Mittag-Leffler Stability and Attractiveness of Pseudo Almost Periodic Solutions for Delayed Cellular Neural Networks

Eidinejad, Zahra; Saadati, Reza; Repovš, Dušan D.

doi:https://doi.org/10.1155/2022/3186963

Journal of Function Spaces

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Abstract Introduction Preliminaries Applications Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 3186963 | https://doi.org/10.1155/2022/3186963

Mittag-Leffler Stability and Attractiveness of Pseudo Almost Periodic Solutions for Delayed Cellular Neural Networks

Zahra Eidinejad,¹Reza Saadati,¹and Dušan D. Repovš^2,3,4

Academic Editor: Umair Ali

Received20 Apr 2022

Revised01 Jul 2022

Accepted05 Jul 2022

Published11 Oct 2022

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 [1–4]. 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 [5–8], 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 [9–11]). 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, 12–14]. 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, 15–20].

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 [21–27].

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 for equation (1).
Given assumptions , , and , we have .

Proof. Let (resp., ). Then, in view of Theorem 19, we have that (resp. ).
Let Using and , we get that so Invoking condition and , we conclude that is a contraction (since is a Banach space, according to condition it is also a contraction on this space). Therefore, we conclude that (or is a unique fixed point for . Also, given (34), (or ) is a unique solution of type - for equation (1).
In the sequel, we shall investigate the Mittag-Leffler stability and the Mittag-Leffler attractiveness for the unique solution of equation (2), which is of type -. First, we state Lemma 22.

Lemma 22. Assuming conditions and , we define by where Then for all and .

Proof. According to condition , function is defined on the interval . Then according to condition , we have Next, we shall show that there exists such that for all . Also, we have If we take and nonnegative numbers then for all and . Now, for every , by continuity, there exists such that the following holds: In the sequel, invoking condition the Lebesgue dominated convergence theorem (LDCT), and the integrability of the function on the interval , we get such that implies If we now take and , we can conclude that for every ,

6. On the Mittag-Leffler Stability and Attractiveness of Unique Solutions

Theorem 23. If we assume conditions , and , then the Mittag-Leffler stability for any solution of equation (2) is established by the initial condition . If we add condition , then the Mittag-Leffler stability for the solution of equation (2) is also established. Also, if we add condition to conditions , , , and , then the Mittag-Leffler stability for the solution of equation (2) is also established.

Proof. Let be a solution of equation (2) with initial value Set Then, Let and be such that Then, for all , we have Given in Lemma 22, we assume that . Then for all , we have Otherwise, for and , given the continuity, we have Now, first, we multiply both sides of (62) by Then, we integrate the obtained equation with respect to on . Finally, we multiply by Therefore, we have Next, by and we obtain that Now, by (64) and (66), we get Since and , we have by virtue of Lemma22, We recall that . This implies that for all . Hence, by (61), we see that which contradicts (66) and we can conclude that what was claimed in (65) is true.
We now assume that is constant and tends to zero. Then we get so the proof is complete.

Corollary 24. If we assume that conditions , and hold, then the Mittag-Leffler attractiveness for unique solution of equation (2) holds.

Corollary 25. If we assume that conditions and hold, then the Mittag-Leffler attractiveness for unique solution of equation (2) holds.

7. Applications

We shall provide two examples (see Figures 1–3).

Example 26. Let
(P₁)We consider Lipschitz functions with the Lipschitz constants .(P₂)Then, is bounded and continuous, (P₃)Furthermore, the next sum is integrable on for (P₄)(see [1]) If where and with then for we have that .(P₅)For we have (P₆)For , we have (P₇)Also, Let where and . Then, all solutions of (1) are in the Mittag-Leffler form and they converge to a unique solution of equation (1) such that , when with convergence rate

Example 27. Assume conditions to hold and consider functions from Example 26. For and Dirac measure , define the following measure: Consider the interval for and . Then for all and , there exist and a bounded interval such that and .
Let where and . Then, all solutions of (1) are in the Mittag-Leffler form and they converge to a unique solution of equation (1) such that , when with convergence rate .

