Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2013, Article ID 502963, 9 pages
http://dx.doi.org/10.1155/2013/502963
Research Article
Positive Periodic Solutions of Cooperative Systems with Delays
and Feedback Controls
Tursuneli Niyaz1 and Ahmadjan Muhammadhaji2
1
2
College of Applied Mathematics, Xinjiang University of Finance and Economics, Ürümqi 830012, China
College of Mathematical Sciences, Xinjiang Normal University, Urumqi 830054, China
Correspondence should be addressed to Ahmadjan Muhammadhaji; ahmadjanm@gmail.com
Received 23 April 2013; Accepted 13 May 2013
Academic Editor: Norio Yoshida
Copyright © 2013 T. Niyaz and A. Muhammadhaji. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
This paper studies a class of periodic n species cooperative Lotka-Volterra systems with continuous time delays and feedback
controls. Based on the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin, some new
sufficient conditions on the existence of positive periodic solutions are established.
1. Introduction
Mathematical ecological system has become one of the
most important topics in the study of modern applied
mathematics. Its dynamical behavior includes persistence,
permanence and extinction of species, global stability of
systems, the existence of positive periodic solutions, positive
almost periodic solutions, and strictly positive solutions. The
existence of positive periodic solutions has already become
one of the most interesting subjects for scholars. In the recent
years, the application of fixed-point theorems to the existence
of positive periodic solutions in mathematical ecology has
been studied extensively, for example, Brouwer’s fixed point
theorem [1–4], Schauder’s fixed-point theorem [5–8], Krasnoselskii’s fixed-point theorem [9–14], Horn’s fixed-point
theorem [15, 16], and Mawhins continuation theorem [17–36],
and so forth. In particular, Mawhins continuation theorem
is a powerful tool for studying the existence of periodic
solutions of periodic high-dimensional time-delayed problems. When dealing with a time-delayed problem, it is very
convenient and the result is relatively simple [30]. Recently,
a considerable number of mathematical models with delays
have been proposed in the study of population dynamics. One
of the most celebrated models for population is the LotkaVolterra system. Subsequently, a lot of the literature related to
the study of the existence of positive periodic solutions for
various Lotka-Volterra-type population dynamical systems
with delays by using the method of continuation theorem
was published and extensive research results were obtained
[17–21, 24–34].
On the other hand, in some situations, people may wish
to change the position of the existing periodic solution but
to keep its stability. This is of significance in the control of
ecology balance. One of the methods for the realization of
it is to alter the system structurally by introducing some
feedback control variables so as to get a population stabilizing
at another periodic solution. The realization of the feedback
control mechanism might be implemented by means of some
biological control scheme or by harvesting procedure [21]. In
fact, during the last decade, the existence of positive periodic
solutions for the population dynamics with feedback control
has been studied extensively [8, 14, 17–21, 24, 29]. To the best
of our knowledge, studies on the existence of positive periodic
solutions for cooperative systems with delays and feedback
controls are fairly rare.
In [21], the authors studied the following neutral LotkaVolterra system with feedback controls:
𝑛
𝑦𝑖̇ (𝑡) = 𝑦𝑖 (𝑡) [𝑟𝑖 (𝑡) − ∑ 𝑎𝑖𝑗 𝑦𝑗 (𝑡)
𝑗=1
[
𝑛
𝑛
𝑗=1
𝑗=1
− ∑𝑏𝑖𝑗 𝑦𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)) − ∑ 𝑐𝑖𝑗 𝑦𝑗̇ (𝑡 − 𝛾𝑖𝑗 (𝑡))
2
International Journal of Differential Equations
2. Preliminaries
−𝑓𝑖 (𝑡) 𝑢𝑖 (𝑡) − 𝑒𝑖 (𝑡) 𝑢𝑖 (𝑡 − 𝜎𝑖 (𝑡)) ]
]
𝑢̇𝑖 (𝑡) = − 𝛼𝑖 (𝑡) 𝑢𝑖 (𝑡) + 𝛽𝑖 (𝑡) 𝑦𝑖 (𝑡)
+ 𝛾𝑖 (𝑡) 𝑦𝑖 (𝑡) (𝑡 − 𝛿𝑖 (𝑡)) ,
𝑖 = 1, 2, . . . , 𝑛.
(1)
By using Mawhin’s continuation theorem, the sufficient conditions on the existence of positive periodic solutions are
established. In [24], the authors considered the following
delay differential system with feedback control:
𝑑𝑥
= 𝐹 (𝑡, 𝑥 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑥 (𝑡 − 𝜏𝑛 (𝑡)) , 𝑢 (𝑡 − 𝛿 (𝑡)))
𝑑𝑡
𝑑𝑢
= −𝜂 (𝑡) + 𝑎 (𝑡) 𝑥 (𝑡 − 𝜎 (𝑡)) .
𝑑𝑡
(2)
A set of natural and easily verifiable sufficient conditions of
the existence of positive periodic solutions are established,
by using Mawhin’s continuation theorem. In [29], the authors
considered the following single-species periodic logistic systems with feedback regulation and infinite distributed delay:
𝑁̇ (𝑡) = 𝑟 (𝑡) 𝑁 (𝑡)
∞
1
× (1 −
∫ 𝐻 (𝑠) 𝑁 (𝑡 − 𝑠) 𝑑𝑠 − 𝑐 (𝑡) 𝑢 (𝑡))
𝐾 (𝑡) 0
∞
𝑢̇ (𝑡) = −𝑎 (𝑡) 𝑢 (𝑡) + 𝑏 (𝑡) ∫ 𝐻 (𝑠) 𝑁2 (𝑡 − 𝑠) 𝑑𝑠.
0
(3)
The sufficient conditions for the existence of positive periodic
solutions are established, based on Mawhin’s continuation
theorem.
Motivated by the above works, in this paper, we investigate the following 𝑛 species periodic Lotka-Volterra-type
cooperative systems with continuous time delays and feedback controls:
𝑚
𝑥𝑖̇ (𝑡) = 𝑥𝑖 (𝑡) [𝑟𝑖 (𝑡) − ∑𝑎𝑖𝑖𝑙 (𝑡) 𝑥𝑖 (𝑡 − 𝜏𝑖𝑖𝑙 (𝑡))
𝑙=1
𝑛
𝑚
+ ∑ ∑𝑎𝑖𝑗𝑙 (𝑡) 𝑥𝑗 (𝑡 − 𝜏𝑖𝑗𝑙 (𝑡))]
𝑗 ≠ 𝑖 𝑙=1
]
(4)
− 𝑑𝑖 (𝑡) 𝑢𝑖 (𝑡) − 𝑒𝑖 (𝑡) 𝑢𝑖 (𝑡 − 𝜀𝑖 (𝑡))
𝑢̇𝑖 (𝑡) = − 𝑏𝑖 (𝑡) 𝑢𝑖 (𝑡) + 𝛽𝑖 (𝑡) 𝑥𝑖 (𝑡)
+ 𝛾𝑖 (𝑡) 𝑥𝑖 (𝑡 − 𝜎𝑖 (𝑡)) ,
𝑖 = 1, 2, . . . , 𝑛.
By using the technique of coincidence degree developed by
Gaines and Mawhin in [36], we will establish some new
sufficient conditions which guarantee that the system has at
least one positive periodic solution.
