Abstract

We consider a class of nonautonomous cellular neural networks (CNNs) with mixed delays, to study the solutions of these systems which are type pseudo almost periodicity. Using general measure theory and the Mittag-Leffler function, we obtain the existence of unique solutions for cellular neural equations and investigate the Mittag-Leffler stability and attractiveness of pseudo almost periodic functions. We also present numerical examples to illustrate the application of our results.

1. Introduction

Due to the many applications of neural cell networks in various fields, these systems have been extensively studied. Image processing, robotics, optimization, etc. are among the fields used by these differential systems [14]. Due to the importance of network systems, stability analysis and synchronization control for these systems have always been considered by many researchers who have studied these systems with different tools. For example, we can mention [58], where Lyapunov functions have been used as a tool for these synchronization analyses.

We shall introduce a neural cellular system and investigate the solutions of this differential equation, which are of the -pseudo almost periodic type (for more details, see [911]). Assume that is a measure, is a positive measurable function in and is singular Lebesgue measure. Here, measure is defined by

The cellular neural system with mixed delay is described by

This system with the initial value is expressed as follows:

The parameters in this equation are as follows: (i) is the -th neuron state(ii) represents the rate of decay,(iii)Real functions are activation functions of the -th neuron(iv) is the input(v) are the delays that are constant(vi) is the transmission delay kernel

Considering a special case of the stated measure, i.e., , the -pseudo almost periodic solutions of the above system are of the weighted pseudo almost periodic functions type.

In the present paper, we shall derive some sufficient conditions for existence and uniqueness results for cellular neural equations [3, 1214]. We first state the basic concepts and then obtain the unique solution for equation (1). In the sequel, we prove our main results, i.e., the Mittag-Leffler stability and attractiveness of -pseudo almost periodic solutions of equation (2), which improves upon and extends [11, 1520].

We conclude the introduction by describing the structure of the paper. In Section 2, we collect the preliminary information. In Section 3, we present several examples of interesting measures. In Section 4, we prove our first main result (Theorem 19). In Section 5, we prove our second main result (Theorem 21). In Section 6, we prove our third main result (Theorem 23). In Section 7, we present some applications.

2. Preliminaries

We denote the space of all positive measures on Lebesgue -field with . If is a positive measure, then we have (i)(ii), for all

Considering as the space of all continuous and bounded functions, as well as the supremum norm , we have a Banach space.

Definition 1. The Mittag-Leffler function is defined by where is a real number, and is a complex variable. The generalization of is defined as where

Definition 2. If and is a complex number, then is called the -order fractional hyperbolic cosine function and is called the -order fractional hyperbolic sine function.

Proposition 3. Assume that Then,

Definition 4. A continuous function is said to be almost periodic if for all .

Definition 5. Let . A bounded continuous function is said to be -ergodic if

Definition 6. Suppose that , , and are almost periodic and -ergodic functions, respectively. Then, is a -pseudo almost periodic function, provided that

We denote the space of all almost periodically functions by , the space of all -ergodic functions by , and the space of all -pseudo almost periodic functions by . All these spaces, equipped with the supremum norm, are Banach spaces. Also, we have ; for more details, see [9].

Definition 7. Let be a solution of equation (2), with initial value . Suppose that for every solution of equation (2) with initial value , there exist constants and such that for all where Then, the property of Mittag-Leffler stability holds for .

We can derive the Mittag-Leffler attractiveness from the Mittag-Leffler stability; for more details, see [2127].

Definition 8. Let be a solution of equation (2), with initial value . Suppose that there exists such that for any solution of equation (2). Then, the property of Mittag-Leffler attractiveness holds for .
If the Mittag-Leffler stability for any solution of equation (2) is established, then depends on its initial value .

Definition 9. The convolution of functions and from to , if any, is defined as follows: where and for and .

Definition 10 (see [28]). Let be a Borel space. If and are measures on , we say that and are mutually singular, if there exist disjoint sets and in such that and

Definition 11 (see [28]). Assume that and are measures on the Borel space . We say that is absolutely continuous relative to provided that for each .

Following Lebesgue-Radon-Nikodym [28], we assume that . We impose the following assumptions for every : (I1) are globaly Lipschitzian with Lipschitz constants , and , respectively(I2) is bounded and continuous(I3)There exists such that is integrable on (I4)For the bounded interval and all , there exists such that when satisfies (I5)There exist such that (I6)There exist and such that (I7)There exist and such that (I8)For all such that , there exist and such that

Hypothesis implies hypothesis whereas the converse is not true. Also, if hypothesis holds, then is a Banach space. If hypothesis holds, then for any , and we have . The proofs can be found in [9].

Theorem 12 (see [29]). For any integrable function such that , we have

Theorem 13 (see [30]). For any on the interval with positive length and any we have where for all . In particular,

Remark 14. For a globally Lipschitzian mapping such that and are Banach spaces and every almost periodic functions , we have , which means that is an almost periodic function.

3. Examples of Measures Satisfying Hypotheses and

Next, we shall introduce three examples of measures which satisfy hypotheses and .

Example 15 (see [9]). We consider a measure which is not absolutely continuous and satisfies . This measure is defined as where is a measure of the Lebesgue type. Also, is the measure on , which in is the -field of the Lebesgue type. This measure is defined as follows:

Example 16. Consider the following measure: where and according to the integer , is a Dirac measure (DM), and is a generalized Dirac comb (GDC). When , this measure is called a Dirac comb (DC).
Since , we shall show that is satisfied for , such that We also have that The conclusion follows with and This means that is a Banach space.
The measure does not satisfy . In the sequel, we shall prove this. Let Then, contains and provided that . Now, Therefore, if , where is a bounded interval, we obtain provided that .

Example 17. We consider the following measure for where is the Dirac measure at and satisfying .
For , let . We can easily see that , and also . Then, The conclusion now follows with

4. On the Integral Solution of Equation (34)

Proposition 18. Assume that and hold. If , then for
Assume further that and hold. If then

Proof. This follows from [15] (Theorem 4.1).
In the sequel, we set . Then, equation (1) is transformed into the following system: Now we show that the integral solutions of equation (34) are mappings of to itself.

Theorem 19. Assuming that and hold, we define the nonlinear mapping on for as follows: If we assume condition along with the other three conditions, then .

Proof. We have (see [29]). According to Proposition 18, for , there exist and such that (1)We claim thatIn fact, According to Theorem 13, and since , for every , there exists a number such as belonging to an interval of positive length such that and for all (see [30]). Then, Let Since using assumption , we obtain that Then, is a continuous and bounded function, given that . Now, by putting equation (39) in equation (38), we have We obtain that (2)Let us show thatAccording to hypothesis , for and for , we have Then, by , Hence, for . Combined with (37), we have

5. Existence and Uniqueness of -Pseudo Almost Periodic Solutions

Assuming that the solution of equation (1) is of the -pseudo almost periodic type, we shall prove the existence and uniqueness of these solutions.

Theorem 21. Given assumptions and , there is a unique solution