenvalues respectively.
Furthermore, the maximums of n eigenvalues ofC(t) andC ′(t) are finite respectively, which are
denoted by λ0 and λ1, and λ0 ≤ Ĉ, λ1 ≤ Ĉ1 from [6]. Throughout this paper, we denote the
eigenvalues ofH(x)C(t)C(t)H(x) andH(x)C ′(t)C ′(t)H(x) by µ1, µ2, . . . , µn and µ′1, µ
′
2, . . . , µ
′
n
respectively, moreover, we assume that µM := maxi∈In{maxx∈Rn,t∈R |µi|} < +∞ and µ′M :=
:= maxi∈In{ max
x∈Rn,t∈R
|µ′i|} < +∞.
3. Main results. Based on properties of the difference operatorD and Mawhin’s continuation
theorem, some results on the existence of periodic solutions of the neutral functional differenti-
al systems (1.1) are obtained.
Theorem 3.1. Under the assumptions (H1) – (H4), Eq. (1.1) has at least one T -periodic soluti-
on if Ĉ <
1
n+ 1
and η >
√
nµ′MT +
√
µM +
√
2Lα +
√
2nλ1αLT +
√
2λ0αL + n
√
λ1L1T
2 +
+
√
nλ0L1T.
Proof. Let Ω1 = {x ∈ D(L) ∩ X : Lx = λNx, λ ∈ (0, 1)}. If x(·) ∈ Ω1, then from (2.2)
and (2.3), we have
d2
dt2
(x(t)− C(t)x(t− r)) + λ
d
dt
gradF (x(t)) + λ gradG(x(t− γ(t))) = λe(t). (3.1)
Integrating both sides of Eq. (3.1) over [0, T ], we obtain
∫ T
0
gradG(x(t− γ(t)))dt = 0, that is,
for all i ∈ In,
T∫
0
∂G(x(t− γ(t)))
∂xi
dt = 0. (3.2)
Now, we claim that there exists a constant t0i ∈ R (which depends on the subscript i) such
that
|xi(t0i)| ≤ M, for all i ∈ In. (3.3)
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188 ZHENGXIN WANG, JINDE CAO, SHIPING LU
In fact, if xi(t− γ(t)) > M ∀t ∈ [0, T ], i ∈ In, then from (H2) we know
T∫
0
∂G(x(t− γ(t)))
∂xi
dt 6= 0,
which is in contradiction to formula (3.2). So there exists a constant ξi ∈ [0, T ] (which is related
to the subscript i) such that
xi(ξi − γ(ξi)) ≤ M ∀i ∈ In. (3.4)
For the same reason, there exists a constant ηi ∈ [0, T ] (which is related to the subscript i) such
that
xi(ηi − γ(ηi)) ≥ −M ∀i ∈ In. (3.5)
Case 1. If xi(ξi− γ(ξi)) < −M, then according to (3.5) and the intermediate value theorem
for a continuous function we know there exists a constant t1i between ξi − γ(ξi) and ηi − γ(ηi)
such that xi(t1i) = −M, i ∈ In. Take t0i = t1i.
Case 2. If xi(ξi − γ(ξi)) ≥ −M, then according to (3.4) we obtain |xi(ξi − γ(ξi))| ≤ M,
i ∈ In. Take t0i = ξi − γ(ξi).
