Research On The Existence Of Solutions For Nonlinear Fourth-order Boundary Value Problem

The BVPs of fourth-order differential equations arise some practical problems of the applied mathematics and physics.In recent decades,with the continuously development of research,a lot of mathematical models are described by boundary value problems(BVPs)of differential equations.The application of elastic beams is particularly widespread.Therefore,it is of great value and practical significance to study the BVPs of fourth-order differential equations.This thesis is mainly divided into four chapters:Chapter 1 is the introduction.We simply introduce the studying background of this thesis.In chapter 2,we use monotone iterative technique and lower and upper solutions of completely continuous operators to get the existence of nontrivial solutions for the following nonlinear fourth-order differential equation two-point boundary value problemwhere f:[0,1]× R? R,g:[0,+?)[0,+?).The results can guarantee the existence of nontrivial sign-changing solutions and positive solutions,and we can construct two iterative sequences for approximating them.In chapter 3,we study the the existence and uniqueness of the monotone positive solutions for the nonlinear fourth-order boundary value problem by using two fixed point theorems for a sum operatorwhere f ? C([0,1]×[0,+?),[0,+?)),g ? C([0,+?),[0,+?)).And we can construct an iterative scheme for approximating the unique solution for any initial value in a special set.In chapter 4.we will use fixed point theorems in partially ordered metric spaces to consider the existence and uniqueness of positive solutions of the following fourth-order two-point boundary value problemwhere f ? C([0,1]× R.R)and ?>0 is a parameter.