By the continuation theorem of coincidence degree and -matrix theory, we obtain some sufficient conditions for the existence and exponential stability of periodic solutions for a class of generalized neural networks with arbitrary delays, which are milder and less restrictive than those of previous known criteria. Moreover our results generalize and improve many existing ones.
1. Introduction
Consider the following generalized neural networks with arbitrary delays:
where ,, , , , is defined by = , , and is a subset of .
System (1.1) contains many neural networks, for examples, the higher-order Cohen-Grossberg type neural networks with delays (see [1])
the Cohen-Grossberg neural networks with bounded and unbounded delays (see [2])
the Cohen-Grossberg neural networks with time-varying delays (see [3])
the celluar neural networks (see [4, Page 193]):
and so on.
Since the model of Cohen-Grossberg neural networks was first introduced by Cohen and Grossberg in [5], the dynamical characteristics (including stable, unstable, and periodic oscillatory) of Cohen-Grossberg neural networks have been widely investigated for the sake of theoretical interest as well as application considerations. Many good results have already been obtained by some authors in [6–15] and the references cited therein. Moreover, the existing results are based on the assumption that demanding either the activation functions, the behaved functions, or delays is bounded in the above-mentioned literature. However, to the best of our knowledge, few authors have discussed the existence and exponential stability of periodic solutions of (1.1). In this paper, by using the continuation theorem of coincidence degree and -matrix theory, we study model (1.1), and get some sufficient conditions for the existence and exponential stability of the periodic solution of system (1.1); our results generalize and improve many existing ones.
Let be two matrices, be two vectors. For convenience, we introduce the following notations.
(i) means that each element is nonnegative (positive) respectively,(ii) means ,(iii) means each element ,(iv) means ,(v).For continuous -periodic function , we denote is the family of continuous functions from to . Clearly, it is a Banach space with the norm , where . The initial conditions of system (1.1) are of the form
where . For , let
Throughout this paper, we assume the following:
() For and are -periodic for their first arguments, respectively, that is, so , for all .() There exists a positive diagonal matrix such that , for all .() There is a positive diagonal matrix such that , and or for all () There exist a nonnegative matrix and a nonnegative vector such that , for all , where .2. Preliminaries
In this section, we first introduce some definitions and lemmas which play an important role in the proof of our main results in this paper.
Definition 2.1. Let be an -periodic solution of system (1.1) with initial value , if there exist two constants and such that for every solution of system (1.1) with initial value (1.6),
Then is said to be globally exponential stable.
Definition 2.2. A real matrix is said to be an -matrix if , and .
Lemma 2.3 (see [15, 16]). Assume that is an -matrix and , then .
Lemma 2.4 (see [15, 16]). Let with , then the following statements are equivalent. (i) is an -matrix.(ii)There exists a positive vector such that .(iii)There exists a positive vector such that .
Lemma 2.5 (see [15, 16]). Let be an matrix and , then , where denotes the identity matrix of size , so is an -matrix.
Now we introduce Mawhin's continuation theorem which will be fundamental in this paper.
Lemma 2.6 (see [17]). Let and be two Banach spaces and a Fredholm mapping of index zero. Assume that is an open bounded set and is a continuous operator which is -compact on . Then has at least one solution in , if the following conditions are satisfied: (1), for all ,(2), for all ,(3).
Let
with the norm defined by , where . Clearly, and are two Banach spaces. Let , we define the linear operator as
and the operators as
It is not difficult to show that and are continuous projectors and the following conditions are satisfied:
Thus, the mapping is a Fredholm mapping of index zero and the isomorphism is the identity operator; the generalized inverse (of exists, which has the form
Therefore
3. Existence of Periodic Solutions
In this section, we shall use Lemma 2.6 to study the existence of at least one periodic solution of system (1.1).
Theorem 3.1. Let ()–() hold. Moveover, suppose that () is a -matrix, where the matrix .then (i)system (1.1) has at least one -periodic solution;(ii)there exists a nonnegative constant such that for all -periodic solution of system (1.1),
Proof. Clearly, and are continuous functions and for every bounded subset , and are bounded. By using the Arzela-Ascoli theorem, and are compact, therefore is -compact on . Consider the following operator equation:
That is,
or
Assume that is a solution of (3.3) for some . Then, for any are all continuous -periodic functions, and there exist , such that
from (), we have
It follows from () that
Thus
we denote the vector , where . It follows from (3.7) that
Since , and application of Lemma 2.3 yields
where satisfies the equation , that is, .
Take
It is easy to see that satisfies condition () in Lemma 2.6.