(a) Numerical solutions of CNNs for

(b) Numerical solutions of CNNs for

(c) Numerical solutions of CNNs for

(a) Numerical solutions of CNNs for

(b) Numerical solutions of CNNs for

(c) Numerical solutions of CNNs for

(a) Numerical solutions of CNNs for

(b) Numerical solutions of CNNs for

(c) Numerical solutions of CNNs for

8. Conclusion

In this work, we considered differential systems of cellular neural networks (CNNs) with mixed delays. We also considered general measurement theory whose general form is . We first investigated the existence of a unique solution of this system and proved that the solutions of equation (1) are -pseudo almost periodic. Then we studied the Mittag-Leffler stability and the Mittag-Leffler attractiveness of these solutions. We obtained our results by considering new conditions and using the fixed point contraction mapping theorem. Also, two examples were given to illustrate our results.

Data Availability

The numerical data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

DD Repovš was supported by the Slovenian Research Agency grants P1-0292, N1-0278, N1-0114, and N1-0083.

References

L. O. Chua and L. Yang, “Cellular neural networks: theory,” IEEE Transactions on Circuits and Systems, vol. 35, no. 10, pp. 1257–1272, 1988.
View at: Publisher Site | Google Scholar
L. O. Chua and L. Yang, “Cellular neural networks: applications,” IEEE Transactions on Circuits and Systems, vol. 35, no. 10, pp. 1273–1290, 1988.
View at: Publisher Site | Google Scholar
B. Liu and C. Tunç, “Pseudo almost periodic solutions for CNNs with leakage delays and complex deviating arguments,” Neural Computing and Applications, vol. 26, no. 2, pp. 429–435, 2015.
View at: Publisher Site | Google Scholar
B. Liu and C. Tunç, “Pseudo almost periodic solutions for a class of nonlinear Duffing system with a deviating argument,” Journal of Applied Mathematics and Computing, vol. 49, no. 1-2, pp. 233–242, 2015.
View at: Publisher Site | Google Scholar
Y. Sun, J. Zhao, and G. M. Dimirovski, “Adaptive control for a class of state-constrained high-order switched nonlinear systems with unstable subsystems,” Nonlinear Analysis: Hybrid Systems, vol. 32, pp. 91–105, 2019.
View at: Publisher Site | Google Scholar
Z. Zhang and H. Wu, “Cluster synchronization in finite/fixed time for semi-Markovian switching T-S fuzzy complex dynamical networks with discontinuous dynamic nodes,” AIMS Mathematics, vol. 7, no. 7, pp. 11942–11971, 2022.
View at: Publisher Site | Google Scholar
J. Bai, H. Wu, and J. Cao, “Secure synchronization and identification for fractional complex networks with multiple weight couplings under DoS attacks,” Computational and Applied Mathematics, vol. 41, no. 4, pp. 1–18, 2022.
View at: Publisher Site | Google Scholar
R. Li, H. Wu, and J. Cao, “Impulsive exponential synchronization of fractional-order complex dynamical networks with derivative couplings via feedback control based on discrete time state observations,” Acta Mathematica Scientia, vol. 42, no. 2, pp. 737–754, 2022.
View at: Publisher Site | Google Scholar
J. Blot, P. Cieutat, and K. Ezzinbi, “New approach for weighted pseudo-almost periodic functions under the light of measure theory, basic results and applications,” Applicable Analysis, vol. 92, no. 3, pp. 493–526, 2013.
View at: Publisher Site | Google Scholar
J. Xu, “Weighted pseudo almost periodic delayed cellular neural networks,” Neural Computing and Applications, vol. 30, no. 8, pp. 2453–2458, 2018.
View at: Publisher Site | Google Scholar
L. Yao, Z. Wang, X. Huang, Y. Li, H. Shen, and G. Chen, “Aperiodic SampledData control for exponential stabilization of delayed neural networks: a refined two-sided looped-functional approach,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 67, no. 12, pp. 3217–3221, 2020.
View at: Publisher Site | Google Scholar
Y. Li, G. Li, and X. Meng, “Weighted pseudo-almost periodic solutions and global exponential synchronization for delayed QVCNNs,” Art, vol. 2019, no. 1, 2019.
