- Research
- Open Access
New studies on dynamic analysis of asymptotically almost periodic recurrent neural networks involving mixed delays
- Yuehua Yu1,
- Shuhua Gong2Email authorView ORCID ID profile and
- Zijun Ning2
https://doi.org/10.1186/s13662-018-1872-8
© The Author(s) 2018
- Received: 25 June 2018
- Accepted: 5 November 2018
- Published: 14 November 2018
Abstract
This paper studies a class of asymptotically almost periodic recurrent neural networks involving mixed delays. By utilizing differential inequality analysis, some novel assertions are gained to validate the asymptotically almost periodicity of the addressed model, which generalizes and refines some recent literature works. In the end, an example with its numerical simulations is carried out to validate the analytical results.
Keywords
- Recurrent neural network
- Asymptotically almost periodicity
- Convergence
- Mixed delay
MSC
- 34C25
- 34K13
- 34K25
1 Introduction
The rest of this paper is arranged as follows. Some preliminaries and lemmas are supplied in Sect. 2. In Sect. 3, some novel sufficient conditions are gained to evidence the asymptotically almost periodicity of system (1.1). In Sect. 4, an illustrative example is presented to validate the correctness of the proposed theory. In the end, a brief conclusion is presented to summarize and evaluate our work.
2 Preliminary results
Notations
Assumptions
- \((U_{0})\) :
-
\(b_{i}(0 )=0\), \(\underline{b}_{i}|u -v |\leq \operatorname {sign}(u-v)(b _{i}(u )-b_{i}(v )) \leq \overline{b}_{i}|u -v | \).
- \((U_{1})\) :
-
\(|f_{j}(u )-f_{j}(v )|\leq L^{f} _{j}|u -v |\), \(|h_{j}(u )-h _{j}(v )| \leq L^{h}_{j}|u -v |\), \(|g_{j}(u )-g_{j}(v )| \leq L^{g}_{j}|u -v | \).
- \((U_{2})\) :
-
\(K_{ij}:\mathbb{R}^{+}\rightarrow \mathbb{R}\) is continuous and absolutely integrable.
- \((U_{3})\) :
-
$$ \begin{aligned} &{-}\bigl[a_{i}^{0}(t)\underline{b}_{i}- \lambda \bigr]\eta_{i}+ \sum_{j=1}^{n} \bigl( \bigl\vert \alpha^{0}_{ij}(t) \bigr\vert + \bigl\vert \alpha^{1}_{ij}(t) \bigr\vert \bigr)L ^{f}_{j}\eta_{j}+ \sum _{j=1}^{n}\bigl( \bigl\vert \beta_{ij}^{0}(t) \bigr\vert + \bigl\vert \beta_{ij}^{1}(t) \bigr\vert \bigr)e^{\lambda \sigma } L ^{h}_{j}\eta_{j} \\ &\quad {}+ \sum_{j=1}^{n}\bigl( \bigl\vert \gamma_{ij}^{0}(t) \bigr\vert + \bigl\vert \gamma_{ij}^{1}(t) \bigr\vert \bigr) \int_{0} ^{+\infty } \bigl\vert K_{ij}(s) \bigr\vert e^{\lambda s}\,ds L^{g}_{j}\eta_{j}< -\xi,\\ &\quad t \in \mathbb{R}^{+}, \sigma =\max_{i, j\in S }\sup _{t\in \mathbb{R}}\sigma_{ij}^{0}(t). \end{aligned} $$
Lemma 2.1
Designate \(x(t) \) to be a solution of the initial value problem \((1.1)^{0}\) and (2.1). If \((U_{0})\), \((U_{1})\), \((U_{2})\), and \((U_{3})\) hold, then \(x(t)\) is bounded and exists on \([0, +\infty )\).
Proof
Remark 2.1
Under the assumptions in Lemma 2.1, an argument similar to that applied in Lemma 2.1 shows that each solution of initial value problem (1.1) and (2.1) is bounded on \([0, +\infty )\).
Lemma 2.2
Proof
3 Asymptotically almost periodicity
Theorem 3.1
If \((U_{0})\), \((U_{1})\), \((U_{2})\), and \((U_{3})\) hold, then every solution of (1.1) with initial condition (2.1) is asymptotically almost periodic on \(\mathbb{R}^{+}\) and converges to an almost periodic function \(x^{*}(t)\) as \(t\rightarrow +\infty \), which is a unique almost periodic solution of system \((1.1)^{0}\).
Proof
Remark 3.1
Under the conditions in Lemma 2.2, from Lemma 2.1 and Lemma 2.2, by applying a similar way as that in Theorem 3.1 of [13], one can show that every solution \(x(t )\) of \((1.1)^{0}\) converges exponentially to \(x ^{* }(t )\) as \(t\rightarrow +\infty \). Since \(\operatorname{AP}(\mathbb{R},\mathbb{R} ) \) is a proper subspace of \(\operatorname{AAP}( \mathbb{R},\mathbb{R} )\), one can easily see that all the results on \((1.1)^{0}\) in [13] are special ones of Theorem 3.1 in this paper. Most recently, the authors in [36] established asymptotically almost periodicity on shunting inhibitory cellular neural networks with time-varying delays and continuously distributed delays. However, the asymptotically almost periodicity on recurrent neural networks without the assumption E and the condition \(b_{i} (u)=u\) has not been explored in [36]. This implies that Theorem 3.1 generalizes and complements the main results of [13, 36].
4 A numerical example
Example 4.1
Numerical solutions of system (4.1) with different initial values
Remark 4.1
5 Conclusions
In this paper, avoiding the exponential dichotomy theory, the asymptotically almost periodicity on recurrent neural networks involving mixed delays has been explored. By combining the Lyapunov function method with differential inequality approach, some sufficient assertions have been gained to validate the global convergence of the addressed model. Particularly, our conditions are easily checked in practice by simple inequality methods, and the approach adopted in this paper provides a possible way to research the topic on asymptotically almost periodic dynamics of other nonlinear neural network models. In future research, we will research the dynamics for asymptotically almost periodic Cohen–Grossberg neural network models.
Declarations
Acknowledgements
The authors would like to express their sincere appreciation to the editors and anonymous reviewers for their constructive comments and suggestions which helped them to improve the present paper. This work was supported by Zhejiang Provincial Natural Science Foundation of China (Grant Nos. LY16A010018, LY18A010019), Zhejiang Provincial Education Department Natural Science Foundation of China (Y201533862), and Natural Scientific Research Fund of Hunan Provincial Education Department of China (Grant No. 17C1076).
Funding
This work was supported by Zhejiang Provincial Natural Science Foundation of China (Grant Nos. LY16A010018, LY18A010019), Zhejiang Provincial Education Department Natural Science Foundation of China (Y201533862), and Natural Scientific Research Fund of Hunan Provincial Education Department of China (Grant No. 17C1076).
Authors’ contributions
YHY, SHG, and ZJN worked together in the derivation of the mathematical results. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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