Positive Solitions Of Boundary Value Problems For Nonlinear Ordinary Differential Equations

In nonlinear functional analysis,the boundary value problem is an extremely active and most researchable and valuable fields.Due to close relation with many problem in the aerospace engineering,physics,chemical,biological and other fields,the existence and multiplicity of solutions for the boundary value problems of differential equations have become an important research topic.And in applied and engineering practice,many mathematical models are usually differential equations.Therefore,it is important to study the boundary value problem of nonlinear differential equation.In this thesis,by using the methods of nonlinear functional analysis,several nonlinear boundary value problems are studies and some new results about the existence and multiplicity of solutions are obtained,which improve or extend some known results in literatures.The paper is divided into four chapters:In Chapter One,the background,significance and research status are introduced.Our work are also stated briefly.In Chapter Two,We mainly discuss the existence of positive solutions for secondorder integral boundary value problem(?)Where (?),By constructing the Green function,we used fixedpoint index theory to establish the existence and multiplicity of positive solutions for the above problem.In Chapter Three,We mainly discuss the existence of positive solutions for generalized Lidstone problems of nonlinear higher order ordinary differential equations.(?)Among them,the two equations studied in (?) can have different orders and their derivatives satisfy different boundary value conditions.Based on a priori estimates,the existence of positive solutions for the above boundary value problems is proved by using the fixed-point index theory.In Chapter Four,We mainly discuss the existence of positive solutions and the existence of multiple positive solutions for the second-order -Laplacian boundary value problem.(?)where φ : R+→ R+is either a convex or concave homeomorphism,and f∈C([0,1]×R2+,R+)(R+ := [0,∞)).Based on a priori estimates achieved by utilizing Jensen’s inequalities for concave and convex functions,we used fixed-point index theory to establish the existence and multiplicity of positive solutions for the above problem.