Combining Cases 1 and 2, it is easy to see that (3.3) holds. Let t2i = kiT + t0i, where ki is
an integer and t2i ∈ [0, T ], i ∈ In, so |xi(t2i)| = |xi(t0i)| ≤ M ∀i ∈ In. Therefore,
|xi(t)| ≤ |xi(t2i)|+
t∫
t2i
|x′i(t)dt| ≤M+
T∫
0
|x′i(t)| dt≤M+
T∫
0
|x′(t)| dt ∀t ∈ [0, T ], i ∈ In. (3.6)
Moreover,
‖x‖0 ≤
√
nM +
√
n
T∫
0
|x′(t)| dt. (3.7)
It follows from Eq. (3.1) that[
d2
dt2
(x(t)− C(t)x(t− r)) + λ
d
dt
gradF (x(t)) +
+ λ gradG(x(t− γ(t))),
d
dt
(x(t)− C(t)x(t− r))
]
=
=
[
λe(t),
d
dt
(x(t)− C(t)x(t− r))
]
. (3.8)
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PERIODIC SOLUTIONS OF SECOND-ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL SYSTEMS . . . 189
As [x′′(t), x′(t)] = 0, [gradG(x(t)), x′(t)] = 0, it follows from (3.8) that
η
T∫
0
|x′(t)|2dt ≤ |[H(x(t))x′(t), x′(t)]| ≤ |[H(x(t))x′(t), C ′(t)x(t− r) + C(t)x′(t− r)]|+
+ |[gradG(x(t− γ(t))), x′(t)− C ′(t)x(t− r)− C(t)x′(t− r)]|+
+ |[e(t), x′(t)− C ′(t)x(t− r)− C(t)x′(t− r)]| ≤
≤ |[H(x(t))x′(t), C ′(t)x(t− r)]|+ |[H(x(t))x′(t), C(t)x′(t− r)]|+
+ |[gradG(x(t− γ(t)))− gradG(x(t)), x′(t)]|+
+ |[gradG(x(t− γ(t)))− gradG(x(t)), C ′(t)x(t− r)]|+
+ |[gradG(x(t)), C ′(t)x(t− r)]|+
+ |[gradG(x(t− γ(t)))− gradG(x(t)), C(t)x′(t− r)]|+
+ |[gradG(x(t)), C(t)x′(t− r)]|+
+ |[e(t), x′(t)]|+ |[e(t), C ′(t)x(t− r)]|+ |[e(t), C(t)x′(t− r)]|. (3.9)
We have
|[H(x(t))x′(t), C ′(t)x(t− r)]| = |[C ′>(t)H(x(t))x′(t), x(t− r)]| =
= |[C ′(t)H(x(t))x′(t), x(t− r)]| ≤
≤ |[C ′(t)H(x(t))x′(t), C ′(t)H(x(t))x′(t)]|
1
2 |[x(t), x(t)]|
1
2 ≤
≤
√
µ′M
T∫
0
|x′(t)|2 dt
1
2
T∫
0
|x(t)|2 dt
1
2
, (3.10)
and
|[H(x(t))x′(t), C(t)x′(t− r)]| = |[C(t)H(x(t))x′(t), x′(t− r)]| ≤
≤ |[C(t)H(x(t))x′(t), C(t)H(x(t))x′(t)]|
1
2 |[x′(t), x′(t)]|
1
2 ≤
≤ √µM
T∫
0
|x′(t)|2 dt. (3.11)
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190 ZHENGXIN WANG, JINDE CAO, SHIPING LU
Let γ(t)−mT = s(t), then by Lemma 2.3 we can obtain the following inequalities:
|[gradG(x(t− γ(t)))− gradG(x(t)), x′(t)]| ≤ L
T∫
0
|x(t− s(t))− x(t)||x′(t)| dt ≤
≤ L
T∫
0
|x(t− s(t))− x(t)|2dt
1
2
×
×
T∫
0
|x′(t)|2dt
1
2
≤
√
2Lα
T∫
0
|x′(t)|2 dt,
(3.12)
|[gradG(x(t− γ(t)))− gradG(x(t)), C ′(t)x(t− r)]| ≤
√
λ1
T∫
0
|x(t)|2dt
1
2
×
×
T∫
0
|gradG(x(t− γ(t)))− gradG(x(t))|2dt
1
2
≤
≤
√
2λ1Lα
T∫
0
|x′(t)|2dt
1
2
T∫
0
|x(t)|2dt
1
2
, (3.13)
and
|[gradG(x(t− γ(t)))− gradG(x(t)), C(t)x′(t− r)]| ≤
√
2λ0Lα
T∫
0
|x′(t)|2 dt. (3.14)
From assumption (H3),
|gradG(x)| ≤ L|x|+ |gradG(0)| := L|x|+ g0 ≤
{
L+ g0 := g1, for |x| ≤ 1,
(L+ g0)|x| := L1|x|, for |x| > 1.
