Research Article | Open Access
Existence and Exponential Stability of Periodic Solution of High-Order Hopfield Neural Network with Delays on Time Scales
On time scales, by using the continuation theorem of coincidence degree theory and constructing some suitable Lyapunov functions, the periodicity and the exponential stability are investigated for a class of delayed high-order Hopfield neural networks (HHNNs), which are new and complement of previously known results. Finally, an example is given to show the effectiveness of the proposed method and results.
Consider the following HHNNs with time-varying delays: where corresponds to the number of units in a neural network, corresponds to the state vector of the th unit 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, , and correspond to the transmission delays, denote the external inputs at time , and and are the activation functions of signal transmission.
Due to the fact that high-order neural networks have stronger approximation property, faster convergence rate, greater storage capacity, and higher fault tolerance than lower-order neural networks, high-order neural networks have been the object of intensive analysis by numerous authors in the recent years. In particular, there have been extensive results on the problem of the existence and stability of equilibrium points and periodic solutions of HHNNs (1.1) in the literature. We refer the reader to [1–9] and the references cited therein. In fact, both continuous and discrete systems are very important in implementing and applications. The theory of calculus on time scales (see [10, 11] and references cited therein) was initiated by Stefan Hilger in his Ph.D. thesis in 1988  in order to unify continuous and discrete analysis, and it has a tremendous potential for application and has recently received much attention since his foundational work. Therefore, it is meaningful to study that on time scales which can unify the continuous and discrete situations.
Our purpose of this paper is to consider the model where is an -periodic time scale which has the subspace topology inherited from the standard topology on is an input periodic vector function with period ; that is, there exists such that for all , and , and is the activation function of the neurons.
System (1.2) is supplemented with initial values given by where denotes continuous -periodic function defined on . To the best of our knowledge, this is the first paper to study the stability and existence of periodic solutions of (1.2).
Throughout this paper, we assume the following.()For are positive continuous periodic functions with period , and is regressive.There exist positive constants such that for .Functions satisfy the Lipschitz condition; that is, there exist constants such that .
In this section, we will first recall some basic definitions and lemmas which are used in what follows.
Let be a nonempty closed subset (time scale) of . The forward and backward jump operators and the graininess are defined, respectively, by
A point is called left-dense if and , left-scattered if , right-dense if and , and right-scattered if . If has a left-scattered maximum , then ; otherwise . If has a right-scattered minimum , then ; otherwise .
A function is right-dense continuous provided that it is continuous at right-dense point in and its left-side limits exist at left-dense points in . If is continuous at each right-dense point and each left-dense point, then is said to be continuous function on .
For and , we define the delta derivative of , to be the number (if it exists) with the property that for a given , there exists a neighborhood of such that for all .
If is continuous, then is right-dense continuous, and if is delta differentiable at , then is continuous at .
A function is called regressive if for all .
If is regressive function, then the generalized exponential function is defined by with the cylinder transformation Let be two regressive functions, we define Then the generalized exponential function has the following properties.
Lemma 2.1 (see ). Assume that are two regressive functions, then (i) and ;(ii);(iii);(iv);(v);(vi);(vii);(viii).
Lemma 2.2 (see ). Assume that are delta differentiable at , then
Let be right-dense continuous. If , then one defines the delta integral by
Lemma 2.3. If and , then (1);(2)If for all , then ;(3)If on , then .
Lemma 2.4. If , and on , then .
Lemma 2.5 (see ). Let , and . If is -periodic, then , and .
Lemma 2.6 (, Cauchy-Schwarz inequality on time scales). Let . For -continuous functions , one has .
Lemma 2.7. Let be right-dense continuous and regressive. , and . The unique solution of the initial value problem is given by
For the sake of convenience, one introduces the following notations:
Obliviously, for system (1.2), finding the periodic solutions is equivalent to finding those of the following boundary-value problem:
Now, one states Mawhin's continuous theorem .
Theorem 2.8. Let and be two Banach spaces and let be a Fredholm mapping of index zero. let be an open bounded set and let be a continuous operator which is -compact on . Assume that (1)for each (2)for each ;(3), where .Then has at least one solution in .
for any (or). Then and are Banach spaces with the norm . Let Then it follows that
is closed in , dim . It is not difficult to show that and are continuous and satisfy . It is easy to see that is closed in , which leads to the following lemma.
Proof. Through an easy computation, we find that the inverse of has the form Thus, the expression of is and then Thus, and are continuous. Since is a Banach space, it is not difficult to show that is compact. Moreover, is bounded. Thus, is -compact on for any open bounded set . The proof of Lemma 2.10 is completed.
3. Existence of Periodic Solution
In this section, we study the existence of periodic solution of (1.2) based on Mawhin's continuation theorem.
Theorem 3.1. Assume that hold, then system (1.2) has at least one -periodic solution.
Proof. Based on the Lemma 2.9 and Lemma 2.10, now, what we need to do is just to search for an appropriate open, bounded subset for the application of the continuation theorem. Corresponding to the operator equation , we have
For the sake of convenience, defined by
for . Suppose that is a solution of system (3.1) for a certain . Integrating (3.1) over , we obtain
Let such that Then by (3.4) andLemma 2.4, we have
Hence By (3.4) and Lemma 2.4, we can also have
Hence From (3.1), (3.4), and Lemma 2.6, we have
From Lemma 2.7 and (3.1), for , we can obtain
Substituting (3.10) into (3.7), we have
From Lemma 2.5, we have
From (3.5), (3.6) and (3.11), there exist positive constants such that for . Clearly, is independent of . Denote , where is taken sufficiently large so that
Now we take . Thus (1) of Theorem 2.8 is satisfied. When is a constant vector in with , then
Therefore Consequently, for This satisfies condition (2) of Theorem 2.8. Define by When is a constant vector in with , we easily have . Therefore Condition (3) of Theorem 2.8 is also satisfied. Thus, by Theorem 2.8 we can obtain that has at least one solution in . That is, system (1.2) has at least one -periodic solution. The proof is complete.
4. Global Exponential Stability of Periodic Solution
In this section, we will construct suitable Lyapunov functions to study the global exponential stability of the periodic solution of (1.2) on time scales. So first we will introduce some definitions.
Definition 4.1 (see ). A function from to is positively regressive if for every .
Denote is the set of positively regressive functions from to , and denote .
Definition 4.2. The periodic solution of system (1.2) is said to be exponentially stable if there exists a positive constant with such that for every , there exists such that the solution of (1.2) through satisfies where .
Theorem 4.3. Assume that hold. Suppose further that there exists positive constants such that then the -periodic solution of system (1.2) is globally exponentially stable.
Proof. According to Theorem 3.1, we know that (1.2) has an -periodic solution . Suppose that is an arbitrary solution of (1.2). Then it follows from system (1.2) that with initial values given by where is defined as in(1.3). If holds, it can always find a smallenough constant satisfying , namely, such that We define a Lyapunov function by . In view of (4.2), weobtain Defining the curve and the set , . It is obvious that if , then . We will prove that the zero solution of (4.2) is exponential stable, namely, there exists a constant