Traveling Wavefronts For The Reaction-diffusion Equations And Multiple Solutions For Some Classes Of Equations

This dissertation mainly deals with the existence of wavefronts for di?usionNicholson blow?ies equation with non-local e?ects, nontrivial solutions of discreteelliptic boundary and multiple periodic solutions of di?erential equations. Thisdissertation contains five chapters.Chapter 1 focuses on the brief introduction of the historic background andsignification for all the investigated problems, and gives out some open problemsto research.Chapter 2 investigates the existence of traveling wavefronts for the di?usiveNicholson blow?ies equations with the delay, non-monotone nonlinearity and non-local e?ects by applying phase-plane technique, singular perturbation theory, Fred-holm theory and fixed-point theorem. The method of this dissertation can dealwith the existence of traveling wavefronts for Nicholson blow?ies models in thediscrete delay case and the distributed delay case, but also in the spatial averagingcase. The chapter improves main results of some relevant literature. Numericalsimulations show that a hump appears when delayτorβis large, which coincideswith the theoretical analysis.Chapter 3 investigates the existence of multiple nontrivial solutions for dis-crete elliptic boundary value problems by using the"Mountain pass theorem".Some conditions are obtained for discrete elliptic boundary value problems to haveat least two nontrivial solutions. These improve the related results in some knownliterature.Chapter 4 is devoted to the existence of periodic solutions for the first orderfunctional di?erential equationsby using Gains and Mawhin's coincidence degree theory. Some su?cient conditionsare obtained to enuring that the above equation has at least two periodic solutions.Chapter 5 is devoted to the existence of the positive periodic solutions for aclass of abstract ecology modelsThe su?cient conditions on the existence of four periodic solutions are obtainedby means of the same theory as that in Chapter 4.