C 1-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 C 1-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.


Introduction
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][2][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][5][6][7][8][9][10][11][12][13][14][15][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][18][19][20][21][22][23][24][25][26]) and the 1almost periodic function is an important subclass of almost periodic functions. However, to the best of our knowledge, few authors have studied problems of 1 -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 1 -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 1 -almost periodic solutions for the following BAM neural networks on time scales: where T is an almost periodic time scale which will be defined in the next section; of R, we denote T = ∩ T. Throughout this paper, we assume the following: ( 1 ) , ∈ (R, R) and there exist positive constants , such that where | |, |V| ∈ R, = 1, 2, . . . , , = 1, 2, . . . , ; where (⋅) denotes a real-valued bounded rd-continuous function defined on [−V, 0] T , and

Preliminaries
In this section, we will first recall some basic definitions and lemmas which are used in what follows. Let T be a nonempty closed subset (time scale) of R. The forward and backward jump operators , : T → T and the graininess : T → R + are defined, respectively, by  If T has a right-scattered minimum , then T = T \ { }; otherwise, T = T.
A function : T → R is right-dense continuous provided it is continuous at right-dense point in T and its leftside limits exist at left-dense points in T. If is continuous at each right-dense point and each left-dense point, then is said to be continuous function on T.
For : T → R and ∈ T , we define the delta derivative of ( ), Δ ( ), to be the number (if it exists) with the property that for a given > 0 there exists a neighborhood of such that If is continuous, then is right-dense continuous, and if is delta differentiable at , then is continuous at .
Let be right-dense continuous. If Δ ( ) = ( ), then we define the delta integral by A function : T → R is called regressive if for all ∈ T . The set of all regressive and rd-continuous functions : T → R will be denoted by R = R(T) = R(T, R). We define the set R + = R + (T, R) = { ∈ R : 1 + ( ) ( ) > 0, ∀ ∈ T}.
If is a regressive function, then the generalized exponential function is defined by with the cylinder transformation Let , : T → R be two regressive functions; we define ⊕ := + + , Then the generalized exponential function has the following properties.
In this section, denotes R or C , denotes an open set in or = , and denotes an arbitrary compact subset of .
satisfies that, for any 1 , 2 ∈ Π, one has 1 ± 2 ∈ Π, where Definition 3. Let T be an almost periodic time scale. For any ∈ T, ∈ Π, we definẽ Obviously, if T is an almost periodic time scale, then inf T = −∞ and sup T = +∞. If there exists a ∈ Π such that T = T, then Definition 2 is equivalent to Definition 3.7 in [31]; otherwise, Definition 2 is more general than Definition 3.7 in [31].
is a relatively dense set in T for all > 0 and for each compact subset of ; that is, for any given > 0 and each compact subset of , there exists a constant ( , ) > 0 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 ( , ) = lim → +∞ (+ , ). From Definitions 2 and 4, one can easily see that all the results obtained in [31] 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 [31], we can obtain the following equivalent definition of uniformly almost periodic functions.
Definition 5. Let ( , ) ∈ (T × , E ), and if for any given sequence ⊂ Π and each compact subset of there exists a subsequence ⊂ such that ( , ) exists uniformly on T × , then ( , ) is called an almost periodic function in uniformly for ∈ . Definition 6. A function ∈ 1 (T, R) is said to be a 1almost periodic function, if , Δ are two almost periodic functions on T.
Definition 7 (see [31]). Let ∈ R , and let ( ) be an × rd-continuous matrix on T; the linear system is said to admit an exponential dichotomy on T if there exist positive constants and , projection , and the fundamental solution matrix ( ) of (15), satisfying where ‖ ⋅ ‖ 0 is a matrix norm on T.
Consider the following linear almost periodic system: where ( ) is an almost periodic matrix function and ( ) is an almost periodic vector function.

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The Scientific World Journal Lemma 10 (see [27]). Every rd-continuous function has an antiderivative. In particular, if 0 ∈ T, then defined by is an antiderivative of .
Lemma 11 (see [27]). If ∈ R and , , ∈ T, then By Lemmas 10 and 11, it is easy to get the following lemma.

Lemma 12. Suppose that ( ) is an -continuous function and ( ) is a positive -continuous function which satisfies that
where 0 ∈ T; then, Lemma 13 (see [31]). If is a real-valued almost periodic function on T and : R → R is a Lipschitz function, then → ( ( )) is an almost periodic function on T.
For ∀ ∈ B, if we define induced modulus where then B is a Banach space.
Theorem 15. Assume that ( 1 ), ( 2 ), and the following hold: The Scientific World Journal then, system (1) has a unique 1 -almost periodic solution in the region Proof. For any given ∈ B, we consider the following almost periodic differential equation: admits an exponential dichotomy on T. Thus, by Lemma 9, we know that system (32) has exactly one almost periodic solution: are almost periodic functions on T; that is, (34) is not only an almost periodic solution of system (32), but also a 1 -almost periodic solution of system (32). First, we define a nonlinear operator on B by Next, we check that Φ( ) ⊂ . For any given ∈ , it suffices to prove that ‖Φ( )‖ B ≤ 0 . By conditions ( 1 )-( 4 ), we have The Scientific World Journal then, it follows from (37) that Therefore, Φ( ) ⊂ .
Taking , ∈ and combining conditions ( 1 ) and ( 4 ), we obtain that Similarly, from (39) it follows that By (40), we obtain that Φ is a contraction mapping from to . Since is a closed subset of B, Φ has a fixed The Scientific World Journal 9 point in , which means that (32) has a unique 1 -almost periodic solution in . Then system (1) has a unique 1almost periodic solution in the region This completes the proof.

Remark 18.
In [17,25,26,29], the existence and stability of almost periodic solutions are studied for several classes of neural networks on almost periodic time scales. However, the almost periodic time scales used in [17,25,26,29] are a kind of periodic time scales. So, the methods and the results of this paper are essentially new.

Some Examples
Consider the following neural network: Thus, ( 4 ) holds for 0 = 1. By Theorems 15 and 17, system (67) has a unique 1 -almost periodic solution in the region which is globally exponentially stable (see Figures 1-4).

Conclusion
In this paper, by using calculus theory on time scales, a fixed point theorem, and differential inequality techniques, some sufficient conditions ensuring the existence and global exponential stability of 1 -almost periodic solutions for a class of neural networks with time-varying delays on a new type of almost periodic time scales are established. To the best of our knowledge, this is the first time to study the existence of 1 -almost periodic solutions of BAM neural networks on time scales. Our methods that are used in this paper can be used to study other types of neural networks, such as Cohen-Grossberg neural networks and fuzzy cellular neural networks.

Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.