Existence And Asymptotic Analysis Of Solution For Several Nonlinear Value Problems

In this paper, by the theory of differential inequalities(upper and lower solutions method), we study the existence(or uniqueness) results of some classes of nonlinear differential equations (without small parameter). Furthermore, we apply the results to singularly perturbed BVPs(Boundary Value Problem for short) which involve small parameters. By the method of boundary layer function, we construct the higher order asymptotic solution and get the error estimate between asymptotic solution and exact solution.We also consider a class of boundary value problems for second-order quasilinear differential equations with singular perturbation on a subinterval.In chapter1, firstly,we introduce the background of Forth-order Differential Equations Value Problems and some important results of former scholars have stud-ied. Secondly, the concept of upper solution,lower solution and Nagumo condition are showed,two principal theorems of second-order differential inequality are intro-duced,and some principal lemmals are cited, which will be utilized in the following chapters.In chapter2, by using the upper and lower solutions method,existence and estimation of solutions to two three-point boundary value problems for forth-order differential equations with nonlinear and mixed boundary conditions are studied.In chapter3, we consider a class of three-point Robin boundary value problem for singularly perturbed similinear fourth-order differential equations. By using the theory of differential inequalities at first, prove the existence of solution, then using the boundary layer function method, the higher order asymptotic solution is constructed,finally,we get the error estimate of the solution by using the theory of differential inequalities.In chapter4, A class of boundary value problems for second-order quasilinear differential equations with singular perturbation on a subinterval is studied. With the method of the boundary layer function and the theory of differential inequalities, we get the existence of solution and the error estimate of the solution.