Abstract
This paper is concerned with antiperiodic solutions for impulsive high-order Hopfield neural networks with leakage delays and continuously distributed delays. By employing a novel proof, some sufficient criteria are established to ensure the existence and global exponential stability of the antiperiodic solution, which are new and complement of previously known results. Moreover, an example and numerical simulations are given to support the theoretical result.
1. Introduction
To describe mathematically a real evolution process with a short-term perturbation, the impulsive differential equations were proposed in many fields such as control theory, physics, chemistry, population dynamics, biotechnologies, industrial robotics, and economics [1–3]. In particular, high-order neural networks with impulses have been studied extensively, and there has been a great deal of the literatures focusing on the existence and stability of equilibrium points, periodic solutions, almost periodic solutions, and antiperiodic solutions [4–13]. Reference [14] has introduced and studied the existence and exponential stability of antiperiodic solutions for the following Hopfield neural networks with time-varying and distributed delays:
Recently, great attention has been paid to neural networks with time delay in the leakage (or forgetting) term (see [15–19]). Specifically, Wang [20] considered the antiperiodic solution of the following impulsive high-order Hopfield neural networks with leakage delays:
Under some reasonable conditions on coefficients of (2) and the following additional conditions: Wang [20] deduced the criteria on the existence and exponential stability of the antiperiodic solution for (2).
However, to the best of our knowledge, few authors have investigated the existence and exponential stability of the antiperiodic solution for impulsive high-order Hopfield neural networks with leakage delays and continuously distributed delays. Motivated by the above arguments, we consider the antiperiodic solution for the impulsive high-order Hopfield neural networks (IHHNNs) with leakage delays and continuously distributed delays as follows: where and is the number of units in a neural network, corresponds to the th unit of the state vector at the time , represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, and are the first- and second-order connection weights of the neural network, respectively, , , and correspond to the leakage delay, the transmission delays, and the transmission delay kernels, respectively, and are the activation functions of signal transmission, and denotes the external input at time . For , we always assume that , , , and , are bounded continuous functions, and is bounded above and below by positive constants. Consider the following: , , , , . are impulsive moments satisfying and . is the initial condition and denotes a real-valued continuous function defined on , . It is easy to see that the system (1) is a special case of system (4) with .
The purpose of this paper is to discuss the existence and exponential stability of antiperiodic solutions for system (4) without the additional condition (3) since it is unduly restrictive and unreasonable. The outline of the paper is as follows. In Section 2, we establish some preliminaries and basic results, which are useful to derive sufficient conditions on the existence and exponential stability of antiperiodic solutions for system (4) in Section 3. In Section 4, we give an example with numerical simulations to illustrate our results.
2. Preliminaries and Basic Results
Throughout this paper, we assume that the following conditions hold.For and , where denotes the set of all positive integers, there exists constant such that where and and are real-valued bounded continuous functions defined on .For , there exist constants , , , , , and such that Consider for and .There exists a such that
For each , the activation functions are continuous and there exist nonnegative constants , , and such that, for all , For all , the delay kernels are continuous, and are integrable on for a certain positive constant .For all and , there exist positive constants and such that andFor ease of notations, let be the set of all real vectors and denote as a column vector, in which the symbol represents the transpose of a vector. As a general rule of the theory of impulsive differential equations, we assume that at the points of discontinuity of the solution . From system (4), it is easy to see that the derivative does not exist in general. On the other hand, according to system (4), there exists the limit . On account of the above convention, we suppose that .
Definition 1. A solution of (4) is said to be -antiperiodic, if
where the smallest positive number is called the antiperiod of function .
In the sequel, we prove some lemmas which will be used to prove our main results in Section 3.
Lemma 2. Suppose that (H2)–(H7) hold. If is a solution of system (4) with initial conditions then, for in the interval of existence and , where
Proof. For in the interval of existence and , let
On the one hand, suppose that (12) holds. Then, for a given in the interval of existence and , we acquire
which combined with implies that (13) holds.
On the other hand, in view of , we have
So, if , then .
