-Almost Periodic Solutions of BAM Neural Networks with Time-Varying Delays on Time Scales
On a new type of almost periodic time scales, a class of BAM neural networks is considered. By employing a fixed point theorem and differential inequality techniques, some sufficient conditions ensuring the existence and global exponential stability of -almost periodic solutions for this class of networks with time-varying delays are established. Two examples are given to show the effectiveness of the proposed method and results.
It is well known that bidirectional associative memory (BAM) neural networks have been extensively applied within various engineering and scientific fields such as pattern recognition, signal and image processing, artificial intelligence, and combinatorial optimization [1–3]. Since all these applications closely relate to the dynamics, the dynamical behaviors of BAM neural networks have been widely investigated. There have been extensive results on the problem of the existence and stability of equilibrium points, periodic solutions, and antiperiodic solutions of BAM neural networks in the literature. We refer the reader to [4–16] and the references cited therein. Moreover, it is known that the existence and stability of almost periodic solutions play a key role in characterizing the behavior of dynamical systems (see [17–26]) and the -almost periodic function is an important subclass of almost periodic functions. However, to the best of our knowledge, few authors have studied problems of -almost periodic solutions of BAM neural networks.
On the other hand, the theory of calculus on time scales (see [27, 28] and references cited therein) was initiated by Hilger in his Ph.D. thesis in 1988 in order to unify continuous and discrete analyses, and it helps avoid proving twice results, once for differential equations and once for difference equations. Therefore, it is significant to study neural networks on time scales (see [5, 29, 30]). In fact, both continuous-time and discrete-time BAM-type neural networks have equal importance in various applications. But it is troublesome to study the existence and stability of almost periodic and -almost periodic solutions for continuous and discrete systems, respectively. Motivated by the above, our purpose of this paper is to study the existence and stability of -almost periodic solutions for the following BAM neural networks on time scales: where is an almost periodic time scale which will be defined in the next section; and correspond to the activation of the th neurons and the th neurons at the time , respectively; are positive functions and they denote the rates with which the cells and reset their potential to the resting state when isolated from the other cells and inputs at time ; and are the connection weights at time ; , are nonnegative, which correspond to the finite speed of the axonal signal transmission at time ; , denote the external inputs at time ; and and are the activation functions of signal transmission. For each interval of , we denote .
Throughout this paper, we assume the following: and there exist positive constants such that where , , ;, , , , , , , are bounded almost periodic functions on , , .
System (1) is supplemented with the initial values given by where denotes a real-valued bounded rd-continuous function defined on , and
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 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 .
Let be right-dense continuous. If , then we define the delta integral by
A function is called regressive if for all . The set of all regressive and rd-continuous functions will be denoted by . We define the set .
If is a 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 1 (see ). Assume that are two regressive functions; then,(i) and ;(ii);(iii);(iv);(v).
In this section, denotes or , denotes an open set in or , and denotes an arbitrary compact subset of .
Definition 2. A time scale is called an almost periodic time scale if satisfies that, for any , one has , where .
Definition 3. Let be an almost periodic time scale. For any , we define where .
Obviously, if is an almost periodic time scale, then and . If there exists a such that , then Definition 2 is equivalent to Definition 3.7 in ; otherwise, Definition 2 is more general than Definition 3.7 in .
Definition 4. Let be an almost periodic time scale. A function is called an almost periodic function in uniformly for if the -translation set of is a relatively dense set in for all and for each compact subset of ; that is, for any given and each compact subset of , there exists a constant such that each interval of length contains a such that is called the -translation number of and and is called the inclusion length of .
For convenience, we introduce some notations. Let and be two sequences. Then means that is a subsequence of . We introduce the translation operator , and means that . From Definitions 2 and 4, one can easily see that all the results obtained in  are still valid under the new concepts of almost periodic time scales and almost periodic functions on time scales. For example, similar to Theorems 3.13 and 3.14 in , we can obtain the following equivalent definition of uniformly almost periodic functions.
Definition 5. Let , and if for any given sequence and each compact subset of there exists a subsequence such that exists uniformly on , then is called an almost periodic function in uniformly for .
Definition 6. A function is said to be a -almost periodic function, if are two almost periodic functions on .
Definition 7 (see ). Let , and let be an rd-continuous matrix on ; the linear system is said to admit an exponential dichotomy on if there exist positive constants and , projection , and the fundamental solution matrix of (15), satisfying where is a matrix norm on .
Consider the following linear almost periodic system: where is an almost periodic matrix function and is an almost periodic vector function.
Lemma 9 (see ). Let be an almost periodic function on , where , , and then, the linear system admits an exponential dichotomy on .
Lemma 10 (see ). Every rd-continuous function has an antiderivative. In particular, if , then defined by is an antiderivative of .
Lemma 11 (see ). If and , then
Lemma 12. Suppose that is an -continuous function and is a positive -continuous function which satisfies that . Let where ; then,
Lemma 13 (see ). If is a real-valued almost periodic function on and is a Lipschitz function, then is an almost periodic function on .
Lemma 14 (see ). If are almost periodic functions, then the following hold:(1) is almost periodic function;(2) is almost periodic function.
3. Existence of -Almost Periodic Solutions
First, for convenience, we introduce some notations. We will use to denote a column vector, in which the symbol denotes the transpose of vector. We let denote the absolute-value vector given by and define .
Let be a bounded real-valued, almost periodic function on , , and For , if we define induced modulus where then is a Banach space.
Theorem 15. Assume that , , and the following hold:, , , , , , ;there exists a constant such that where then, system (1) has a unique -almost periodic solution in the region
Proof. For any given , we consider the following almost periodic differential equation:
Since , it follows from Lemma 10 that the linear system
admits an exponential dichotomy on . Thus, by Lemma 9, we know that system (32) has exactly one almost periodic solution:
By Lemmas 13 and 14, we have that
are almost periodic functions on ; that is, (34) is not only an almost periodic solution of system (32), but also a -almost periodic solution of system (32). First, we define a nonlinear operator on by
Next, we check that . For any given , it suffices to prove that . By conditions , we have
then, it follows from (37) that
Taking and combining conditions and , we obtain that Similarly, from (39) it follows that
By (40), we obtain that is a contraction mapping from to . Since is a closed subset of , has a fixed point in , which means that (32) has a unique -almost periodic solution in . Then system (1) has a unique -almost periodic solution in the region This completes the proof.
4. Exponential Stability of the -Almost Periodic Solution
Definition 16. The -almost periodic solution of system (1) with initial value is said to be globally exponentially stable. There exist a positive constant with and such that every solution of system (1) with any initial value satisfies where
Theorem 17. Suppose that hold and ; then, system (1) has a unique -almost periodic solution which is globally exponentially stable.
Proof. According to Theorem 15, we know that (1) has a -almost periodic solution
with initial value . Suppose that
is an arbitrary solution of (1) with initial value
Then it follows from system (1) that
where , and , , and the initial conditions of (49) and (50) are
Let and be defined by By , we get Since are continuous on and , as , there exist such that , , and for and for . By choosing