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PERIODIC SOLUTIONS OF SECOND-ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL SYSTEMS . . . 191
Let41 = {t ∈ [0, T ] : |x(t)| ≤ 1},42 = {t ∈ [0, T ] : |x(t)| > 1}, then
|[gradG(x(t)), C ′(t)x(t− r)]| ≤ |[gradG(x(t)), gradG(x(t))]|
1
2 |[C ′(t)x(t− r), C ′(t)x(t− r)]|
1
2 ≤
≤
√
λ1
T∫
0
|x(t)|2dt
1
2
∫
41
+
∫
42
|gradG(x(t))|2dt
1
2
≤
≤
√
λ1
T∫
0
|x(t)|2dt
1
2
g21T + L2
1
T∫
0
|x(t)|2dt
1
2
≤
≤ g1
√
λ1T
T∫
0
|x(t)|2dt
1
2
+ L1
√
λ1
T∫
0
|x(t)|2 dt, (3.15)
and
|[gradG(x(t)), C(t)x′(t− r)]| ≤
√
λ0
T∫
0
|x′(t)|2dt
1
2
∫
41
+
∫
42
|gradG(x(t))|2 dt
1
2
≤
≤
√
λ0
T∫
0
|x′(t)|2 dt
1
2
g21T + L2
1
T∫
0
|x(t)|2 dt
1
2
≤
≤ g1
√
λ0T
T∫
0
|x′(t)2 dt
1
2
+
+ L1
√
λ0
T∫
0
|x′(t)|2dt
1
2
T∫
0
|x(t)|2dt
1
2
. (3.16)
Furthermore,
|[e(t), x′(t)]|+ |[e(t), C ′(t)x(t− r)]|+ |[e(t), C(t)x′(t− r)]| ≤
≤ ẽ
T∫
0
|x′(t)2 dt
1
2
+ ẽ
√
λ1
T∫
0
|x(t)|2 dt
1
2
+ ẽ
√
λ0
T∫
0
|x′(t)|2 dt
1
2
, (3.17)
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192 ZHENGXIN WANG, JINDE CAO, SHIPING LU
where ẽ =
(∫ T
0
|e(t)2 dt
) 1
2
. Substituting (3.7) and (3.10) – (3.17) into (3.9) we obtain
η
T∫
0
|x′(t)|2dt≤
(
T
√
nµ′M +
√
µM +Lα
√
2+αLT
√
2nλ1+αL
√
2λ0+nL1T
2
√
λ1+L1T
√
nλ0
)
×
×
T∫
0
|x′(t)|2dt+
(
M
√
nµ′MT + αLM
√
2nTλ1 + 2nML1T
3
2
√
λ1 + g1T
3
2
√
nλ1 +
+ g1
√
λ0T + L1M
√
nTλ0 + ẽ+ ẽT
√
nλ1 + ẽ
√
λ0
) T∫
0
|x′(t)|2 dt
1
2
+
+ nTM2L1
√
λ1 + g1MT
√
nλ1 + ẽM
√
nλ1T . (3.18)
By η > T
√
nµ′M +
√
µM +Lα
√
2+αLT
√
2nλ1 +αL
√
2λ0 +nL1T
2
√
λ1 +L1T
√
nλ0 and (3.18),∫ T
0
|x′(t)|2 dt is bounded. So there exists a positive constant M1 such that
T∫
0
|x′(t)|2 dt ≤ M1, (3.19)
which, together with (3.7), gives
‖x‖0 ≤
√
nM +
√
nTM1. (3.20)
From the definition of the operator D we know that, for x ∈ D(L),
[Dx′′](t) = [Dx]′′(t) + 2C ′(t)x′(t− r) + C ′′(t)x(t− r),
which, together with Eq. (3.1), yields
x′′(t) +D−1
[
λ
d
dt
gradF (x(t)) + λ gradG(x(t− γ(t)))
]
=
= D−1[λe(t) + 2C ′(t)x′(t− r) + C ′′(t)x(t− r)],
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PERIODIC SOLUTIONS OF SECOND-ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL SYSTEMS . . . 193
so by Lemma 2.4 we obtain
T∫
0
|x′′(t)|2 dt≤ 1
1− Ĉ
T∫
0
‖H(x(t))‖F |x′(t)| dt+
T∫
0
|gradG(x(t− γ(t)))| dt+
T∫
0
|e(t)| dt +
+ 2
T∫
0
‖C ′(t)‖F |x′(t− r)|dt+
T∫
0
‖C ′′(t)‖F |x(t− r)|dt
≤
≤ 1
1− Ĉ
[H0
√
TM1 +G0T + ẽ
√
T + 2Ĉ1
√
TM1Ĉ2(
√
nM +
√
nTM1)T ] := M2,
(3.21)
where H0 and G0 are the maximums of ‖H(x(t))‖F and |gradG(x(t − γ(t)))| on {x : ‖x‖0 ≤
≤
√
nM +
√
nTM1}. Therefore,
|x′(t)| =
(
n∑
i=1
|x′i(t)|2
) 1
2
≤
T n∑
i=1
T∫
0
|x′′i (t)|2dt
1
2
≤
√
nTM2 ∀t ∈ [0, T ],
that is, ‖x′‖0 ≤
√
nTM2, which together with (3.20), shows that ‖x‖ ≤
√
nM +
√
nTM1 +
+
√
nTM2.
