Research On The Interior Layer Problem For Delayed Singularly Perturbed Problems

By applying the boundary layer function method, patching connections, implicit function theorem and other methods, this dissertation mainly aims to construct the asymptotic expansion of the solution for four kinds of singu-larly perturbed problems. The sufficient conditions to obtain the existence of the solution are also found. We consider the region and reason for the inner layer, and give the proof of the uniformly valid asymptotic expansion. Meanwhile, we give the corresponding application models in the system of control theory, chemical kinetics, semiconductor theory and the other fields. The result obtained in the thesis generalizes and extends the corresponding known results.The whole thesis contains fives chapters. The first chapter introduces concisely the background of the subjects relevant to this dissertation and some definitions and theorems, emphatically introducing the main work to be done in this thesis.Chapter2and Chapter5mainly consider a kind of quasi-linear singu-larly perturbed differential difference equation with initial values and bound-ary values and a class of predator-prey models. By using the boundary layer function method, the asymptotic solutions of these problems are constructed. The existence of the solution to the original problem has also been proved by successive approximation method and the contraction mapping theorem.Chapter3is devoted to the problems for vector singularly perturbed delay-differential equations. This Chapter contains two sections. The first section narrate the internal layer problem of a kind of fast system. In the sec-ond section, we study the singularly perturbed delayed systems of Tikhonov's type with fast and slow variables. By means of the boundary layer function method, patching connections and implicit function theorem we prove the existence of solution of our problems near the degenerate solution for suffi-ciently small μ and obtain the uniformly valid solution in the whole interval.Chapter4is focused on singularly perturbed delayed boundary value problems in the critical case. We consider the internal layers of a weakly nonlinear initial value problem and high dimensional differential equations respectively. We not only construct the asymptotic expansion of the solu-tion for the original equation but also give the proof of the uniformly valid asymptotic expansion by using successive approximations and the contrac-tion mapping theorem. Meanwhile, we give examples to demonstrate our results.