In system (4), we have that 𝑥𝑖 (𝑡) (𝑖 = 1, 2, . . . 𝑛) represent
the density of 𝑛 cooperative species 𝑥𝑖 (𝑖 = 1, 2, . . . 𝑛)
at time 𝑡, respectively; 𝑟𝑖 (𝑡) (𝑖 = 1, 2, . . . 𝑛) represent the
intrinsic growth rate of species 𝑥𝑖 (𝑖 = 1, 2, . . . 𝑛) at time 𝑡,
respectively; 𝑎𝑖𝑖𝑙 (𝑡) (𝑖 = 1, 2, . . . , 𝑛, 𝑙 = 1, 2, . . . , 𝑚) represent
the intrapatch restriction density of species 𝑥𝑖 (𝑖 = 1, 2, . . . 𝑛)
at time 𝑡, respectively; 𝑎𝑖𝑗𝑙 (𝑡) (𝑙 = 1, 2, . . . , 𝑚, 𝑖 ≠ 𝑗, 𝑖, 𝑗 =
1, 2, . . . , 𝑛) represent the cooperative coefficients between 𝑛
species 𝑥𝑖 (𝑖 = 1, 2, . . . 𝑛) at time 𝑡, respectively. 𝑢𝑖 (𝑡) (𝑖 =
1, 2, . . . 𝑛) represent the indirect feedback control variables
[21] at time 𝑡, respectively. 𝛽𝑖 (𝑡), 𝑒𝑖 (𝑡), 𝑏𝑖 (𝑡), 𝑑𝑖 (𝑡), and 𝛾𝑖 (𝑡) (𝑖 =
1, 2, . . . 𝑛) represent the feedback control coefficients at time
𝑡, respectively. In this paper, we always assume that
(H1) 𝜏𝑖𝑗𝑙 (𝑡) (𝑙 = 1, 2, . . . , 𝑚, 𝑖, 𝑗 = 1, 2, . . . , 𝑛), 𝜎𝑖 (𝑡), 𝜀𝑖 (𝑡), 𝑟𝑖
(𝑡) (𝑖 = 1, 2, . . . 𝑛) are continuous 𝜔-periodic func𝜔
󸀠
(𝑡) < 1 and ∫0 𝑟𝑖 (𝑡)𝑑𝑡 > 0. 𝑎𝑖𝑗𝑙 (𝑡) (𝑖, 𝑗 =
tions with 𝜏𝑖𝑗𝑙
1, 2, 𝑙 = 1, 2, . . . , 𝑚), 𝛽𝑖 (𝑡), 𝑒𝑖 (𝑡), 𝑏𝑖 (𝑡), 𝑑𝑖 (𝑡), and
𝛾𝑖 (𝑡) (𝑖 = 1, 2, . . . 𝑛) are continuous, positive 𝜔periodic functions.
From the viewpoint of mathematical biology, in this
paper for system (4) we only consider the solution with the
following initial conditions:
𝑥𝑖 (𝑡) = 𝜙𝑖 (𝑡) ,
∀𝑡 ∈ [−𝜎, 0] , 𝑖 = 1, 2, . . . 𝑛,
𝑢𝑖 (𝑡) = 𝜓𝑖 (𝑡) ,
∀𝑡 ∈ [−𝜎, 0] , 𝑖 = 1, 2, . . . 𝑛,
(5)
where 𝜙𝑖 (𝑡), 𝜓𝑖 (𝑡) (𝑖 = 1, 2, . . . 𝑛) are nonnegative continuous
functions defined on [−𝜎, 0] satisfying 𝜙𝑖 (0) > 0, 𝜓𝑖 (0) >
0 (𝑖 = 1, 2, . . . 𝑛) with 𝜎 = max𝑡∈[0,𝜔] {𝜏𝑖𝑗𝑙 (𝑡), 𝜎𝑖 (𝑡), 𝜀𝑖 (𝑡) (𝑖, 𝑗 =
1, 2, 𝑙 = 1, 2, . . . , 𝑚)}.
In this paper, for any 𝜔-periodic continuous function 𝑓(𝑡)
we denote
𝑓𝐿 = min 𝑓 (𝑡) ,
𝑡∈[0,𝜔]
𝑓=
𝑓𝑀 = max 𝑓 (𝑡) ,
𝑡∈[0,𝜔]
1 𝜔
∫ 𝑓 (𝑡) 𝑑𝑡.
𝜔 0
(6)
In order to obtain the existence of positive 𝜔-periodic
solutions of system (4), we will use the continuation theorem
developed by Gaines and Mawhin in [36]. For the reader’s
convenience, we will introduce the continuation theorem in
the following.
Let 𝑋 and 𝑍 be two normed vector spaces. Let 𝐿 :
Dom 𝐿 ⊂ 𝑋 → 𝑍 be a linear operator and let 𝑁 : 𝑋 → 𝑍
be a continuous operator. The operator 𝐿 is called a Fredholm
operator of index zero, if dimKer 𝐿 = codimIm 𝐿 < ∞ and
Im 𝐿 is a closed set in 𝑍. If 𝐿 is a Fredholm operator of index
zero, then there exist continuous projectors 𝑃 : 𝑋 → 𝑋
and 𝑄 : 𝑍 → 𝑍 such that Im 𝑃 = Ker 𝐿 and Im 𝐿 =
Ker 𝑄 = Im (𝐼 − 𝑄). It follows that 𝐿 | Dom 𝐿 ∩ Ker 𝑃 :
Dom 𝐿 ∩ Ker 𝑃 → Im 𝐿 is invertible and its inverse is
denoted by 𝐾𝑃 ; denote by 𝐽 : Im 𝑄 → Ker 𝐿 an isomorphism
of Im 𝑄 onto Ker 𝐿. Let Ω be a bounded open subset of 𝑋; we
say that the operator 𝑁 is 𝐿-compact on Ω, where Ω denotes
International Journal of Differential Equations
3
the closure of Ω in 𝑋, if 𝑄𝑁(Ω) is bounded and 𝐾𝑃 (𝐼 − 𝑄)𝑁 :
Ω → 𝑋 is compact.
Lemma 1 (see [35]). Suppose 𝜏 ∈ 𝐶1 (𝑅, 𝑅) with 𝜏(𝑡+𝜔) ≡ 𝜏(𝑡)
and 𝜏󸀠 (𝑡) < 1, ∀𝑡 ∈ [0, 𝜔]. Then the function 𝑡 − 𝜏(𝑡) has a
unique inverse function 𝜇(𝑡) satisfying 𝜇 ∈ 𝐶(𝑅, 𝑅), 𝜇(𝑢 + 𝜔) =
𝜇(𝑢) + 𝜔, ∀𝑢 ∈ 𝑅.
From (9), we obtain
𝑢𝑖 (𝑡 + 𝜔) = (∫
𝑡+𝜔
0
(𝛽𝑖 (𝑠) 𝑥𝑖∗ (𝑠) + 𝛾𝑖 (𝑠) 𝑥𝑖∗ (𝑠 − 𝜎𝑖 (𝑠)))
𝑠
× exp {∫ 𝑏𝑖 (𝜃) 𝑑𝜃} 𝑑𝑠 + 𝜓𝑖 (0))
0
Lemma 2 (see [36]). Let 𝐿 be a Fredholm operator of index
zero and let 𝑁 be 𝐿-compact on Ω. If
× exp {− ∫
𝑡+𝜔
0
𝑏𝑖 (𝜃) 𝑑𝜃} ,
𝑖 = 1, 2, . . . , 𝑛,
𝑡
(a) for each 𝜆 ∈ (0, 1) and 𝑥 ∈ 𝜕Ω ∩ Dom 𝐿, 𝐿𝑥 ≠ 𝜆𝑁𝑥;
𝑢𝑖∗ (𝑡) = (∫ (𝛽𝑖 (𝑠) 𝑥𝑖∗ (𝑠) + 𝛾𝑖 (𝑠) 𝑥𝑖∗ (𝑠 − 𝜎𝑖 (𝑠)))
0
(b) for each 𝑥 ∈ 𝜕Ω ∩ Ker 𝐿, 𝑄𝑁𝑥 ≠ 0;
𝑠
(c) deg {𝐽𝑄𝑁, Ω ∩ Ker 𝐿, 0} ≠ 0,
× exp {∫ 𝑏𝑖 (𝜃) 𝑑𝜃} 𝑑𝑠 + 𝜓𝑖 (0))
0
then the operator equation 𝐿𝑥 = 𝑁𝑥 has at least one solution
lying in Dom 𝐿 ∩ Ω.
