Entire Solutions in Lattice Delayed Differential Equations with Nonlocal Interaction: Bistable Cases

¹ School of Mathematics and Statistics, Lanzhou University Lanzhou, Gansu 730000, People’s Republic of China
² College of Mathematics and Information Science, Shaanxi Normal University Xi’an, Shaanxi 710062, People’s Republic of China
³ Department of Mathematics, University of Miami P. O. Box 249085, Coral Gables, FL 33124-4250, USA

^⋆ Corresponding author. E-mail: wangzhch@lzu.edu.cn

Abstract

This paper is concerned with entire solutions of a class of bistable delayed lattice differential equations with nonlocal interaction. Here an entire solution is meant by a solution defined for all (n,t) ∈ ℤ × ℝ. Assuming that the equation has an increasing traveling wave front with nonzero wave speed and using a comparison argument, we obtain a two-dimensional manifold of entire solutions. In particular, it is shown that the traveling wave fronts are on the boundary of the manifold. Furthermore, uniqueness and stability of such entire solutions are studied.