By using the coincidence degree theorem and differential inequality
techniques, sufficient conditions are obtained for the existence and global exponential
stability of periodic solutions for general neural networks with time-varying (including
bounded and unbounded) delays. Some known results are improved and some new
results are obtained. An example is employed to illustrate our feasible results.
1. Introduction
In recent
years, the delayed cellular neural networks (DCNNs) have been
extensively studied because of their immense potentials of
application perspective in different areas such as pattern
recognition, optimization, and signal and image processing [1–3]. Hence, they have been the object of
intensive analysis by numerous authors, and some interesting results
on the existence and stability of periodic and almost periodic
solutions have been obtained [4–12]. To our knowledge, few
authors have considered global stability of periodic solutions for
the neural networks with bounded and unbounded time-varying
delays. In theory and application, global stability of periodic
solutions of DCNNs is of great importance since the
global stability of equilibrium points can be considered as a
special case of periodic solution with zero period [8]. Hence, in this paper, we will study the
existence and global exponential stability of periodic solutions of
the following general neural networks with time-varying
delays:where is the state of
neuron, , and are connection
matrices, is the input
function, and is the
activation function of the neurons.
DCNNs in [4–12] and the references cited therein are
special cases of (1.1). In particular, when are constants,
the authors of [9] considered the
existence and global exponential stability for (1.1) with periodic
impulses. The methods used in [9] are
Mawhin's coincidence degree theorem [13] and Lyapunov functions. In [14], by using Mawhin's coincidence degree theorem [13], the authors investigated the global
existence of positive periodic solutions of mutualism systems with
bounded and unbounded time-varying delays, and some sufficient
conditions are obtained. In [12], the
authors considered (1.1) when are constants.
We assume what follows.
(H1) are continuous -periodic
functions, and are continuous -periodic
in the first variable. .(H2)
There exist
positive constants and such that , and , for all (H3). There are positive constants , , such that , for all (H4)
The delay
kernels are continuous,
integrable, and satisfy(H5)
There exists a
constant such
that
The organization of this paper is as follows. In
Section 2, we introduce some notations and definitions, and state
some preliminary results needed in later sections. We then study, in
Section 3, the existence of periodic solutions of system (1.1) by
using the continuation theorem of coincidence degree theorem
proposed by Gains and Mawhin [13]. In
Section 4, by constructing Lyapunov function we will derive
sufficient conditions for the global exponential stability of the
periodic solution of system (1.1). At last, an example is employed
to illustrate the feasible results of this paper.
2. Preliminaries
For convenience, we use to denote the
maximums of , respectively.
We also use symbols
to denote a
column vector, in which the symbol denotes the
transpose of a vector. denotes the
identity matrix of size . A matrix or vector means that all
entries of are greater
than or equal to zero (resp., ).
For matrices or vectors and , (resp., )
means that (resp., ).
The initial condition of (1.1) is of
the formwhere , are continuous -periodic
solutions.
Definition 2.1. Let be an -periodic
solution of (1.1) with initial value . If there exist constants and such that for
every solution of (1.1) with
initial value ,where , then is said to be
globally exponentially stable.
Definition 2.2. (See [15, 16]). A real matrix is said to be a
nonsingular -matrix
if , and , where denotes the
inverse of .
Lemma 2.3 (See [15, 16]). Let with . Then the following statements are equivalent:
(1) is a
nonsingular -matrix,(2)
there exists a
vector such that ,(3)
there exists a
vector such that .
Lemma 2.4 (See [16]). Let be an matrix and , then there exists a vector such that , where denotes the
spectral radius of .
To end this section, we introduce Mawhin's
continuation theorem [13, page 40] as follows.
Consider an abstract equation in a Banach space , where is a Fredholm
operator with index-zero, and is a parameter.
Let and denote two
projectors, and such that .
Lemma 2.5. Let be a Banach
space. Suppose that is a Fredholm
operator with index-zero, let be an open
bounded set, and let be a continuous
operator which is -compact
on . Moreover, assume that the following conditions are
satisfied:
(a)
for each ,(b)
for each ,(c).
Then, has at least
one solution in .
3. Existence of Periodic Solutions
Theorem 3.1. Let hold. Assume
that the following condition is satisfied:
there exists a
vector such that where
Then, (1.1) has at least one -periodic
solution.
Proof. For convenience, we introduce the
following notations: In order to use Lemma 2.5, we take ; then is a Banach
space with the norm Set
Obviously, and So, is closed in . It is easy to show that and are continuous
projectors satisfying Hence, is a Fredholm
mapping of index-zero. Furthermore, through an easy computation, we
find that the generalized inverse of has the
formThus,
Clearly, and are continuous.
Using the Arzela-Ascoli theorem, it is not difficult to show that
, are relatively
compact for any open bounded set . Therefore, is -compact
on for any open
bounded set .
Now, we reach the position to search for an
appropriate open bounded subset for the
application of Lemma 2.5. Corresponding to the operator equation
(2.3), we haveLet be a solution
of system (3.7) for some . Then, for any are all
continuously differentiable. Thus, there exist such that . Hence, From (3.7), we
haveIn view of , we have
Set . Clearly, (3.9) implies thatThus,together with , we haveTherefore,Again from , it follows from Lemma 2.3 that is a
nonsingular -matrix,
and there exists a vector such that , which implies that we can choose a constant such that and and We takewhich satisfies Condition (a) of Lemma 2.5.
If , then is a constant
vector in , and there exists some such that . We claim thatBy way of contradiction, suppose
that then there
exists some such
thatwhich implies
thathenceThis implies that , which contradicts (3.14). Therefore, (3.16) holds, and
hence, Condition of Lemma 2.5 is
satisfied.
Furthermore, we define a continuous function byfor all and . It follows that If , then is a constant
vector in , and there exists some such that . We claim thatBy way of contradiction, suppose
that then there
exists some such
thatthat is,Now, we will consider the
following two cases.
Case 1. If , from , we have