𝑡
× exp {− ∫ 𝑏𝑖 (𝜃) 𝑑𝜃} ,
0
𝑖 = 1, 2, . . . , 𝑛.
(10)
3. Main Results
In order to obtain the existence of positive periodic solutions
of system (4), firstly, we introduce the following lemma.
Lemma 3. Suppose that (𝑥1∗ (𝑡), 𝑥2∗ (𝑡), . . ., 𝑥𝑛∗ (𝑡), 𝑢1∗ (𝑡),
𝑢2∗ (𝑡), . . . , 𝑢𝑛∗ (𝑡)) is an 𝜔-periodic solution of (4) and (5); then
(𝑥1∗ (𝑡), 𝑥2∗ (𝑡), . . ., 𝑥𝑛∗ (𝑡), 𝑢1∗ (𝑡), 𝑢2∗ (𝑡), . . . , 𝑢𝑛∗ (𝑡)) satisfies the
system
Considering that (𝑥1∗ (𝑡), 𝑥2∗ (𝑡), . . . , 𝑥𝑛∗ (𝑡), 𝑢1∗ (𝑡), 𝑢2∗ (𝑡),. . .,
𝑢𝑛∗ (𝑡)) is an 𝜔-periodic solution of system (4) and (5), we
obtain
𝑡
(∫ (𝛽𝑖 (𝑠) 𝑥𝑖∗ (𝑠) + 𝛾𝑖 (𝑠) 𝑥𝑖∗ (𝑠 − 𝜎𝑖 (𝑠)))
0
𝑚
2
𝑥𝑖̇ (𝑡) = 𝑥𝑖 (𝑡) (𝑟𝑖 (𝑡) − ∑ (−1)𝑖+𝑗 ∑𝑎𝑖𝑗𝑙 (𝑡) 𝑥𝑗 (𝑡 − 𝜏𝑖𝑗𝑙 (𝑡)))
𝑗=1
𝑙=1
= (∫
𝑢𝑖 (𝑡) = ∫
𝑡
𝑡+𝜔
0
− 𝑑𝑖 (𝑡) 𝑢𝑖 (𝑡) − 𝑒𝑖 (𝑡) 𝑢𝑖 (𝑡 − 𝜀𝑖 (𝑡)) ,
𝑡+𝜔
𝑠
𝑡
0
0
× exp {∫ 𝑏𝑖 (𝜃) 𝑑𝜃} 𝑑𝑠 + 𝜓𝑖 (0)) exp {− ∫ 𝑏𝑖 (𝜃) 𝑑𝜃}
(𝛽𝑖 (𝑠) 𝑥𝑖∗ (𝑠) + 𝛾𝑖 (𝑠) 𝑥𝑖∗ (𝑠 − 𝜎𝑖 (𝑠)))
𝑠
× exp {∫ 𝑏𝑖 (𝜃) 𝑑𝜃} 𝑑𝑠 + 𝜓𝑖 (0))
[𝛽𝑖 (𝑠) 𝑥𝑖 (𝑠) + 𝛾𝑖 (𝑠) 𝑥𝑖 (𝑠 − 𝜎𝑖 (𝑠))] 𝐺𝑖 (𝑡, 𝑠) 𝑑𝑠,
0
𝑡+𝜔
× exp {− ∫
𝑖 = 1, 2, . . . , 𝑛,
(7)
where
0
𝑏𝑖 (𝜃) 𝑑𝜃} ,
𝑖 = 1, 2, . . . , 𝑛.
(11)
Then
𝑠
𝐺𝑖 (𝑡, 𝑠) =
exp {∫𝑡 𝑏𝑖 (𝜃) 𝑑𝜃}
𝜔
exp {∫0 𝑏𝑖 (𝜃) 𝑑𝜃} − 1
,
𝑖 = 1, 2, . . . , 𝑛.
(8)
𝑡
(∫ (𝛽𝑖 (𝑠) 𝑥𝑖∗ (𝑠) + 𝛾𝑖 (𝑠) 𝑥𝑖∗ (𝑠 − 𝜎𝑖 (𝑠)))
0
The converse is also true.
𝑠
𝜔
0
0
× exp {∫ 𝑏𝑖 (𝜃) 𝑑𝜃} 𝑑𝑠 + 𝜓𝑖 (0)) exp {∫ 𝑏𝑖 (𝜃) 𝑑𝜃}
Proof. By (4), (5), and the variation constants formula in
ordinary differential equations, we have
𝑡+𝜔
= (∫
0
𝑡
𝑠
𝑢𝑖 (𝑡) = (∫ (𝛽𝑖 (𝑠) 𝑥𝑖 (𝑠) + 𝛾𝑖 (𝑠) 𝑥𝑖 (𝑠 − 𝜎𝑖 (𝑠)))
× exp {∫ 𝑏𝑖 (𝜃) 𝑑𝜃} 𝑑𝑠 + 𝜓𝑖 (0))
0
0
𝑠
× exp {∫ 𝑏𝑖 (𝜃) 𝑑𝜃} 𝑑𝑠 + 𝜓𝑖 (0))
0
𝑡
× exp {− ∫ 𝑏𝑖 (𝜃) 𝑑𝜃} ,
0
(𝛽𝑖 (𝑠) 𝑥𝑖∗ (𝑠) + 𝛾𝑖 (𝑠) 𝑥𝑖∗ (𝑠 − 𝜎𝑖 (𝑠)))
(9)
𝑡
= ∫ (𝛽𝑖 (𝑠) 𝑥𝑖∗ (𝑠) + 𝛾𝑖 (𝑠) 𝑥𝑖∗ (𝑠 − 𝜎𝑖 (𝑠)))
0
𝑖 = 1, 2, . . . , 𝑛.
𝑠
× exp {∫ 𝑏𝑖 (𝜃) 𝑑𝜃} 𝑑𝑠 + 𝜓𝑖 (0)
0
4
International Journal of Differential Equations
+∫
𝑡+𝜔
𝑡
By a direct calculation, we have
(𝛽𝑖 (𝑠) 𝑥𝑖∗ (𝑠) + 𝛾𝑖 (𝑠) 𝑥𝑖∗ (𝑠 − 𝜎𝑖 (𝑠)))
𝑠
× exp {∫ 𝑏𝑖 (𝜃) 𝑑𝜃} 𝑑𝑠,
𝑑𝑢∗ (𝑡)
𝑑𝑡
𝑖 = 1, 2, . . . , 𝑛,
0
(12)
= −𝑏𝑖 (𝑡) ∫
𝑡+𝜔
𝑡
(𝛽𝑖 (𝑠) 𝑥𝑖∗ (𝑠) + 𝛾𝑖 (𝑠) 𝑥𝑖∗ (𝑠 − 𝜎𝑖 (𝑠)))
𝑠
which implies
×
𝑡
(∫ (𝛽𝑖 (𝑠) 𝑥𝑖∗ (𝑠) + 𝛾𝑖 (𝑠) 𝑥𝑖∗ (𝑠 − 𝜎𝑖 (𝑠)))
𝑠
0
𝑡
𝑡+𝜔
(13)
(𝛽𝑖 (𝑠) 𝑥𝑖∗ (𝑠) + 𝛾𝑖 (𝑠) 𝑥𝑖∗ (𝑠 − 𝜎𝑖 (𝑠)))
𝑠
× exp {∫ 𝑏𝑖 (𝜃) 𝑑𝜃} 𝑑𝑠,
0
×
exp {∫𝑡
That is,
𝑏𝑖 (𝜃) 𝑑𝜃}
𝜔
exp {∫0 𝑏𝑖 (𝜃) 𝑑𝜃} − 1
1
𝜔
exp {∫0 𝑏𝑖 (𝜃) 𝑑𝜃} − 1
= −𝑏𝑖 (𝑡) 𝑢𝑖 (𝑡) + 𝛽𝑖 (𝑡) 𝑥𝑖∗ (𝑡) + 𝛾𝑖 (𝑡) 𝑥𝑖∗ (𝑡 − 𝜎𝑖 (𝑡)) ,
𝑡
∫ (𝛽𝑖 (𝑠) 𝑥𝑖∗ (𝑠) + 𝛾𝑖 (𝑖) 𝑥𝑖∗ (𝑠 − 𝜎𝑖 (𝑠)))
𝑖 = 1, 2, . . . , 𝑛.
