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.
This work investigates traveling waves for a class of delayed cellular neural networks with nonmonotonic output functions on the one-dimensional integer latticeZ. The dynamics of each given cell depends on itself and its nearestmleft orlright neighborhood cells with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Our technique is to construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions. Then we construct a suitable wave profiles set and derive the existence of traveling wave solutions by using Schauder's fixed point theorem.
The goal of this paper is to study the stability and traveling waves of stage-structured predator-prey reaction-diffusion systems of Beddington-DeAngelis functional response with both nonlocal delays and harvesting. By analyzing the corresponding characteristic equations, the local stability of various equilibria is discussed. We reduce the existence of traveling waves to the existence of a pair of upper-lower solutions by using the cross iteration method and the Schauder's fixed point theorem. The existence of traveling waves connecting the zero equilibrium and the positive equilibrium is then established by constructing a pair of upper-lower solutions.
MSC: 35K57; 92D25
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