Positive almost periodicity on SICNNs incorporating mixed delays and D operator*

In the early 1990s, Bouzerdout and Pinter first established the shunt inhibitory cellular neural networks (SICNNs) system [2], which has attracted extensive attention because the lateral inhibition of shunting not only can greatly enhance the edge and contrast, but also be of significant influence in vision. With the increasing improvement of neu- ral networks, the above systems and their variants are widespread used in the fields of pronunciation, robotics, associative memories, psychophysics and optimization [1]. It has been discovered that time delay is inevitable and it may be one of the important reasons leading to the instability and terrible efficiency of the system [7, 10, 17, 28]. Hence, numerous scholars focus on the dynamic researches of cellular neural networks and biomathematical models accompanying bounded time-varying delays, and many in- teresting findings have been published in [8, 9, 11, 18]. In addition, in the large-scale networks models, since the occurrence of many parallel ways containing a various of axon lengths and sizes, it is meaningful to reveal the dynamical characteristics of neural networks incorporating continuously distributed delays [3, 30]. Because neural cells have complex dynamic characteristics in the real world, in order to further simulate the dynamics of this complex neural reactions, the neural networks system should contain some messages on derivatives of past states [13, 14], which inspired people to study the neutral- type systems. Generally speaking, neutral-type neural network systems can be expressed as non-D operator and D operator, and the cellular neural networks accompanying D operator are more practical than those ones touching non-D operator [23, 24]. Usually, let , and be the units amount, the neutral-type SICNNs incorporating D operators, continuously distributed delays and time varying delays are often modelled as the neutral-type functional differential equations

Here labels the cell at lattice is designated as the neighborhood. is identically declared designates the neuron state, is the decay rate, and denote thejointing or coupling intensity of postsynaptic action of the cell conveyed to the cell are transfer delay functions and stand for the activationfunctions substituting the firing rate or output of the cell is correlating to theexternal input, and one can consult [16, 26] for more detailed biological explanations.

Furthermore, the initial value conditions of SICNNs (1) are denoted as

During recent decades, great efforts have been put into the periodicity and almost periodicity of the population and ecology models [25, 29]. More precisely, many biolog- ical and cognitive activities need to be repeated, for example, oscillators [20], which are essential in many electronic circuits, usually generate almost periodic signals. Meanwhile, almost periodicity can better describe the changes in the natural environment and has a significant impact in illustrating the behavior of nonlinear dynamic systems [15, 17, 21, 22, 28]. It is worth pointing out that, in neural networks dynamics involving the field of biomathematics, the relevant state variables are currently treated as light intensity levels, proteins and electric or molecules charge, and they are surely positive restraints [19]. Such biological systems are often handled as positive systems [4]. However, the positive almost periodic stability for neutral-type SICNNs incorporating D operator has not been involved, which needs further research.

Inspired by the above considerations, in this article, we focus on the positive al- most periodic stability on SICNNs system. Briefly speaking, the main contributions and highlights of this article can be summarized as below. (i) The positiveness of bounded solutions of SICNNs (1) are demonstrated with the help of some differential inequality methods; (ii) Under certain hypothesises, by exploiting the fixed point theory and Lya- punov functional approach, the positiveness and global exponential stability for the almost periodic solutions of SICNNs (1) are proved for the first time; (iii) Numerical simulations accompanying comparison discussions are supplied to validate the effectiveness of our theoretical findings.

The rest framework of this article is outlined as below. In Section 2, we shall present some definitions and preliminary results. In Section 3, we afford the main theorems and their comprehensive proofs. Section 4 furnishes a numerical example to check the advantage and validity of our results. We terminate this article by a concise conclusion in Section 5.

In what follows, a few definitions, lemmas and presumptions are provided, which are advantageous in the following verification process of the main findings.

Notations. For convenience and simplicity, the -dimensional real vectors assemble is denoted by . For each . For a real function , define

Letting the supremum norm , the bounded and continuous functions collection is a Banach space.

Definition 1. (See [5].) Let , and the assemble be relative density, that is to say, for each , one can find a constant agreeing that every interval of length contains a such that for arbitrary . Then is said as an almost periodic function in.

Denote as the assemble of the almost periodic functions from .

For all , we also presume that

The following hypotheses will be adopted later.

(S0) For arbitrary , we can take constants satisfying

(S1) , and for a positive number .

(S2) There are positive numbers obeying that

and

First, on account of , employing a discussion similar to that used in Lemma 2.2 of [26], one can reveal the global existence and uniqueness on all solutions of the initialvalue problem (1)–(2).

Lemma 1. Suppose that (S0).(S2) are obeyed. Then every solution for system (1)involving initial values (2)possesses global existence and uniqueness on .

Hereafter, to show the positiveness of bounded solutions on SICNNs (1), we make the following hypotheses:

and

Then, for all , one can select a constant agreeing with

Adopting the above assumptions of external inputs, some positive solutions of the addressed system can be presented as follows.

Lemma 2.Let (S0).(S2), (3)and(4) be satisfied. In addition, mark as the solution of(1)with

Then

Proof. Striving for a contradiction, suppose that (8) is not true. We shall deal with two scenarios as follows.

