Qualitative Analysis of Differential, Difference Equations, and Dynamic Equations on Time ScalesView this Special Issue
Antiperiodic Solutions to Impulsive Cohen-Grossberg Neural Networks with Delays on Time Scales
We use the method of coincidence degree and construct suitable Lyapunov functional to investigate the existence and global exponential stability of antiperiodic solutions of impulsive Cohen-Grossberg neural networks with delays on time scales. Our results are new even if the time scale or . An example is given to illustrate our feasible results.
It is well known that Cohen-Grossberg neural networks (CGNNs) include many models from different research fields, such as neurobiology, population biology, and evolutionary theory, as well as Hopfield neural networks and other recurrent neural network models. Over the past few years, a large number of scholars have extensively studied the dynamical behaviors, in particular, the existence and stability of the equilibrium point and periodic and almost-periodic solutions of Cohen-Grossberg neural networks. There have been considerable results on CGNNs (e.g., see [1–16]). In contrast, however, very few results are available on the existence and exponential stability of antiperiodic solutions for neural networks, while the existence of antiperiodic solutions plays a key role in characterizing the behavior of nonlinear differential equations (see [17–22]).
In , the authors studied the existence and exponential stability of antiperiodic solutions for the following Cohen-Grossberg neural networks with bounded and unbounded delays: where corresponds to the number of units in the neural networks, denotes the potential (or voltage) of cell at time , represents an amplification function, is an appropriately behaved function, , , and denote the strengths of connectivity between cells and at time , respectively. The activation functions , , and show how the th neuron reacts to the input, corresponds to the time delay required in processing and transmitting a signal from the th cell to the th cell at time , is the kernel, and denotes the th component of an external input source introduced from outside the network to cell at time .
In fact, both continuous and discrete systems are very important in implementation and application. Therefore, the study of dynamic equations on time scales has received much attention (see [18, 19, 23–28]) which displays a combination of characteristics of both continuous-time and discrete-time system. For example, in , the authors extended the almost-periodic theory on time scales with the delta derivative to that with the nabla derivative and then derived some sufficient conditions ensuring the existence, uniqueness, and exponential stability of almost-periodic solutions for a class of cellular neural networks with time-varying delays in leakage terms on time scales.
Also, differential equations with impulses provide an adequate mathematical model of many evolutionary processes that suddenly change their states at certain moments. For example,  applied the method of coincidence degree to investigate the existence of antiperiodic solutions to the following impulsive shunting inhibitory cellular neural networks on time scales:
Motivated by the abovementioned works, in this paper, we will apply the method of coincidence degree and construct suitable Lyapunov functional to investigate the existence and global exponential stability of antiperiodic solutions to the following impulsive CGNN model with delays on time scales: where is an -periodic time scale which has the subspace topology inherited from the standard topology on , , , represent right and left limit of in the sense of time scales, is a sequence of real numbers such that as , and there exists a positive integer p such that , , . Without loss of generality, we also assume that .
The initial conditions associated with system (3) are of the form where is a bounded continuous function on .
Throughout this paper, we make the following assumptions:(), , , , and , , , , , and , ;(), , and there exist positive constants , , such that and for all ;() is delta differential and , , , ;(there exists a positive constant such that for all ;(), , and , , , , , and there exist positive constants , , , , , and such that , , , , , and , for all ;(), ;() and there exist positive constants such that for all , , and .
For the sake of convenience, we introduce some notations: where .
The organization of the rest of this paper is as follows. In Section 2, we introduce some definitions and make some preparations for later sections. In Sections 3 and 4, we establish our main results for the existence and exponential stability of antiperiodic solutions of (3). Finally, we present an example to illustrate the feasibility and effectiveness of our results obtained in previous sections.
In this section, we recall some basic definitions and lemmas which are used in what follows.
Definition 1 (see [24, 26]). A time scale is an arbitrary nonempty closed subset of the real set with the topology and ordering inherited from . The forward and backward jump operators and the graininess are defined, respectively, by The point is called left-dense, left-scattered, right-dense, or right-scattered if , , , or , respectively. Points that are right-dense and left-dense at the same time are called dense. If has a left-scattered maximum , define ; otherwise, .