View at: Publisher Site | Google Scholar
B. Liu, “Pseudo almost periodic solutions for neutral type CNNs with continuously distributed leakage delays,” Neurocomputing, vol. 148, pp. 445–454, 2015.
View at: Publisher Site | Google Scholar
M. Miraoui and N. Yaakoubi, “Measure pseudo almost periodic solutions of shunting inhibitory cellular neural networks with mixed delays,” Numerical Functional Analysis and Optimization, vol. 40, no. 5, pp. 571–585, 2019.
View at: Publisher Site | Google Scholar
D. Bekolle, K. Ezzinbi, S. Fatajou, D. E. Houpa Danga, and F. M. Besseme, “Attractiveness of pseudo almost periodic solutions for delayed cellular neural networks in the context of measure theory,” Neurocomputing, vol. 435, pp. 253–263, 2021.
View at: Publisher Site | Google Scholar
T. Diagana, “Les fonctions pseudo presque periodiques avec poids et applications,” Comptes Rendus Mathematique, vol. 343, no. 10, pp. 643–646, 2006.
View at: Publisher Site | Google Scholar
B. Ghanmi and M. Miraoui, “Stability of unique pseudo almost periodic solutions with measure,” Applications of Mathematics, vol. 65, no. 4, pp. 421–445, 2020.
View at: Publisher Site | Google Scholar
J. Jia, X. Huang, Y. Li, J. Cao, and A. Alsaedi, “Global stabilization of fractional-order memristor-based neural networks with time delay,” IEEE Transactions on Neural Networks and Learning Systems, vol. 31, no. 3, pp. 997–1009, 2019.
View at: Publisher Site | Google Scholar
Y. Yu, “Exponential stability of pseudo almost periodic solutions for cellular neural networks with multi-proportional delays,” Neural Processing Letters, vol. 45, no. 1, pp. 141–151, 2017.
View at: Publisher Site | Google Scholar
H. Zhang, “Existence and stability of almost periodic solutions for CNNs with continuously distributed leakage delays,” Neural Computing and Applications, vol. 24, no. 5, pp. 1135–1146, 2014.
View at: Publisher Site | Google Scholar
E. Graily, S. M. Vaezpour, and Y. J. Cho, “Generalization of fixed point theorems in ordered metric spaces concerning generalized distance,” Fixed Point Theory and Applications, vol. 2011, no. 1, 2011.
View at: Publisher Site | Google Scholar
M. Miraoui and D. D. Repovš, “Dynamics and oscillations of models for differential equations with delays,” Boundary Value Problems, vol. 2020, 17 pages, 2020.
View at: Publisher Site | Google Scholar
M. Miraoui and D. D. Repovš, “Existence results for integro-differential equations with reflection,” Numerical Functional Analysis and Optimization, vol. 42, no. 8, pp. 919–934, 2021.
View at: Publisher Site | Google Scholar
Q. Zhou and J. Shao, “Weighted pseudo-anti-periodic SICNNs with mixed delays,” Neural Computing and Applications, vol. 29, no. 10, pp. 865–872, 2018.
View at: Publisher Site | Google Scholar
B. Liu, “Global exponential convergence of non-autonomous cellular neural networks with multi-proportional delays,” Neurocomputing, vol. 191, pp. 352–355, 2016.
View at: Publisher Site | Google Scholar
B. Liu, “Exponential convergence of SICNNs with delays and oscillating coefficients in leakage terms,” Neurocomputing, vol. 168, pp. 500–504, 2015.
View at: Publisher Site | Google Scholar
A. Zhang, “Pseudo almost periodic solutions for SICNNs with oscillating leakage coefficients and complex deviating arguments,” Neural Processing Letters, vol. 45, no. 1, pp. 183–196, 2017.
View at: Publisher Site | Google Scholar
H. L. Royden, Real Analysis, Macmillan Publishing Company, New York, Third edition edition, 1988.
A. M. Fink, Almost periodic differential equations, Springer-Verlag, Berlin-New York, 1974.
G. M. N’Guérékata, Topics in Almost Automorphy, Springer-Verlag, New York, 2005.

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Copyright © 2022 Zahra Eidinejad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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