Let Ω2 = {x ∈ X : QNx = 0, x ∈ kerL}. If x ∈ Ω2, then x = Ψ(t)K,K ∈ Rn satisfying
− 1
T
T∫
0
gradG(x(t− γ(t)))dt ≡ − 1
T
T∫
0
1
1− γ′(µ(t))
gradG(Ψ(t)K) dt = 0,
where µ(s) is the inverse of t− γ(t).
In view of ψi(t) = [D−1ei](t) = ei +
∑∞
j=1
∏j
k=1C(t− (k − 1)r)ei, we have
|kiψii(t)| =
∣∣∣∣∣∣ki +
∞∑
j=1
ki
(
j∏
k=1
C(t− (k − 1)r)ei
)
i
∣∣∣∣∣∣ ≥
≥ |ki| − |ki|
∞∑
j=1
j∏
k=1
Ĉ = |ki| − |ki|
Ĉ
1− Ĉ
:= (1− ρ)|ki|,
and
|ψil(t)|=
∣∣∣∣∣∣
∞∑
j=1
(
j∏
k=1
C(t− (k − 1)r)ei
)
l
∣∣∣∣∣∣≤
∞∑
j=1
j∏
k=1
Ĉ =
Ĉ
1− Ĉ
:= ρ,
where l 6= i, for all t ∈ [0, T ]. Then |ki|−ρ
∑n
j=1 |kj | ≤ M.Otherwise, if |ki|−ρ
∑n
j=1 |kj | > M,
then
|(Ψ(t)K)i|=
∣∣∣∣∣∣
n∑
j=1
kjψji(t)
∣∣∣∣∣∣ =
∣∣∣∣∣∣kiψii(t)−
n∑
j 6=i
kjψji(t)
∣∣∣∣∣∣≥
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194 ZHENGXIN WANG, JINDE CAO, SHIPING LU
≥ |ki| − ρ|ki| − ρ
n∑
j 6=i
|kj | = |ki| − ρ
n∑
j=1
|kj | > M,
for all t ∈ [0, T ], and
T∫
0
1
1− γ′(µ(t))
gradG(Ψ(t)K)dt 6= 0,
by assumption (H2). This is a contradiction. Therefore, |ki| − ρ
∑n
j=1 |kj | ≤ M, which together
with
∑n
j=1 |kj | ≤
√
n|K|, gives |ki| −
√
nρ|K| ≤ M, that is, |ki| ≤
√
nρ|K|+M. Furthermore,
|K| =
n∑
j=1
|kj |2
1
2
≤
√
n(
√
nρ|K|+M) = nρ|K|+
√
nM,
therefore, |K| ≤
√
nM
1− nρ
. On the other hand,
Ψ(t) =
(
[D−1e1](t), [D
−1e2](t), . . . , [D
−1en](t)
)
=
=
e1 +
∞∑
j=1
j∏
k=1
C(t− (k − 1)r)e1, . . . , en +
∞∑
j=1
j∏
k=1
C(t− (k − 1)r)en
=
= I +
∞∑
j=1
j∏
k=1
C(t− (k − 1)r)I,
where I is the identity matrix, moreover,
‖Ψ‖m = max
t∈[0,T ]
‖Ψ(t)‖F ≤ ‖I‖F +
∞∑
j=1
j∏
k=1
max
t∈[0,T ]
‖C(t− (k − 1)r)‖F ‖I‖F ≤
≤
√
n+
∞∑
j=1
j∏
k=1
Ĉ
√
n =
√
n
1
1− Ĉ
=
√
nρ
Ĉ
.
Then ‖x‖0 = maxt∈[0,T ] |Ψ(t)K| ≤ ‖Ψ‖m|K| ≤
√
nρ
Ĉ
√
nM
1− nρ
=
nρM
Ĉ(1− nρ)
:= M3.