0
𝑠
× exp {∫ 𝑏𝑖 (𝜃) 𝑑𝜃} 𝑑𝑠 + 𝜓𝑖 (0)
This completes the proof.
0
It is easy to see that system (7) is equivalent to the following system:
𝑡
= 𝑢𝑖∗ (𝑡) exp {∫ 𝑏𝑖 (𝜃) 𝑑𝜃}
0
=
𝑡+𝜔
𝑒𝑏𝜔 − 1
∫
𝑡
(𝛽𝑖 (𝑠) 𝑥𝑖∗
(17)
− (𝛽𝑖 (𝑡) 𝑥𝑖∗ (𝑡) + 𝛾𝑖 (𝑡) 𝑥𝑖∗ (𝑡 − 𝜎𝑖 (𝑡)))
𝑖 = 1, 2, . . . , 𝑛.
×
1
𝑑𝑠
+𝛾𝑖 (𝑡 + 𝜔) 𝑥𝑖∗ (𝑡 + 𝜔 − 𝜎𝑖 (𝑡 + 𝜔)))
× exp {∫ 𝑏𝑖 (𝜃) 𝑑𝜃} 𝑑𝑠 + 𝜓𝑖 (0)) (𝑒𝑏𝜔 − 1)
𝑡+𝜔
𝜔
exp {∫0 𝑏𝑖 (𝜃) 𝑑𝜃} − 1
+ (𝛽𝑖 (𝑡 + 𝜔) 𝑥𝑖∗ (𝑡 + 𝜔)
0
=∫
exp {∫𝑡 𝑏𝑖 (𝜃) 𝑑𝜃}
(𝑠) +
𝛾𝑖 (𝑠) 𝑥𝑖∗
𝑠
× exp {∫ 𝑏𝑖 (𝜃) 𝑑𝜃} 𝑑𝑠,
0
2
𝑚
𝑗=1
𝑙=1
𝑥𝑖̇ (𝑡) = 𝑥𝑖 (𝑡) (𝑟𝑖 (𝑡) − ∑(−1)𝑖+𝑗 ∑𝑎𝑖𝑗𝑙 (𝑡) 𝑥𝑗 (𝑡 − 𝜏𝑖𝑗𝑙 (𝑡)))
(𝑠 − 𝜎𝑖 (𝑠)))
− 𝑑𝑖 (𝑡) 𝑢𝑖 (𝑡) − 𝑒𝑖 (𝑡) 𝑢𝑖 (𝑡 − 𝜀𝑖 (𝑡)) ,
𝑖 = 1, 2, . . . , 𝑛.
𝑢𝑖 (𝑡) = ∫
(14)
𝑡+𝜔
𝑡
𝐾 (𝑥𝑖 ) 𝐺𝑖 (𝑡, 𝑠) 𝑑𝑠,
𝑖 = 1, 2, . . . , 𝑛,
(18)
Hence
𝑢𝑖∗
(𝑡) = ∫
𝑡+𝜔
𝑡
where
(𝛽𝑖 (𝑠) 𝑥𝑖∗
(𝑠) +
𝛾𝑖 (𝑠) 𝑥𝑖∗
𝑠
(𝑠 − 𝜎𝑖 (𝑠)))
𝐺𝑖 (𝑡, 𝑠) =
𝑠
×
exp {∫𝑡 𝑏𝑖 (𝜃) 𝑑𝜃}
𝜔
exp {∫0 𝑏𝑖 (𝜃) 𝑑𝜃} − 1
𝑑𝑠,
𝑖 = 1, 2, . . . , 𝑛.
𝑡+𝜔
𝑡
(𝛽𝑖 (𝑠) 𝑥𝑖∗ (𝑠) + 𝛾𝑖 (𝑠) 𝑥𝑖∗ (𝑠 − 𝜎𝑖 (𝑠)))
×
exp {∫𝑡 𝑏𝑖 (𝜃) 𝑑𝜃}
𝜔
exp {∫0 𝑏𝑖 (𝜃) 𝑑𝜃} − 1
𝑑𝑠,
𝑖 = 1, 2, . . . , 𝑛.
(16)
,
𝐾 (𝑥𝑖 ) = 𝛽𝑖 (𝑠) 𝑥𝑖 (𝑠) + 𝛾𝑖 (𝑠) 𝑥𝑖 (𝑠 − 𝜎𝑖 (𝑠)) ,
(19)
𝑖 = 1, 2, . . . , 𝑛.
It is clear that in order to prove that systems (4) and (5) have
at least one 𝜔-periodic solution, we only need to prove that
system (18) has at least one 𝜔-periodic solution.
Now, for convenience of statements we denote the functions
𝑚
𝑠
𝜔
exp {∫0 𝑏𝑖 (𝜃) 𝑑𝜃} − 1
(15)
On the other hand, assume that (𝑥1∗ (𝑡), 𝑥2∗ (𝑡), . . . , 𝑥𝑛∗ (𝑡), 𝑢1∗ (𝑡),
𝑢2∗ (𝑡),. . . , 𝑢𝑛∗ (𝑡)) is an 𝜔-periodic solution of system (7), then
𝑢𝑖∗ (𝑡) = ∫
exp {∫𝑡 𝑏𝑖 (𝜃) 𝑑𝜃}
𝑎𝑖𝑗 (𝑡) = ∑𝑎𝑖𝑗𝑙 (𝑡) ,
𝑖, 𝑗 = 1, 2, . . . , 𝑛.
(20)
𝑙=1
The following theorem is about the existence of positive
periodic solutions of system (4).
International Journal of Differential Equations
5
Theorem 4. Suppose that assumption (H1) holds and there
exists a constant 𝜃𝑖 > 0, 𝜁𝑖 > 0, 𝑖 = 1, 2, . . . , 𝑛, such that
𝐶(𝑅, 𝑅𝑛 ) denote the space of all continuous functions 𝑦(𝑡) =
(𝑦1 (𝑡), 𝑦2 (𝑡), . . . , 𝑦𝑛 (𝑡)) : 𝑅 → 𝑅𝑛 . We take
𝑋=𝑍
𝑛
}
{𝑚
min {∑ [𝛿𝑖𝑖𝑙 (𝑡) 𝜃𝑖 − ∑ 𝛿𝑗𝑖𝑙 (𝑡) 𝜃𝑗 ]} =: 𝜁𝑖 > 0,
𝑡∈[0,𝜔]
𝑗 ≠ 𝑖
]}
{𝑙=1 [
(21)
𝑖 = 1, 2, . . . , 𝑛,
= {𝑦 (𝑡) ∈ 𝐶 (𝑅, 𝑅𝑛 ) : 𝑦 (𝑡) is an 𝜔-periodic function} ,
(28)
with norm
𝑛
󵄩󵄩 󵄩󵄩
󵄨
󵄨
󵄩󵄩𝑦󵄩󵄩 = ∑ max 󵄨󵄨󵄨𝑦𝑖 (𝑡)󵄨󵄨󵄨 .
