Traveling waves for nonlinear cellular neural networks with distributed delays

Abstract

In this paper, we will establish the existence and nonexistence of traveling waves for nonlinear cellular neural networks with finite or infinite distributed delays. The dynamics of each given cell depends on itself and its nearest m left or l right neighborhood cells where delays exist in self-feedback and left or right neighborhood interactions. Our approach is to use Schauder's fixed point theorem coupled with upper and lower solutions of the integral equation in a suitable Banach space. Further, we obtain the exponential asymptotic behavior in the negative infinity and the existence of traveling waves for the minimal wave speed by the limiting argument. Our results improve and cover some previous works.

Traveling Waves for Delayed Cellular Neural Networks with Nonmonotonic Output Functions

“…But under what conditions the wave will oscillate on the nontrivial equilibrium at infinity is an interesting problem. On the other hand, similarly to the proofs of nonexistence of traveling waves in [26], we can also obtain the same conclusions for nonmonotonic DCNN models.…”

“…the authors of [10] also derived the existence of traveling wave solutions. Recently, when the propagation of signal is very slow, authors of [26] considered CNN models with infinite time delays. More precisely, Yu et al [26] investigated the existence and nonexistence of monotonic traveling waves for the following more general CNN models with infinite distributed time delay terms:…”

“…Here the dynamics of each given cell depends on itself and its nearest left or right neighborhood cells with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Moreover, we point out that the CNN models with discrete time delays or no delays case can be included in this model by choosing suitable kernel functions (see [26]).…”

“…Since the output functions are nonmonotonic, our main idea is to construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions. Then we can apply the results in [26] and Schauder's fixed point theorem to derive the existence of traveling wave solutions. Now we state our main results.…”

Entire solutions of non-quasi-monotone delayed reaction—diffusion equations with applications

“…, n 0 are given constants. The output function f is continuous such that f (0) = 0, (a + α + β)f (K) = K and (a + α + β)f (u) > u for some constant K > 0, where α := m0 k=1 α k and β := n0 l=1 β l .If the output function f (u) is non-decreasing in u ∈ [0, K], Yu et al [36]. and Wu and Hsu[31] considered the travelling wavefronts and entire solutions of (3.1), respectively.…”