Now, if we let Ω = {x : x ∈ X, ‖x‖ ≤
√
nTM1+
√
nTM2+M3+
√
nM}, then Ω1∪Ω2 ⊂ Ω.
So the condition (1) and condition (2) of Lemma 2.1 are satisfied.
If x ∈ ∂Ω ∩ kerL, then x(t) = Ψ(t)K, K ∈ Rn with ‖x‖0 >
√
nM, hence there exists an
i ∈ In such that |(Ψ(t)K)i| > M, t ∈ [0, T ]. Otherwise, |(Ψ(t)K)i| ≤ M, t ∈ [0, T ], for all
i ∈ In, then ‖x‖0 = maxt∈[0,T ]
(∑n
i=1 |(Ψ(t)K)i|2
) 1
2 ≤
√
nM. It is a contradiction. It follows
from |(Ψ(t)K)i| > M, t ∈ [0, T ] that QNx = − 1
T
∫ T
0
1
1− γ′(µ(t))
gradG(Ψ(t)K)dt 6= 0 and
sgn
(
xi
∂G(x)
∂xi
)
= (−1)si , si ∈ {0, 1}, i ∈ In.
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PERIODIC SOLUTIONS OF SECOND-ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL SYSTEMS . . . 195
Let
H(µ, x) = µSx+ (1− µ)JQNx,
where S = diag {(−1)s1 , (−1)s2 , . . . , (−1)sn} and J(x) = x for all x ∈ Rn. Therefore, for all
(µ, x) ∈ [0, 1]× ∂Ω ∩ kerL, H(µ, x) 6= 0 and
deg{JQN,Ω ∩ kerL, 0}= deg{H(0, ·),Ω ∩ kerL, 0}= deg{H(1, ·),Ω ∩ kerL, 0}=
= deg{id,Ω ∩ kerL, 0} 6= 0.
Now, we know that conditions in Lemma 2.1 are all satisfied, therefore Lx = Nx has at least
one solution in X.
Theorem 3.1 is proved.
If we replace the assumption (H3) with the following assumption:
(H5) There is a positive constant ` such that |gradG(x)| ≤ `|x|+ w, for all x ∈ Rn.
Then the following theorem holds.
Theorem 3.2. Under the assumptions (H1), (H2) and (H5), Eq. (1.1) has at least one T -
periodic solution if Ĉ <
1
n+ 1
and η >
√
nµ′MT +
√
µM + `1T + `1nT
2
√
λ1 + `1T
√
λ0.
Proof. From assumption (H5) we obtain
|gradG(x)| ≤ `|x|+ w ≤
{
`+ w := w1, for |x| ≤ 1,
(`+ w)|x| := `1|x|, for |x| > 1.
(3.22)
It follows from (3.8) that
η
T∫
0
|x′(t)|2dt ≤ |[H(x(t))x′(t), x′(t)]| ≤ |[H(x(t))x′(t), C ′(t)x(t− r) + C(t)x′(t− r)]|+
+ |[gradG(x(t− γ(t))), x′(t)− C ′(t)x(t− r)− C(t)x′(t− r)]|+
+ |[e(t), x′(t)− C ′(t)x(t− r)− C(t)x′(t− r)]| ≤
≤ |[H(x(t))x′(t), C ′(t)x(t− r)]|+ |[H(x(t))x′(t), C(t)x′(t− r)]|+
+ |[gradG(x(t− γ(t))), x′(t)]|+ |[gradG (x(t− γ(t))), C ′(t)x(t− r)]|+
+ |[gradG(x(t− γ(t))), C(t)x′(t− r)]|+ |[e(t), x′(t)]|+
+ |[e(t), C ′(t)x(t− r)]|+ |[e(t), C(t)x′(t− r)]|. (3.23)
Let∇1 = {t ∈ [0, T ] : |x(t− γ(t))| ≤ 1},∇2 = {t ∈ [0, T ] : |x(t− γ(t))| > 1}. Then it follows
from (3.7) and (3.22) that
|[gradG(x(t− γ(t))), x′(t)]| ≤
T∫
0
|x′(t)|2dt
1
2
∫
∇1
+
∫
∇2
gradG(x(t− γ(t)))|2dt
1
2
≤
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 2
196 ZHENGXIN WANG, JINDE CAO, SHIPING LU
≤
T∫
0
|x′(t)|2dt
1
2 (
w2
1T + `21‖x‖20T
) 1
2 ≤
≤
T∫
0
|x′(t)|2dt
1
2
w1
√
T + `1M
√
nT + `1T
T∫
0
|x′(t)|2dt
1
2
≤
≤
(
w1
√
T + `1M
√
nT
) T∫
0
|x′(t)|2dt
1
2
+ `1T
T∫
0
|x′(t)|2 dt. (3.24)