𝑡∈[0,𝜔]
where
𝛿𝑖𝑗𝑙 (𝑡) =
𝑎𝑖𝑗𝑙 (𝜑𝑖𝑗𝑙 (𝑡))
󸀠 (𝜑
1 − 𝜏𝑖𝑗𝑙
𝑖𝑗𝑙 (𝑡))
,
(22)
𝑖, 𝑗 = 1, 2, . . . , 𝑛, 𝑙 = 1, 2, . . . , 𝑚,
It is obvious that 𝑋 and 𝑍 are the Banach spaces. We define
a linear operator 𝐿 : Dom 𝐿 ⊂ 𝑋 → 𝑍 and a continuous
operator 𝑁 : 𝑋 → 𝑍 as follows:
𝐿𝑦 (𝑡) = 𝑦̇ (𝑡) ,
and the algebraic equation
𝑁𝑦 (𝑡) = (𝑁𝑦1 (𝑡) , 𝑁𝑦2 (𝑡) , . . . , 𝑁𝑦𝑛 (𝑡)) ,
𝑛
𝑟𝑖 − 𝐻𝑖 − 𝑎𝑖𝑖 V𝑖 + ∑ 𝑎𝑖𝑗 V𝑗 = 0,
(29)
𝑖=1
𝑖 = 1, 2, . . . , 𝑛,
𝑗 ≠ 𝑖
(30)
where
(23)
𝑚
𝑁𝑦𝑖 (𝑡) = 𝑟𝑖 (𝑡) − ∑𝑎𝑖𝑖𝑙 (𝑡) exp {𝑦𝑖 (𝑡 − 𝜏𝑖𝑖𝑙 (𝑡))}
𝑙=1
where
𝑛
+ ∑ ∑𝑎𝑖𝑗𝑙 (𝑡) exp {𝑦𝑗 (𝑡 − 𝜏𝑖𝑗𝑙 (𝑡))}
𝜔
𝐻𝑖 = ∫ (𝛽𝑖 (𝑠) + 𝛾𝑖 (𝑠))
𝑗 ≠ 𝑖 𝑙=1
0
× 𝑒𝑖 (𝑡) 𝑢𝑖 (𝑡 − 𝜀𝑖 (𝑡)) .
𝑖 = 1, 2, . . . , 𝑛,
(24)
has a unique positive solution. Then system (4) has at least one
positive 𝜔-periodic solution.
Proof. For system (18) we introduce new variables 𝑦𝑖 (𝑡) (𝑖 =
1, 2, . . . , 𝑛) such that
𝑖 = 1, 2, . . . , 𝑛.
(25)
𝑚
𝑗=1
𝑙=1
𝑦𝑖̇ (𝑡) = 𝑟𝑖 (𝑡) − ∑ (−1)𝑖+𝑗 ∑𝑎𝑖𝑗𝑙 (𝑡) exp {𝑦𝑗 (𝑡 − 𝜏𝑖𝑗𝑙 (𝑡))}
− exp {−𝑦𝑖 (𝑡)} 𝑑𝑖 (𝑡) 𝑈𝑖 (𝑡) ,
(26)
− exp {−𝑦𝑖 (𝑡)} 𝑒𝑖 (𝑡) 𝑈𝑖 (𝑡 − 𝜀𝑖 (𝑡))
𝑈𝑖 (𝑡) = ∫
𝑡+𝜔
𝑡
𝐾 (𝑒𝑦𝑖 ) 𝐺𝑖 (𝑡, 𝑠) 𝑑𝑠,
Further, we define continuous projectors 𝑃 : 𝑋 → 𝑋 and
𝑄 : 𝑍 → 𝑍 as follows:
𝑃𝑦 (𝑡) =
1 𝜔
∫ 𝑦 (𝑡) 𝑑𝑡,
𝜔 0
𝑄V (𝑡) =
1 𝜔
∫ V (𝑡) 𝑑𝑡.
𝜔 0
(32)
𝜔
We easily see Im 𝐿 = {V ∈ 𝑍 : ∫0 V(𝑡)𝑑𝑡 = 0} and Ker 𝐿 = 𝑅𝑛 .
It is obvious that Im 𝐿 is closed in 𝑍 and dimKer 𝐿 = 𝑛. Since
for any V ∈ 𝑍 there are unique V1 ∈ 𝑅𝑛 and V2 ∈ Im 𝐿 with
1 𝜔
(33)
V2 (𝑡) = V (𝑡) − V1 ,
∫ V (𝑡) 𝑑𝑡,
𝜔 0
such that V(𝑡) = V1 + V2 (𝑡), we have codimIm 𝐿 = 𝑛. Therefore,
𝐿 is a Fredholm mapping of index zero. Furthermore, the
generalized inverse (to 𝐿) 𝐾𝑝 : Im 𝐿 → Ker 𝑃 ∩ Dom 𝐿 is
given in the following form:
V1 =
Then system (18) is rewritten in the following form:
2
(31)
− exp {−𝑦𝑖 (𝑡)} 𝑑𝑖 (𝑡) 𝑢𝑖 (𝑡) − exp {−𝑦𝑖 (𝑡)}
× (𝑑𝑖 (𝑡) 𝐺𝑖 (𝑡, 𝑠) + 𝑒𝑖 (𝑡) 𝐺𝑖 (𝑡 − 𝜀𝑖 (𝑡) , 𝑠)) 𝑑𝑠,
𝑥𝑖 (𝑡) = exp {𝑦𝑖 (𝑡)} ,
𝑚
𝑖 = 1, 2,
𝑡
1 𝜔 𝑡
(34)
∫ ∫ V (𝑠) 𝑑𝑠 𝑑𝑡.
𝜔 0 0
0
For convenience, we denote 𝐹(𝑡) = (𝐹1 (𝑡), 𝐹2 (𝑡), . . . , 𝐹𝑛 (𝑡)) as
follows:
𝐾𝑝 V (𝑡) = ∫ V (𝑠) 𝑑𝑠 −
𝑚
𝐹𝑖 (𝑡) = 𝑟𝑖 (𝑡) − ∑𝑎𝑖𝑖𝑙 (𝑡) exp {𝑦𝑖 (𝑡 − 𝜏𝑖𝑖𝑙 (𝑡))}
where
𝑙=1
𝐾 (𝑒𝑦𝑖 ) = 𝛽𝑖 (𝑠) exp {𝑦𝑖 (𝑠)} + 𝛾𝑖 (𝑠) exp {𝑦𝑖 (𝑠 − 𝜎𝑖 (𝑠))} ,
𝑖 = 1, 2.
(27)
In order to apply Lemma 2 to system (26), we introduce the normed vector spaces 𝑋 and 𝑍 as follows. Let
𝑛
𝑚
+ ∑ ∑𝑎𝑖𝑗𝑙 (𝑡) exp {𝑦𝑗 (𝑡 − 𝜏𝑖𝑗𝑙 (𝑡))}
𝑗 ≠ 𝑖 𝑙=1
− exp {−𝑦𝑖 (𝑡)} 𝑑𝑖 (𝑡) 𝑢𝑖 (𝑡) − exp {−𝑦𝑖 (𝑡)}
× 𝑒𝑖 (𝑡) 𝑢𝑖 (𝑡 − 𝜀𝑖 (𝑡)) .
