Stability of traveling wave solutions for nonlinear cellular neural networks with distributed delays
type: Item
instance of: scholarly article
title: Stability of traveling wave solutions for nonlinear cellular neural networks with distributed delays
zbMATH Open document ID: 1432.34091
author: Cheng-Hsiung Hsu
author: Jian-Jhong Lin
published in: Journal of Mathematical Analysis and Applications
publication date: 2018-10-19T00:00:00Z
review text: In this paper, the authors consider the stability of traveling wave solutions for the following class of nonlinear cellular neural networks with distributed time delays: \begin{align*} x_n'(t)=-x_n(t)&+\sum_{i=1}^m \alpha_i\int_0^{\tau}J_i(y)f(x_{n-i}(t-y))\,dy+a \int_0^{\tau}J_{m+1}(y)f(x_{n}(t-y))\,dy\\ &+\sum_{j=1}^{\ell} \beta_j\int_0^{\tau}J_{m+1+j}(y)f(x_{n+j}(t-y))\,dy, \forall n\in \mathbb{Z}, \: m,\ell\in \mathbb{N}, \end{align*} where \(x_n\) represents the \(n\)-th cell of the neural, \(f\in C([0, +\infty), [0, +\infty))\) denotes the output function of the cells, the nonnegative constants \(a\), \(\alpha_i\) \((i =1,\ldots,m)\) and \(\beta_j\) (\(j = 1,\ldots,\ell)\) are related to the synaptic weights of self-feedback and its neighbors to the cells, \(\tau>0\) is a constant delay, and the functions \(J_i:[0,\tau]\to[0,+\infty)\) are piecewise continuous density functions describing the distributed delay effects from the cells. A travelling solution for this infinite system is a solution such that there exist \(c > 0\) (called wave speed) and a function \(\phi: \mathbb{R} \to \mathbb{R}\) smooth enough satisfying that \(x_n(t)=\phi(n-ct),\) for every \(t\in\mathbb{R}\) and \(n\in\mathbb{Z}\). The work is based on the existence of nondecreasing travelling wave solutions satisfying the conditions \( \lim_{\xi\to-\infty} \phi(\xi)=0,\) and \( \lim_{\xi\to+\infty} \phi(\xi)=K.\) To study the stability, the authors prove two kinds of comparison results for the Cauchy problem, and the global existence of solution for two Cauchy problems. Under the appropriate hypotheses, and using some decay estimates, they show that the solutions to the Cauchy problems converge exponentially to the traveling wave solutions when the initial perturbations around the traveling wave solutions belong to suitable weighted spaces.
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 92B20
Mathematics Subject Classification ID: 34K10
zbMATH DE Number: 6958717
zbMATH Keywords: cellular neural networks
zbMATH Keywords: traveling waves
zbMATH Keywords: comparison theorem
zbMATH Keywords: stability results
MaRDI profile type: Publication
full work available at URL: j.jmaa.2018.10.010
OpenAlex ID: W2895351049
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DOI: 10.1016/J.JMAA.2018.10.010
description: scientific article; zbMATH DE number 6958717