Nonlinear Ordinary Differential Equation Boundary Value Problems And Applications

Nonlinear analysis is an important branch of mordern analysis mathe-matics, because it can explain all kinds of natural phenomena, more and moremathematicians are devoting their time to it. Among them, the nonlinearboundary value problem comes from a lot of branches of applied mathematicsand physics, it is at present one of the most active fields that is studied inanalyst mathematics.The present paper employs the cone theory, fixed point index theory andso on, to investigate the existence of positive solutions to boundary value prob-lem of several kinds of nonlinear differential equations. The thesis is dividedinto four sections according to contents.Chapter 1 is the introduction of this paper, which introduce the develop-ment of nonlinear functional analysis and the main results in this paper.In chapter 2 ,we consider the first-order periodic boundary value problemswith impulse:Here 0=t₁2<…m=N,λ∈R\{θ},N>0,I∈C(R,R), f:[0,N]×R→R is continuous on (t,x)∈[0,N]\{t₁,…,t_m}×R, f(t_k^-,x), f(t_k⁺,x)exist and f(t_k^-,x) = f(t_k,x).â–³x|_t=tk=x(t_k⁺)-x(t_k^-) where x(t_k⁺) (respectivelyx(t_k^-)) denote the right limit (respectively left limit) of x(t) at t = t_k.By using an effective operator: B : y→max{y,z₀} (here z₀ is a nonneg-ative function), we define an operator from cone to cone and then the fixedpoint index theorem can be applied. In this way, the existence of the positivesolutions to the boundary value problem are obtained.In chapter 3, we investigate the first-order periodic boundary value prob-lems with impulse fatherly and obtain the multiplicity results for the problem.In chapter4 ,we consider the following fourth-order singular semi-positone boundary value problem:Here f: (0,1)×[0, +∞)×[0, +∞)→[0, +∞) is continuous, and f(x) is allowedto be singular at t = 0 and t = 1. q:(0,1)→(-∞, +∞) is Lebesgue integrableand has finite singular points in [0,1].By the theory of fixed point and Monotone operator, We study the fourth-order singular semi-positone boundary value problem and obtain the existenceof C²[0, 1]∩C⁴(0,1) positive solution for the problem under certain hypotheses.