(35)
6
International Journal of Differential Equations
Thus, we have
×∫
𝑡−𝜀𝑖 (𝑡)
1 𝜔
∫ 𝐹 (𝑡) 𝑑𝑡,
𝜔 0
𝑄𝑁𝑦 (𝑡) =
𝑡−𝜀𝑖 (𝑡)+𝜔
𝐾 (𝑒𝑦𝑖 ) 𝐺𝑖 (𝑡 − 𝜀𝑖 (𝑡) , 𝑠) 𝑑𝑠 𝑑𝑡
𝜔
𝑡+𝜔
0
𝑡
+ ∫ 𝑑𝑖 (𝑡) exp {−𝑦𝑖 (𝑡)} ∫
𝐾𝑝 (𝐼 − 𝑄) 𝑁𝑢 (𝑡) = 𝐾𝑝 𝐼𝑁𝑢 (𝑡) − 𝐾𝑝 𝑄𝑁𝑢 (𝑡)
(36)
1 𝜔 𝑡
= ∫ 𝐹 (𝑠) 𝑑𝑠 − ∫ ∫ 𝐹 (𝑠) 𝑑𝑠 𝑑𝑡
𝜔 0 0
0
𝐾 (𝑒𝑦𝑖 ) 𝐺𝑖 (𝑡, 𝑠) 𝑑𝑠 𝑑𝑡,
𝑖 = 1, 2, . . . , 𝑛.
(39)
𝑡
𝜔
1 𝑡
+ ( − ) ∫ 𝐹 (𝑠) 𝑑𝑠.
2 𝜔 0
From formulas (36), we easily see that 𝑄𝑁 and 𝐾𝑝 (𝐼 − 𝑄)𝑁
are continuous operators. Furthermore, it can be verified that
𝐾𝑝 (𝐼 − 𝑄)𝑁(Ω) is compact for any open bounded set Ω ⊂ 𝑋
by using the Arzela-Ascoli theorem and 𝑄𝑁(Ω) is bounded.
Therefore, 𝑁 is 𝐿-compact on Ω for any open bounded subset
Ω ⊂ 𝑋.
Now, we reach the position to search for an appropriate
open bounded subset Ω for the application of the continuation theorem (Lemma 2) to system (26).
Corresponding to the operator equation 𝐿𝑦(𝑡) = 𝜆𝑁𝑦(𝑡)
with parameter 𝜆 ∈ (0, 1), we have
𝑦𝑖̇ (𝑡) = 𝜆𝐹𝑖 (𝑡) ,
𝑖 = 1, 2, . . . , 𝑛,
From the continuity of 𝑦(𝑡) = (𝑦1 (𝑡), 𝑦2 (𝑡), . . . , 𝑦𝑛 (𝑡)), there
exist constants 𝜉𝑖 , 𝜂𝑖 ∈ [0, 𝜔] (𝑖 = 1, 2, . . . , 𝑛) such that
𝑦𝑖 (𝜉𝑖 ) = max 𝑦𝑖 (𝑡) ,
𝑚
0
𝑙=1
𝜔
𝑟𝑖 𝜔 ≤ ∫ 𝑎𝑖𝑖 (𝑡) exp {𝑦𝑖 (𝜉𝑖 )} 𝑑𝑡
0
+ ∫ 𝐴 𝑖 (𝑡) exp {𝑦𝑖 (𝜉𝑖 )} 𝑑𝑡,
where
𝐴 𝑖 (𝑡) = 𝑒𝑖 (𝑡) ∫
− 𝑒𝑖 (𝑡) exp {−𝑦𝑖 (𝑡)} ∫
𝑡−𝜀𝑖 (𝑡)+𝜔
𝑡−𝜀𝑖 (𝑡)
−𝑑𝑖 (𝑡) exp {−𝑦𝑖 (𝑡)} ∫
𝑡+𝜔
𝑡
𝐾 (𝑒𝑦𝑖 ) 𝐺𝑖 (𝑡 − 𝜀𝑖 (𝑡) , 𝑠) 𝑑𝑠
𝑡−𝜀𝑖 (𝑡)
+ 𝑑𝑖 (𝑡) ∫
𝑡+𝜔
𝑡
𝐺𝑖 (𝑡 − 𝜀𝑖 (𝑡) , 𝑠) (𝛽𝑖 (𝑠) + 𝛾𝑖 (𝑠)) 𝑑𝑠
𝐺𝑖 (𝑡, 𝑠) (𝛽𝑖 (𝑠) + 𝛾𝑖 (𝑠)) 𝑑𝑠,
𝑖 = 1, 2, . . . , 𝑛.
(42)
𝑟𝑖
𝑎𝑖𝑖 + 𝐴𝑖
),
𝜔
0
𝜔−𝜏𝑖𝑗𝑙 (𝜔)
=∫
−𝜏𝑖𝑗𝑙 (0)
󸀠 (𝜑
1 − 𝜏𝑖𝑗𝑙
𝑖𝑗𝑙 (𝑡))
0
𝑙=1
+ ∫ 𝑒𝑖 (𝑡) exp {−𝑦𝑖 (𝑡)}
exp {𝑦𝑖 (𝑡)} 𝑑𝑡.
One can see that
󸀠 (𝜑
1 − 𝜏𝑖𝑗𝑙
𝑖𝑗𝑙 (𝑡))
𝑚
= ∫ [∑𝑎𝑖𝑖𝑙 (𝑡) exp {𝑦𝑖 (𝑡 − 𝜏𝑖𝑖𝑙 (𝑡))}] 𝑑𝑡
(44)
𝑎𝑖𝑗𝑙 (𝜑𝑖𝑗𝑙 (𝑡))
𝑎𝑖𝑗𝑙 (𝜑𝑖𝑗𝑙 (𝑡))
𝑚
(43)
∫ 𝑎𝑖𝑗𝑙 (𝑡) exp {𝑦𝑖 (𝑡 − 𝜏𝑖𝑗𝑙 (𝑡))} 𝑑𝑡
Consequently,
𝑟𝑖 𝜔 + ∫ [ ∑ ∑𝑎𝑖𝑗𝑙 (𝑡) exp {𝑦𝑗 (𝑡 − 𝜏𝑖𝑗𝑙 (𝑡))}] 𝑑𝑡
0
]
[𝑗 ≠ 𝑖 𝑙=1
𝑖 = 1, 2, . . . , 𝑛.
Let 𝑠𝑖𝑗𝑙 (𝑡) = 𝑡 − 𝜏𝑖𝑗𝑙 (𝑡) (𝑖, 𝑗 = 1, 2, . . . , 𝑛, 𝑙 = 1, 2, . . . , 𝑚);
then from Lemma 1 and (H1) we get that function 𝑠𝑖𝑗𝑙 (𝑡) has
a unique 𝜔 periodic inverse function 𝜑𝑖𝑗𝑙 (𝑡); then, for every
𝑖, 𝑗 = 1, 2, . . . , 𝑛, 𝑙 = 1, 2, . . . , 𝑚, we have
𝐾 (𝑒𝑦𝑖 ) 𝐺𝑖 (𝑡, 𝑠) 𝑑𝑠] 𝑑𝑡 = 0,
𝑖 = 1, 2, . . . , 𝑛.
(38)
0
𝑡−𝜀𝑖 (𝑡)+𝜔
𝑦𝑖 (𝜉𝑖 ) ≥ ln (
𝑗 ≠ 𝑖 𝑙=1
𝜔
𝑖 = 1, 2, . . . , 𝑛,
0
Therefore, we further have
+ ∑ ∑𝑎𝑖𝑗𝑙 (𝑡) exp {𝑦𝑗 (𝑡 − 𝜏𝑖𝑗𝑙 (𝑡))}
𝜔
(41)
𝜔
(37)
𝑛 𝑚
𝑛
(40)
By (39) and (40) we obtain
∫ [𝑟𝑖 (𝑡) − ∑𝑎𝑖𝑖𝑙 (𝑡) exp {𝑦𝑖 (𝑡 − 𝜏𝑖𝑖𝑙 (𝑡))}
𝜔
𝑡∈[0,𝜔]
𝑖 = 1, 2, . . . , 𝑛.
where 𝐹𝑖 (𝑡) (𝑖 = 1, 2, . . . , 𝑛) are given in (35).
