The Existence Of Solutions For Singular Boundary Value Problems With The Derivative In Nonlinearities

In recent years, because of the high practical value in the fields of astronomy,fluidmechanics, engineering mechanics, biology,economics and so on, singular boundaryvalue problem with the derivative in nonlinearities becomes one of the important prob-lems that attracts the attention of mathematicians and other technicians gradually.Along with the problem study thoroughly, the method of upper and lower solutions, thevariational method, topological degree and cone theory were gradually used to demon-strate the existence results of positive solutions of singular boundary value problemwith the derivative in nonlinearities.This paper deeply discusses the existence of solutions for singular boundary valueproblems with the derivative in nonlinearities by making use of fixed point indextheory,cone compression and expansion fixed point theorem, upper and lower solutionsmethod.The dissertation contains four chapters:In Chapter 1, we consider the existence of singular semipositone boundary valueproblem of impulsive diferential equationsWhere 0 = t₀< t₁< t₂... < t_m< t_m+1= 1, I = [0, 1], J = （0, 1）, R⁺= [0, +∞);f∈C（J×R⁺×R, R） may be singular at t = 0, t = 1; J_k∈C（R⁺, R⁺）, k = 1, 2...m,â–³x （t_k）denotes the jump of x （t） at t = t_k, i.e.â–³x （t_k） = x （t⁺k） x （t^-k）, in which x （t⁺k）,x（t^-k） denote the right and left limits of x（t） at t = t_k respectively. Byconstructing a special cone and using the fixed point index theory,we get the existenceof one and two positive solutions. In chapter 2, we investigate the existence of positivesolutions of third order singular boundary value problem with parametersWhereλis a positive parameter; h : （0, 1）â†'[0, +∞), f : [0, 1]×[0, +∞)×[0, +∞)â†'[0, +∞) are continuous funtions and h（t） may be singular at t = 0, t = 1; a, b≥0with 1 + a > b > 1. By using cone compression and expansion fixed point theorem, wepresent the existence of one,two,even infinitely many positive solutions of boundaryvalue problems.In chapter 3, we study the existence of solutions of fourth order singular boundaryvalue problemWhere f : （0, 1）×R4â†'R is a continuous function and may be singular at t =0, t = 1, b≥a≥0.Here R = （∞, +∞）. By using the method of upper and lowersolutions and the fixed point theorem of Schauder, we get the existence of one andthree solutions.In chapter 4, we consider the existence of positive solutions for the followingsingular boundary value problems of fourth order diferential systemsWhere f1, f2∈（0, 1）×[0, +∞)×[0, +∞)×(∞, 0]×（∞, 0]â†'[0, +∞） are continuousfunctions and may be singular at t = 0, t = 1. What is more, f1, f2involve the second derivative.By using cone compression and expansion point theorem, we obtain theexistence of one and two positive solutions.