Case 1. There arise and obeying that

and

Case 2. There are agreeing with (9), and

If Case 1 holds, one can assert that for arbitrary

(10) On the contrary, suppose that (10) is false. Then there are satisfying that for

Hence it follows that

which contradicts the positiveness of and proves (10). Consequently,

and

Meanwhile

and

In view of (1), (5), (11), (12) and (S0), we gain

which is absurd.

If Case 2 holds, we can also deduce (11) and (12), which, together with (1), (3) and (S0), results in

which produces a conflict and demonstrates that (8) is correct. This verifies Lemma 2.

Remark 1. When the hypotheses adopted in Lemma 2 are obeyed, (8), (11) and (12) entail that for any solution of (1) incorporating assumptions (6), (7),

Proposition 1. (See [28, Prop. 3.1].) For

Theorem 1. Under the presumptions of (S0).(S2), (3) and (4), SICNNs (1) possesses just one almost periodic solution , which is positive and globally exponentially stable. Moreover, for arbitrary solution of (1) incorporating the initial values (2), it can be discovered two constants and satisfying

and

where can be found in (7).

Proof. Firstly, we evidence that the possible existing almost periodic solution has even- tual positiveness. To do this, assume that the SICNNs (1) possesses a globally exponentially stable almost periodic solution and denote by an arbitrary solution of (1) incorporating assumptions (6), (7). It follows from (13) and (14) that

By Proposition 1, one can acquire

which reveals that is positive on .

Secondly, we shall verify that SICNNs (1) possesses an almost periodic solution.

Denote , we gain

Given and Lemma 2.2 in [12] we acquire

which, combined with (S0) and the argument process of Lemma 2.3 in [15], indicates that

and

Now, we take into consideration the following auxiliary equations:

Combining and Theorem 2.3 in [27], weknow that (15) possesses a sole almost periodic solution:

Manifestly,

and is the solealmost periodic solution of

We set

If , then

here . In addition, we set a mapping

Next, it will be proven that for any . Indeed, with the help of (S0),(S2), (16) and (17), one can discover that

which indicates that

Moreover, we show that is a contractive mapping. In fact, (S0), (S2), (16) and (17)yield

This leads to

Therefore, in view of Theorem 0.3.1 of [6] and , we know that owns a sole fixed point satisfying that

and

which, together with (16), results in

This entails that is the almost periodic solution of SICNNs (1).

Eventually, we verify the globally exponential stability of .

Denote by a solution of SICNNs (1) incorporating (2), and let

Then, for

From (S1) and (S2) one can discover a positive number

satisfying that for all , there holds and

Set

For arbitrary , one can obtain

thus, one can take a sufficiently large constant satisfying that

Hereafter, we validate

By way of contradiction, there must be obeying that

and

Moreover,

where , which indicates that

Note that

which means that

Consequently, owing to (S0), (S2), (17), (18), (19) and (23), we obtain

This causes a conflict with (21). Consequently, (20) holds. Letting , we have

Then, using a similar discussion with (22) and (23), we get from (24) that

and

where . This assures Theorem 1.

Remark 2. In Theorem 1, we firstly set up the positive stability of delayed almostperiodic SICNNs with D operator. So far, many achievements on the exponential convergenceor stability of delayed cellular neural network models have been revealed, see,e.g., [16, 23, 24, 26] and the related references. However, as far as we know, there isno result exploring the positive almost periodicity of SICNNs with D operator. Ourresults complement and improve some corresponding ones of the existing publicationsin [23, 24, 26].

Regard the neutral-type SICNNs incorporating D operator:

where

and

Take

Obviously, all requirements of Theorem 1 are obeyed in (25). Thus, SICNNs (25) pos- sesses a sole one globally exponentially stable almost periodic solution, which has posi- tiveness (see Fig. 1).

Remark 3. It should be noted that the positive almost periodicity of SICNNs incorporating D operator and mixed delays has not been touched in the previous publications [4, 9]. Thus, the corresponding conclusions of the above mentioned literatures are ineffective to reflect the positive almost periodic stability of SICNNs (25).

In this work, we obtain some results involving the existence and global exponential stability of the positive almost periodic solution for a kind of shunting inhibitory cellular neural networks incorporating mixed delays and D operator with the help of some analysis methods and inequality techniques. Because neutral-type operator exists in the neural networks system, the existing methods are no longer applicable to show the positiveness of the almost periodic solutions, we have developed novel techniques and mathematical approaches for overcoming the obstacles coming from neutral-type operator. Lemma 2 is important for the judgment of the prior boundedness of the operator equation. Moreover, numerical examples are worked out to demonstrate the advantages of our results. The strategy adopted in this work can be also applied to explore other types of D operator cellular neural networks, such as neural networks systems involving parameters uncertainties and impulse disturbance, neural networks accompanying neutral-type mixed delays and so on. This is our future investigation direction.

The authors wish to thank the anonymous reviewers for their careful work and thoughtful suggestions that have helped improve this paper substantially. In particular, the authors express the sincere gratitude to Prof. Gang Yang (Hunan University of Technology and Business) for the helpful discussion.