Assume that 𝑦(𝑡) = (𝑦1 (𝑡), 𝑦2 (𝑡), . . . , 𝑦𝑛 (𝑡)) ∈ 𝑋 is a
solution of system (37) for some parameter 𝜆 ∈ (0, 1). By
integrating system (37) over the interval [0, 𝜔], we obtain
𝜔
𝑦𝑖 (𝜂𝑖 ) = min 𝑦𝑖 (𝑡)
𝑡∈[0,𝜔]
=: 𝛿𝑖𝑗𝑙 (𝑡) ,
(45)
𝑖, 𝑗 = 1, 2, . . . , 𝑛, 𝑙 = 1, 2, . . . , 𝑚,
are 𝜔 periodic functions. Then, for every 𝑖, 𝑗 = 1, 2, . . . , 𝑛, 𝑙 =
1, 2, . . . , 𝑚, we have
𝜔
𝜔
0
0
∫ 𝑎𝑖𝑗𝑙 (𝑡) exp {𝑦𝑖 (𝑡 − 𝜏𝑖𝑗𝑙 (𝑡))} 𝑑𝑡 = ∫ 𝛿𝑖𝑗𝑙 (𝑡) exp {𝑦𝑖 (𝑡)} 𝑑𝑡.
(46)
International Journal of Differential Equations
7
From (39) and (46) we further obtain
On the other hand, directly from system (26) we have
𝜔
𝑛
𝜔
𝑚
0
𝑙=1
󵄨
󵄨
∫ 󵄨󵄨󵄨𝑦𝑖̇ (𝑡)󵄨󵄨󵄨 𝑑𝑡
0
∑ (∫ (∑𝛿𝑖𝑖𝑙 (𝑡) exp {𝑦𝑖 (𝑡)}
𝑖=1
𝜔
+ 𝑑𝑖 (𝑡) exp {−𝑦𝑖 (𝑡)} ∫
𝑡+𝜔
𝑡
󵄨
󵄨
≤ ∫ 󵄨󵄨󵄨𝑟𝑖 (𝑡)󵄨󵄨󵄨 𝑑𝑡
0
𝐾 (𝑒𝑦𝑖 ) 𝐺𝑖 (𝑡, 𝑠) 𝑑𝑠
+ 𝑒𝑖 (𝑡) exp {−𝑦𝑖 (𝑡)}
×∫
𝑡−𝜀𝑖 (𝑡)+𝜔
𝑡−𝜀𝑖 (𝑡)
𝜔
𝑚
0
𝑙=1
+ ∫ (∑𝑎𝑖𝑖𝑙 (𝑡) exp {𝑦𝑖 (𝑡 − 𝜏𝑖𝑖𝑙 (𝑡))}
𝑛
𝐾 (𝑒𝑦𝑖 ) 𝐺𝑖 (𝑡 − 𝜀𝑖 (𝑡) , 𝑠) 𝑑𝑠) 𝑑𝑡)
𝑚
+ ∑ ∑𝑎𝑖𝑗𝑙 (𝑡) exp {𝑦𝑗 (𝑡 − 𝜏𝑖𝑗𝑙 (𝑡))}) 𝑑𝑡
𝑗 ≠ 𝑖 𝑙=1
𝑛
𝑛
𝜔
𝑚
𝜔
󵄨
󵄨
≤ ∫ 󵄨󵄨󵄨𝑟𝑖 (𝑡)󵄨󵄨󵄨 𝑑𝑡
0
= ∑ (𝑟𝑖 𝜔 + ∫ [ ∑ ∑𝛿𝑖𝑗𝑙 (𝑡) exp {𝑦𝑗 (𝑡)}] 𝑑𝑡) .
0
𝑖=1
]
[𝑗 ≠ 𝑖 𝑙=1
(47)
𝜔 𝑚
𝑛
0 𝑙=1
𝑗 ≠ 𝑖
+ ∫ ∑ (𝛿𝑖𝑖𝑙 (𝑡) + ∑ 𝛿𝑗𝑖𝑙 (𝑡)) exp {𝑦𝑖 (𝑡)} 𝑑𝑡
From the above equality we have
𝑛
𝑚
𝜔
𝑛
𝜔
𝜔
󵄨
󵄨
≤ ∫ 󵄨󵄨󵄨𝑟𝑖 (𝑡)󵄨󵄨󵄨 𝑑𝑡 + 𝐷𝑖𝑀 ∫ exp {𝑦𝑖 (𝑡)} 𝑑𝑡
0
0
𝐷𝑀 ∑ 𝑛 𝑟 𝜔
󵄨 󵄨
≤ 󵄨󵄨󵄨𝑟𝑖 󵄨󵄨󵄨 𝜔 + 𝑖 𝑖=1 𝑖 ,
𝜁𝑖
𝑚
∑ (∫ [∑𝛿𝑖𝑖𝑙 (𝑡) − ∑ ∑𝛿𝑗𝑖𝑙 (𝑡)] exp {𝑦𝑖 (𝑡)} 𝑑𝑡)
0
𝑖=1
𝑗 ≠ 𝑖 𝑙=1
[𝑙=1
]
𝑖 = 1, 2, . . . , 𝑛,
(52)
𝑛
≤ ∑𝑟𝑖 𝜔,
where
𝑖=1
𝑚
𝜔
(48)
𝑛
𝑚
∫ [∑𝛿𝑖𝑖𝑙 (𝑡) − ∑ ∑𝛿𝑗𝑖𝑙 (𝑡)] exp {𝑦𝑖 (𝑡)} 𝑑𝑡
0
𝑗 ≠ 𝑖 𝑙=1
[𝑙=1
]
𝑛
≤ ∑𝑟𝑖 𝜔,
𝑚
𝑛
𝑙=1
𝑗 ≠ 𝑖
𝐷𝑖 (𝑡) = ∑ (𝛿𝑖𝑖𝑙 (𝑡) + ∑ 𝛿𝑗𝑖𝑙 (𝑡)) ,
𝑖 = 1, 2, . . . , 𝑛.
(53)
From (43), (51), and (52), we have for any 𝑡 ∈ [0, 𝜔]
𝑖 = 1, 2, . . . , 𝑛.
𝑖=1
𝜔
From the assumptions of Theorem 4 and (48) we can obtain
𝜔
𝑛
𝜁𝑖 ∫ exp {𝑦𝑖 (𝑡)} 𝑑𝑡 ≤ ∑𝑟𝑖 𝜔,
0
󵄨
󵄨
𝑦𝑖 (𝑡) ≤ 𝑦𝑖 (𝜂𝑖 ) + ∫ 󵄨󵄨󵄨𝑦𝑖̇ (𝑡)󵄨󵄨󵄨 𝑑𝑡
0
≤ ln (
𝑖 = 1, 2, . . . , 𝑛.
