Positive Solution For Elastic Beam Equation Boundary Value Problem

In this thesis,the existence of positive solutions for a class of fourth-order p-Laplace elastic beam equation boundary value problem is studied.The existence conditions of the positive solution is obtained by using a new fixed point theorem and a new numerical iterative method,and the existence condition of the positive solution is obtained by applying the numerical method to a class of three order two point boundary value problems.According to the content,the full thesis is divided into five chapters.In the first chapter,we introduce the research background and development of the elastic beam equation boundary value problem.In the second chapter,the existence of positive solutions for a class of fourth-order p-Laplace boundary value problems is studied.Using a new fixed point theorem on the cone,the existence of the positive solution is proved.The advantage of the new fixed point theorem is that each variable in the nonlinear term can be discussed at the same time in a cone.And the norm estimation of the solution u(t)and the derivative u"(t)is given.Then,we give a weaker conditions of the existence of positive solutions through the properties of Green functions.Finally,the example verification results are given.In the third chapter,we get the existence of the iterative solution of a class of fourth-order p-Laplace boundary value problems.Since the p-Laplace operator is a nonlinear operator,it is difficult to obtain the corresponding maximum principle and it is difficult to use the upper and lower solutions method to obtain the iterative solution.In this chapter,a new numerical iterative method is proposed,we construct a closed ball in the Banach space,and the existence of the iterative solution is obtained by using the fixed point theorem of Banach,and the limitation of numerical iteration method to deal with such boundary value problems is analyzed.Then matlab is used to carry out numerical simulation to analyze the image properties.The fourth chapter deals with a class of third order two point boundary value problems with each order derivative in the nonlinear term by the new iterative method proposed in the third chapter.Since the third-order differential operator is a linear operator,we get the convergence rate linear estimation of the iterative solution,which can not be obtained in the p-Laplace boundary value problem.Finally,the example verification results are given.The fifth chapter is the summary and prospect of this thesis.