For all , is a constant vector in and there exists some such that , we claim that
We firstly claim that (1)if , then ,(2)if , then .
We only prove (1), since the proof of (2) is similar. If , we have
Therefore
Thus (3.11) is valid.
Next, we define continuous functions , by
respectively. If , from (i) we have
If , from () we can get
Using the homotopy invariance theorem, we obtain if ,
or if ,
To summarize, satisfies all the conditions of Lemma 2.6. This completes the proof of (i).
For all -periodic solution of system (1.1), from (3.3)–(3.7) we have
where , Notes , thus , for all . This completes the proof of (ii).
From the proof of Theorem 3.1, we can easily obtain the following corollary.
Corollary 3.2. Suppose that ()–() hold, and in , then system (1.1) has only one -periodic solution .
Some special cases of Theorem 3.1 are in what follows.
Corollary 3.3. Equation (1.3) has at least one -periodic solution, if the following conditions are satisfied. () For are continuous -periodic () functions.() For , are positive, and there exist such that .() For there exist such that
() For there exist such that
().
Proof. It is clear that
Thus
where . Therefore, by using Lemma 2.5 and Theorem 3.1, we know that (1.3) has an -periodic solution. The proof is complete.
Remark 3.4. For [1, Equation ()], are continuous differentiable -periodic solutions and , this implies that are constant functions, thus . It is not difficult to verify that all of conditions of Corollary 3.3 are satisfied under the conditions of [1, Theorem 1] moreover the other requirements of [1, Theorem 1] are more restrictive than ours. Therefore, Corollary 3.3 improves the corresponding result obtained in [1].
Corollary 3.5. If the following conditions are satisfied: () for are continuous -periodic () functions, are continuous functions on , and are -periodic for their first arguments, respectively,() for there exist positive constants such that , for all ,() for there exist positive constants such that , for all ,() there exist nonnegative constants such that
() the delay kernels satisfy
(), where then (1.3) has at least one -periodic solution.
Remark 3.6. In [2, Theorem 3.1], the activation functions are required to be Lipschitzian, which implies that condition in Corollary 3.5 holds. Therefore, Corollary 3.5 improves Theorem 3. In 2.
Corollary 3.7. Assume that the following conditions are satisfied: () are continuous -periodic () functions, are continuous functions on , and are periodic in the first variable,() there exist positive constants such that
() there exist positive constants such that
() There exist nonnegative constants such that
(), where and .Then (1.4) has at least one -periodic solution.
Remark 3.8. In [3, Theorem 3.1], the activation functions are Lipschitzian (which also implies that condition in Corollary 3.7 holds) and the behaved functions are required to satisfy that there exist positive constants such that for all , which are more restrictive than that of Corollary 3.7.
Corollary 3.9. Assume that the following conditions are satisfied () For are continuous -periodic solution functions.() For are continuous functions and there exist nonnegative constants such that
() and ,then (1.5) has at least one -periodic solution.
The proofs of Corollaries 3.5–3.9 are the same as that of Corollary 3.3.
4. Uniqueness and Exponential Stability of Periodic Solution
In this section, we establish some results for the uniqueness and exponential stability of the -periodic solution of (1.1).
Theorem 4.1. Assume that is a bounded subset of , and ()–() and hold. Suppose also the following conditions are satisfied. There exists a nonnegative matrix such that
where , .(), are Lipschitzian with Lipschitz constants , and there exist such that
() For all , there exist positive constants such that
() For , set , and assume that is an -matrix, where , and
Proof. Obviously, implies , since ()–() hold, it follows from Theorem 3.1 that system (1.1) has at least one -periodic solution
with the initial value . Let
be an arbitrary solution of system (1.1) with the initial value (1.6), set . Then for
Thus, for ,
for and Lemma 2.4, there exist a positive constant and a positive constant vector such that . Hence
where . Moreover for all ,
Since, is a bounded subset of , we can choose a positive constant , such that
and also can choose a positive constant such that
Set, for all , for all ,
It follows from (4.11) and (4.13) that
Thus
where , from (4.12) and (4.13), we can get
We claim that
Suppose that it is not true, then there exits some and such that
Thus
It follows from (4.8), (4.15), and (4.18) that
which contradicts to (4.19), thus (4.17) holds. Set and , from (4.17), we have
where . This completes the proof of Theorem 4.1.
5. Conclusion
In this paper, a class of generalized neural networks with arbitrary delays have been studied. Some sufficient conditions for the existence and exponential stability of the periodic solutions have been established. These obtained results are new and they improve and complement previously known results.
Acknowledgment
This work is supported by the National Natural Sciences Foundation of China under Grant 10971183.