Thus, considering the above two cases, it is sufficient to prove (12). We proceed this by contradiction. Suppose that (12) does not hold; then there exist and such that
It follows that (13) holds for all and . By virtue of (4), we obtain
This, together with (H2), (H5)–(H7), (18), and the fact that (13) holds for all and , we obtain
It contradicts . Hence, (12) holds. The proof is now completed.
Remark 3. Under conditions (H2)–(H7), the solution of system (4) always exists (see [1, 2]). On account of the boundedness of this solution, it follows from the theory of impulsive differential equations in [1] that the solution of system (4) can be defined on .
Lemma 4. Suppose that (H1)–(H7) are true. Let be the solution of system (4) with initial value , and let be the solution of system (4) with initial value . Then, there exists a positive constant such that
Proof. With the help of and a similar discussion as that in the proof of in [20], we can select and such that , and
Let and , . Then,
which leads to
We define a positive constant as follows:
There is a positive number such that
We assert that
Clearly, (28) holds for . We first prove that (28) is true for . Otherwise, there exist and such that
It follows that, for and ,
and thus
Calculating the upper left derivative of , together with (22), (24), (29), (31), (H2), and (H5)–(H7), we get
which is a contradiction. Therefore, (28) holds for . From (25), (28), (31), and (H3), we know that
Thus, using the same argument as the above procedure, we can obtain
Further, we have
That is,
Remark 5. If is an -antiperiodic solution of system (4), it follows from Lemma 4 that is globally exponentially stable.
3. Main Results
In this section, we will study the existence and global exponential stability of the antiperiodic solution for system (4).
Theorem 6. Suppose that all conditions in Lemma 4 are satisfied. Then, system (4) has exactly one -antiperiodic solution . Moreover, is globally exponentially stable.
Proof. Let be a solution of system (4). By Remark 3, the solution can be defined for all . By hypotheses , we have, for any natural number and ,
Further, by hypothesis of , we obtain
Hence, for any natural number , we obtain that is a solution of system (4) for all . Hence, is also a solution of (4) with initial values:
Then, by the proof of Lemma 4, for , there exists a constant such that, for any natural number ,
Furthermore, for any natural number and , we can obtain
Due to (40)−(41), we know that converges uniformly to a piecewise continuous function on any compact set of .
Next, we show that is an -antiperiodic solution of system (4). It is easy to see that is -antiperiodic, since
where . Observing that the right side of (4) is piecewise continuous, together with (37) and (38), we find that converges uniformly to a piecewise continuous function on any compact set of . Therefore, letting on both sides of (37) and (38), we get
Thus, is an -antiperiodic solution of system (4).
Finally, adopting the same procedure as in the proof of Lemma 4, we can show that is globally exponentially stable. This completes the proof.
4. An Example
In this section, we provide an example to demonstrate the results obtained in the previous sections.
Example 1. Consider the following IHHNNs consisting of two neurons with leakage delays and continuously distributed delays:
Here, it is assumed that the activation functions
Note that
Then, we obtain
It follows that system (44) satisfies all the conditions in Theorem 6. Therefore, system (44) has exactly one 1-antiperiodic solution. Moreover, the 1-antiperiodic solution is globally exponentially stable. The fact is verified by the numerical simulations in Figure 1.
(a)
(b)
Remark 7. Since [4–13] mainly obtained the stability results on the impulsive Hopfield neural networks without leakage delays and [15–19] dealt with nonimpulsive neural networks, it can be observed that all the results in the literature and the references therein cannot be applicable to prove the existence and exponential stability of 1-antiperiodic solution for IHHNNs (44). Furthermore, is possible for some , ; one can find that the results in [20] are invalid. This implies that the results of this paper are essentially new.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of P.R. China (Grants nos. 11301341, 11201184), the Natural Scientific Research Fund of Zhejiang Provincial of P.R. China (Grant no. LY12A01018), Innovation Program of Shanghai Municipal Education Commission (Grant no. 13YZ127), and Shanghai Fiscal Special Fund (Grant no. A4-4902-13-12).