Existence And Uniqueness Of Solutions For Second Order Singularity Differential Equations With Integral Boundary Value

Nonlinear functional analysis is a subject in applied mathematics that have both profound theory and widespread application. It takes the nonlinear problems appearing in natural sciences and mathematics as background to establish some general theories and methods to handle nonlinear problem.In recent decades, the differential equations with integral boundary value conditions produced in physics, mathematics and engineering, etc.. Recently, many authors established the existence and uniqueness of the solution,which is a very important ?eld in the study of the differential equation. In this article mainly uses Banach contraction mapping principle and the mixed monotone operator theorem to study the existence and uniqueness of positive solutions of the nonlinear differential equations with integral boundary conditions. We also use the corresponding example to show the validity of the theorem.The thesis is divided into three chapters according to contents.In Chapter 1, by using the cone theory and Banach contraction mapping principle, the existence and uniqueness theorem of ?xed points for a kind of two-component operator in ordered Banach spaces are investigated under more general condition. As an application, an existence and uniqueness theorem of solutions for mixed boundary value problems of second order nonlinear singular equations in Banach spaces is given.?In Chapter 2, by using the cone theory and Banach contraction mapping principle, research for a class of second order singular differential equations with integral boundary conditions, under the more general condition to get the existence and uniqueness of solutions of equations and applied to speci?c examples.?where λ > 0 is a parameter, α, γ ≥ 0, β, δ > 0 are constants such thatρ = αγ + αδ + βγ > 0, and the integrals in(2.1.1) are given by Stieltjes integral with a signed measure, where ξ, η are suitable functions of bounded variation. f :(0, 1) ×(0, ∞) ×(0, ∞) �' [0, ∞) and f ∈ L1[0, 1], f(t, x, x)may be singular at t = 0, t = 1 and x = 0. Furthermore f(t, x, y) is mixed non-monotone about x, y.In Chapter 3, using the generalized theory of the mixed monotone operator to get the existence and uniqueness of positive solutions of equation(3.1.1) and its dependence on the parameter λ under certain conditions.where λ > 0 is a parameter, α, γ ≥ 0, β, δ > 0 are constants such thatρ = αγ + αδ + βγ > 0, and the integrals in(2.1.1) are given by Stieltjes integral with a signed measure, where ξ, η are suitable functions of bounded variation. f :(0, 1) ×(0, ∞) ×(0, ∞) �' [0, ∞) is continuous, f(t, x, x) may be singular at t = 0, t = 1, and x = 0. Furthermore f can be written as f(t, u(t), u(t)) = g(t, u(t), u(t)) + h(t, u(t)).