The Existence Of Solutions To The Boundary Value Problem Of A Nonlinear Differential Equation With A P-Laplacian

In this paper,we study the existence of positive solutions to the boundary value problem of a fourth-order di?erential equation with a p-Laplacian and the existence of solutions to the multi-point boundary value problem with a p(t)-Laplacian.Firstly,it is primarily shown,by the fixed point theorem of cone expansion-compression type with norm,that there exist at least one or two positive solutions to this problem under some conditions.Then,by Leray-Schauder degree method and the fixed point theorem,we prove the existence of solutions to the multi-point boundary value problem with a p(t)-Laplacian.The paper is organized as follows,In chapter 1,we outline the historical background,some essential concepts,and the main results of this paper.Some useful information about p-Laplacian and fixed point theorems,which will be applied in the following chapters,is also included.In chapter 2,we use the fixed point theorem of cone expansion-compression type with norm to prove the existence of solutions to the boundary value problem of a fourth-order di?erential equation with a p-Laplacian.Then,we show that under some conditions,there exist at least one or two positive solutions which depend on a positive parameter.Furthermore,we give a corresponding example to illustrate its validity.In chapter 3,we prove,by Leray-Schauder degree method and the fixed point theorem,that under given conditions,the solutions to the multi-point boundary value problem with a p(t)-Laplacian exist.Then,we provide two criteria to determine whether solutions to equations of this form exist,and give two additional examples.