∑𝑛𝑖=1 𝑟𝑖
𝐷𝑀 ∑𝑛 𝑟 𝜔
(54)
󵄨 󵄨
) + 󵄨󵄨󵄨𝑟𝑖 󵄨󵄨󵄨 𝜔 + 𝑖 𝑖=1 𝑖 =: 𝑀𝑖 ,
𝜁𝑖
𝜁𝑖
(49)
𝑖 = 1, 2, . . . , 𝑛,
𝑖=1
𝜔
󵄨
󵄨
𝑦𝑖 (𝑡) ≥ 𝑦𝑖 (𝜉𝑖 ) − ∫ 󵄨󵄨󵄨𝑦𝑖̇ (𝑡)󵄨󵄨󵄨 𝑑𝑡
0
Consequently,
𝜔
∫ exp {𝑦𝑖 (𝑡)} 𝑑𝑡 ≤
0
∑𝑛𝑖=1 𝑟𝑖 𝜔
,
𝜁𝑖
≥ ln (
𝑖 = 1, 2, . . . , 𝑛.
(50)
𝑖 = 1, 2, . . . , 𝑛.
(55)
From (40) and (50), we further obtain
∑𝑛 𝑟
𝑦𝑖 (𝜂𝑖 ) ≤ ln ( 𝑖=1 𝑖 ) ,
𝜁𝑖
𝐷𝑀 ∑𝑛 𝑟 𝜔
󵄨 󵄨
) − 󵄨󵄨󵄨𝑟𝑖 󵄨󵄨󵄨 𝜔 − 𝑖 𝑖=1 𝑖 =: 𝑁𝑖 ,
𝜁𝑖
𝑎𝑖𝑖 + 𝐴𝑖
𝑟𝑖
𝑖 = 1, 2, . . . , 𝑛.
Therefore, from (54) and (55), we have
(51)
󵄨 󵄨 󵄨 󵄨
󵄨
󵄨
max 󵄨󵄨󵄨𝑦𝑖 (𝑡)󵄨󵄨󵄨 ≤ max {󵄨󵄨󵄨𝑀𝑖 󵄨󵄨󵄨 , 󵄨󵄨󵄨𝑁𝑖 󵄨󵄨󵄨} =: 𝐵𝑖 ,
𝑡∈[0,𝜔]
𝑖 = 1, 2, . . . , 𝑛.
(56)
8
International Journal of Differential Equations
It can be seen that the constants 𝐵𝑖 (𝑖 = 1, 2, . . . , 𝑛) are
independent of parameter 𝜆 ∈ (0, 1). For any 𝑦 =
(𝑦1 , 𝑦2 , . . . , 𝑦𝑛 ) ∈ 𝑅𝑛 , from (31) we can obtain
𝑄𝑁𝑦 = (𝑄𝑁𝑦1 , 𝑄𝑁𝑦2 , . . . , 𝑄𝑁𝑦𝑛 )
𝑛
𝑄𝑁𝑦𝑖 = 𝑟𝑖 − 𝐻𝑖 − 𝑎𝑖𝑖 exp {𝑦𝑖 } + ∑ 𝑎𝑖𝑗 exp {𝑦𝑗 } ,
𝑗 ≠ 𝑖
𝑖 = 1, 2, . . . , 𝑛,
𝐻𝑖 (𝑡) = ∫ (𝛽𝑖 (𝑠) + 𝛾𝑖 (𝑠)) (𝑑𝑖 (𝑡) 𝐺𝑖 (𝑡, 𝑠))
0
𝑖 = 1, 2, . . . , 𝑛.
We consider the following algebraic equation:
𝑛
𝑟𝑖 − 𝐻𝑖 − 𝑎𝑖𝑖 V𝑖 + ∑ 𝑎𝑖𝑗 V𝑗 = 0,
𝑖 = 1, 2, . . . , 𝑛.
𝑗 ≠ 𝑖
(58)
From the assumption of Theorem 4, the equation has
a unique positive solution V∗ = (V1∗ , V2∗ , . . . , V𝑛∗ ). Hence,
the equation 𝑄𝑁𝑦 = 0 has a unique solution 𝑦∗ =
(𝑦1∗ , 𝑦2∗ , . . . , 𝑦𝑛∗ ) = (ln V1∗ , ln V2∗ , . . . , ln V𝑛∗ ) ∈ 𝑅𝑛 .
Choosing constant 𝐵 > 0 large enough such that |𝑦1∗ | +
|𝑦2∗ | + ⋅ ⋅ ⋅ + |𝑦𝑛∗ | < 𝐵 and 𝐵 > 𝐵1 + 𝐵2 + ⋅ ⋅ ⋅ + 𝐵𝑛 , we define a
bounded open set Ω ⊂ 𝑋 as follows:
󵄩 󵄩
Ω = {𝑦 ∈ 𝑋 : 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 < 𝐵} .
(59)
It is clear that Ω satisfies conditions (𝑎) and (𝑏) of Lemma 2.
On the other hand, by directly calculating we can obtain
deg {𝐽𝑄𝑁, Ω ∩ Ker 𝐿, (0, 0, . . . , 0)}
󵄨󵄨𝑓1
󵄨󵄨 𝑦1
󵄨󵄨
󵄨󵄨 2
󵄨
= sgn 󵄨󵄨󵄨𝑓𝑦1
󵄨󵄨
󵄨󵄨 ⋅ ⋅ ⋅
󵄨󵄨 𝑛
󵄨󵄨𝑓𝑦1
𝑓𝑦12 ⋅ ⋅ ⋅ 𝑓𝑦1𝑛 󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨
2
2 󵄨󵄨󵄨
𝑓𝑦2 ⋅ ⋅ ⋅ 𝑓𝑦𝑛 󵄨󵄨 ,
󵄨
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 󵄨󵄨󵄨
𝑛
𝑛 󵄨󵄨
𝑓𝑦2 ⋅ ⋅ ⋅ 𝑓𝑦𝑛 󵄨󵄨
(60)
where
𝑓𝑦𝑖 𝑗 = −𝑎𝑖𝑗 exp {𝑦𝑗∗ } ,
𝑓𝑦𝑖 𝑗 = 𝑎𝑖𝑗 exp {𝑦𝑗∗ } ,
𝑖=𝑗
𝑖 ≠ 𝑗 𝑖, 𝑗 = 1, 2, . . . , 𝑛.
From the assumption of Theorem 4, we have
󵄨󵄨𝑓1 𝑓1 ⋅ ⋅ ⋅ 𝑓1 󵄨󵄨
󵄨󵄨 𝑦1 𝑦2
𝑦𝑛 󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨 2
󵄨󵄨𝑓 𝑓2 ⋅ ⋅ ⋅ 𝑓2 󵄨󵄨󵄨
𝑦𝑛 󵄨󵄨 ≠ 0.
󵄨󵄨󵄨 𝑦1 𝑦2
󵄨󵄨 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 󵄨󵄨󵄨
󵄨
󵄨󵄨 𝑛 𝑛
󵄨󵄨𝑓 𝑓 ⋅ ⋅ ⋅ 𝑓𝑛 󵄨󵄨󵄨
󵄨 𝑦1 𝑦2
𝑦𝑛 󵄨
(61)
(62)
From this, we finally have
deg {𝐽𝑄𝑁, Ω ∩ Ker 𝐿, (0, 0, . . . , 0)} ≠ 0.
This work was supported by the Natural Science Foundation
of Xinjiang (Grants no. 2012223177, 200721104) and the
National Natural Science Foundation of China (Grants no.
10661010, 11261061).
References
(57)
𝜔
+ 𝑒𝑖 (𝑡) 𝐺𝑖 (𝑡 − 𝜀𝑖 (𝑡) , 𝑠) 𝑑𝑠,
Acknowledgments
(63)
This shows that Ω satisfies condition (𝑐) of Lemma 2.
Therefore, system (26) has an 𝜔-periodic solution 𝑦∗ (𝑡) =
(𝑦1∗ (𝑡), 𝑦2∗ (𝑡), . . . , 𝑦𝑛∗ (𝑡)) ∈ Ω. Finally, we have system that
(4) has a positive 𝜔-periodic solution. This completes the
proof.
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