We can also obtain the following two inequalities:
|[gradG(x(t− γ(t))), C ′(t)x(t− r)]| ≤
√
λ1(w1TM
√
n+ n`1TM
2)+
+
√
λ1
(
w1T
3
2
√
n+ 2n`1MT
3
2
) T∫
0
|x′(t)|2 dt
1
2
+
+ `1nT
2
√
λ1
T∫
0
|x′(t)|2 dt, (3.25)
and
|[gradG(x(t− γ(t))), C(t)x′(t− r)]| ≤
√
λ0
(
w1
√
T + `1M
√
nT
) T∫
0
|x′(t)|2dt
1
2
+
+ `1T
√
λ0
T∫
0
|x′(t)|2dt. (3.26)
Substituting (3.7), (3.10), (3.11), (3.17), (3.24), (3.25) and (3.26) into (3.23) we obtain
η
T∫
0
|x′(t)|2dt ≤ (T
√
nµ′M +
√
µM + `1T + `1nT
2
√
λ1 + `1T
√
λ0)
T∫
0
|x′(t)|2dt+
+ (M
√
nµ′MT + w1
√
T + `1M
√
nT + w1T
3
2
√
nλ1 + 2n`1MT
3
2
√
λ1+
+ w1
√
Tλ0 + `1M
√
nTλ0)
T∫
0
|x′(t)|2dt
1
2
+
+
√
λ1(w1TM
√
n+ n`1TM
2) + ẽM
√
nλ1T . (3.27)
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PERIODIC SOLUTIONS OF SECOND-ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL SYSTEMS . . . 197
By η > T
√
nµ′M +
√
µM +`1T+`1nT
2
√
λ1+`1T
√
λ0 and (3.27) we know there exists a positive
constant M3 such that
T∫
0
|x′(t)|2dt ≤ M3. (3.28)
The remainder proof is similar to Theorem 3.1.
Theorem 3.2 is proved.
Remark 3.1. It is easy to see from assumption (H2) that the sign of xi
∂G
∂xi
is different for
every i ∈ In.
Remark 3.2. As the coefficient matrix C(t) is not a constant matrix C, the results in this
paper can not be obtained from [1, 2, 4, 5, 8 – 11].
4. A numerical example. In this section, we consider a example and give two simulations to
illustrate the theoretical results of Section 3.
Example 4.1. Consider the following systems:
x(t)−
1
64
sin t
1
64
cos t
1
64
cos t
1
64
sin t
x(t− 2)
′′
+
d
dt
gradF (x(t))+
+ gradG(x(t− γ(t))) =
(
sin t
cos t
)
, (4.1)
where x(t) = (x1(t), x2(t))
>, F (x) = x21 + x22, G =
1
32
(x21 + x22) and γ(t) =
1
16
cos t. By
calculation, we obtain λ0 = λ1 = 0.0221, α = L = L1 =
1
16
, T = 2π, η = 2, µM =
= µ′M = 0.002. It is easy to verify that (H1) – (H4) hold. Furthermore, Ĉ =
√
2
64
<
1
3
and
η = 2 >
√
nµ′MT+
√
µM +
√
2Lα+
√
2nλ1αLT+
√
2λ0αL+n
√
λ1L1T
2+
√
nλ0L1T = 1.5196.
Therefore, it follows from Theorem 3.1 that Eq. (4.1) has at least one 2π-periodic solutions.
Simulation results are shown in Figures 4.1 and 4.2. As showed in Figures 4.1 and 4.2, there
exists a periodic solution, and solutions (x1(t), x2(t)) which start from 4 initial values tend to it.
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 2
198 ZHENGXIN WANG, JINDE CAO, SHIPING LU
Fig. 4.1. Evolution of (a) the x1(t) and (b) the x2(t) with 4 initial values.
Fig. 4.2. Phase plane of systems (4.1) with 4 initial points.
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Received 29.03.12
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