Study On The Solution Of Boundary Value Problems For Several Classes Of Differential Equations

The boundary value problem of nonlinear differential equations is an important research direction in nonlinear theory. It has been widely used in bridge engineering, aerospace technology, biotechnology, applied mathematics and other fields. Therefore, by studying the boundary value problems of nonlinear differential equations we will solve the common problems in these fields. Moreover, in practical research, the mathematical model of differential equations is usually nonlinear.Therefore, it is very important to study the boundary value problems of nonlinear differential equations.In chapter one, a brief introduction of the history and present situation of the study about the boundary value problems of the differential equation is given, the purpose and significance of the research are described. We show that the research of the existence of solutions for boundary value problems of differential equations not only has important theory significance, but also has vital practical significance.In chapter two, we investigate the uniqueness of solution for singular boundary value problems of fourth-order differential equationwhere h(t) is allowed to be singular at t = 1. The main novelty of this chapter is that the Lipschitz constant is related to the first eigenvalues corresponding to the relevant operators. We show the uniqueness of solution by applying the u0 - norm and contraction mapping principle.In chapter three, we discuss the boundary value Problem for fractional differential equationhas at least one positive solution. where 2<p?3 is a real number. The result are obtained by using the first eigenvalue of the relevant linear operator and fixed point index theory.In chapter four, the existence of solution for fourth-order two point boundary value problem are obtained by using the monotone iterative technique and the method of lower and upper solutions, where f:[0,1]×R?R is continuous. The novelty of this chapter is to establish a comparison result for the boundary value problem by using the properties of linear operators, and the existence of extremal solution is studied.