A Class Of Nonlinear Ordinary Differential Equation Boundary Value Problem Of The Solution And Application

Nonlinear ordinary differential equation boundary value problems derive from applied mathematics, physics, chemistry, biology, medicine, economics, engineering, and many other scientific fields. In19th century, the rapid development of analysis mathematics promoted to research ordinary differential equation boundary value problem. In the20th century, important development of nonlinear functional analysis promoted the study of nonlinear ordinary differential equation boundary value problem. For initial value prob-lem, the study of first order differential equation is more fully, the study of boundary value problem and high order equation boundary value problem is much more difficult than second-order equation boundary value problem. A large number of nonlinear problems in practical problems need to depict the nonlinear differential equation. As a result, Devel-oped for decades in the nonlinear analysis using various advanced analysis tools study the existence of solutions of nonlinear boundary value problem and the solution of the exact number, and the applied to the specific practical problems is a very important research subject.This article is divided into four chapters. In the first chapter of this paper, we intro-duce the development and status of the nonlinear ordinary differential equation boundary value problem to be solved.In the second chapter, we consider the existence of positive solutions and exact number of solutions for the following second-order ordinary differential equations boundary value problem existence of positive solutions and exact number of solutions, where0<α<1<β, η>0is a parameter. Using the transform analysis skills and ideas, we obtain the above boundary value problems has no solutions if η>η*; the above boundary value problem has a positive solutions if η=η*; the above boundary value problems have two positive solutions yη,1(t),yη,2(t) if0<η<η*; withyη,1(t)<yη2(t).In the third chapter, we consider the existence of positive solutions for the following p-laplacian differential equations boundary value problem wherep∈(1,2],α>0,λ>0is a parameter, using the quadrature method. We prove the existence of multiple solutions of p-Laplacian forα≥0, and determine their exact number fora=0.In the fourth chapter, we consider the existence of positive solutions for integral boundary value conditions p-laplacian differential equations boundary value problemwhere(?)p(t)=|t|p-2t, p≥2,(?)q=(?)p-1,1/p+1/q=1,ω(t)∈L1[0,1]. Using the method of the degree theory, we prove the existence of positive solution to boundary value problems and the existence of multiple solutions.