Existence Of Traveling Wave Solutions In Two Lattice Differential Systems With Delay

Many models in areas of physics, chemistry, biology can biol down to reaction-diffusion equations. The study of reaction-diffusion equations is also concerned by excepts.In the discussion of reaction-diffusion equations, there is a kind of important solution, that istraveling wave solution. This solutions are highly attended. Because this kind of solutionsdescribes a course of transition from a equilibrium to another equilibrium. Many mathe-matician, physicist and chemist are interested in the study on traveling wave solutions andobtain many theoretical and realistic results. Firstly people are concerned about continuousreaction-diffusion equations. With the development of computers, people begin to study spa-tially discrete reaction-diffusion equation,that is lattice diffusion equation. Recently peopleare strongly interested in lattice differential equations, especially lattice differential equa-tions with delay.The present paper mainly investigates existence of traveling wave solutionsto lattice differential systems.In the first chapter, we introduce the historical background of problems and our workon this field.In the second chapter, we consider the existence of traveling wave fronts for the latticedifferential equation with delay. By monotone iteration method and the upper-lower solutiontechnique,the existence theorem of traveling wavefronts of this model is established.In the last chapter, we discuss the existence problem of traveling wave solution toa kind of lattice differential equations with delay. Firstly, we change the problem into aoperator's fixed point problem. Secondly,by choosing a properly subset equipped withnorm, which is obtained from a pair of upper and lower solutions, Schauder's fixed pointtheorem is applied to the operator to prove the existence of traveling wave solution in latticedifferential equations.