Definition 2 (see [24, 26]). A vector function is rd-continuous provided it is continuous at each right-dense point in and has a left-sided limit at each left-dense point in . The set of rd-continuous functions will be denoted by .
Definition 3 (see [24, 26]). For a function (the range of may be actually replaced by Banach space), the (delta) derivative is defined by if is continuous at and is right-scattered. If is not right-scattered, then the derivative is defined by provided that this limit exists.
Definition 5 (see [23, 24]). If , , and is rd-continuous on , then we define the improper integral by provided that this limit exists, and one says that the improper integral converges in this case. If this limit does not exist, then one says that the improper integral diverges.
Definition 6 (see [24, 26]). A function is called regressive if for all , where is the graininess function. The set of all regressive rd-continuous functions is denoted by , while the set is given by for all . Let . The exponential function is defined by with the cylinder transformation
Definition 7 (see [18, 24]). For each , let be a neighborhood of . Then one defined the generalized derivative (or Dini derivative) to mean that, given , there exists a right neighborhood of such that
for each , .
In case is right-scattered and is continuous at , this reduces to where the upper right Dini derivative is defined as
Definition 8 (see ). One says that a time scale is periodic if there exists such that if , then . For , the smallest positive is called the period of the time scale. Let be a periodic time scale with period . We say that the function is -antiperiodic if there exists a natural number such that , for all and is the smallest number such that . If , one says that is -antiperiodic if is the smallest positive number such that for all .
Lemma 10 (see ). If , then
Definition 13 (see ). The antiperiodic solution of system (3) with initial value is said to be globally exponentially stable if there exist positive constants and , for any solution of system (3) with the initial value , such that where The following fixed point theorem of coincidence degree is crucial in the arguments of our main results.
Lemma 14 (see [18, 29]). Let be two Banach spaces and let be open bounded and symmetric with . Suppose that is a linear Fredholm operator of index zero with and is L-compact. Further, one also assumes that(H) for all .
Then has at least one solution on .
3. Existence of Antiperiodic Solutions
In this section, by using fixed point theorem of coincidence degree, we will study the existence of at least one antiperiodic solution for system (3).
Theorem 16. Assume that (H1)–(H7) hold. Suppose further that , . Then system (3) has at least one -antiperiodic solution.
Proof. Let be a piecewise continuous map with first-class discontinuous points in and at each discontinuous point it is continuous on the left}, . Let
be two Banach spaces equipped with the norms
in which , and is any norm of . Set
for . It is easy to see that
Thus, , and is a linear Fredholm operator of index zero.
Define the continuous projector and the averaging projector by Hence, and . Denoting by the inverse of , we have in which for all .
Similar to , it is not difficult to show that , are relatively compact for any open bounded set . Therefore, is -compact on for any open bounded set .
In order to apply Lemma 14, we need to find an appropriate open bounded subset in . Corresponding to the operator equation , we have where for .
Set , . Then, by (31), (), (), (), (), and Lemma 15, we obtain that Integrating (31) from to , we have by (33) In view of (34), (), and Lemma 15, we get for . In addition, from Lemma 12, for any , we have where . Dividing by on the two sides of (36) and (37), respectively, we obtain that where .
Let such that , , by the arbitrariness of ; we get from (33)–(39) that Thus, we have from (40) that From (41), we have Then, by the assumption of Theorem 16 and (42), we have Let Clearly, is independent of . Then take It is clear that satisfies all the requirements in Lemma 14 and condition (H) is satisfied. In view of all the discussions above, we conclude from Lemma 14 that system (3) has at least one -antiperiodic solution. This completes the proof.
4. Global Exponential Stability of Antiperiodic Solutions
In this section, we will construct some suitable Lyapunov functions to study the global exponential stability of antiperiodic solutions of system (3).
Theorem 17. Assume that (H1)–(H7) hold. Suppose further the following.()The impulsive operators satisfy ()For , there exist constants and such that Then the -antiperiodic solution of system (3) is globally exponentially stable.
Proof. According to Theorem 16 and its proof, we know that system (3) has an -antiperiodic solution with the initial value and , and suppose that is an arbitrary solution of system (3) with the initial value . Set . Then it follows from system (3) and () that
In view of the above system and ()–(), for , , , , similar to , we have For any , we consider the following Lyapunov function: For , , , calculating the upper right derivative of by (48)–(50), we have