The Initial And Boundary Value Problems For A Kind Of Second-order Quasilinear Singularly Perturbed Equations In Critical Case

In this paper,we discuss a kind of second-order quasilinear system : ey" = A(y,t)y' + (?)f(y,t) (*) which is in the critical case. In section l,we simply introduce some developments of singular perturbation of ODE in the critical case,and list the preparing knowledge as well as some corresponding conceptions needed later . In section 2, for a kind of special two-order square matrix A(y₁,y₂, t) ,we study the case of infinite initial values y(0, (?)) = y°, y'(0, (?)) = z^-1/(?) firstly, and technically find the differential equations and initial values in detail to which regular items and boundary functions should be matched ,and explain their solvability at the same time.Then,we complete the construction of uniformly asymptotic solution.In section 3,we discuss the case in which equation (*) should satisfy the boundary conditions y(0, (?)) — y°, y(1,(?)) = y¹ .We mainly base on the results we have gotten in section 2, i,e, we consider the known initial problem as auxiliary one ,and z^-1(i = 1,2) as parameters .We focusly prove the fact that y(1,(?)) = y¹ is a equation of these parameters,and the determinant of its coefficient does not equal to zero.Therefore,we solve the boundary value problem.When come to section 4,we extend the foregoing problems forward ,and we offer an abstract way for the reader to think about and work it out.