The Existence Of Solutions Of Several Of Classes Of Different Systems

In recent years, because of the high practical value in the fields of gas dynamics fluid mechanics, the theory of boundary layer, nonlinear optics and so on, singular boundary value problem becomes one of the important problems that attract the attention of mathematicians and other technicians gradually. Along with the problem study thoroughly, the method of upper and lower solutions, topological degree and cone theory or method of approximation were gradually used to demonstrate the existence results of positive solution of singular boundary value problem. This paper attempts to make use of fixed point theorem, fixed point index, cone compression and expansion fixed point theorem, upper and lower solutions, Leggett-Williams theorem to discuss such problems more generally on the basis of above references.Chapter 1 investigates the existence of positive solutions for the following singular boundary value problems of second ordinary different systems. where f(t,x(t),y(t)),g(t,x(t),y(t)) is singularity at t=0,1,X=0,y=0, andα1,β1,γ1,δ1,α2,β2,γ2,δ2≥0,ρ=αiγi+αiδi+βiγi>0,i=1,2. By constructing an approximating of (1.1.1) and using the fixed point theorem, we get the existence of multiple solutions. In chapter 2, we consider the existence of positive solutions to the following sys-tems: where p,q is positive integer.f, g may be singular at t=0,1 and u=0,v=0, by using Schauder theorem and fixed point theory in cone. And we also consider the solutions with f, g singular at u=0,v=0, by using fixed index theory.In chapter 3, we study the 2n-order boundary value problems on the basis of chapter 2. where f∈C((0,1)×(0,+∞), [0,+∞)), g E C([0,1]×[0,+∞), [0,+∞)). By construct-ing upper and lower solutions and using fixed point theorem, a sufficient conditions for the existence of positive is established.Problems with integral boundary conditions arise naturally in thermal conduc-tion problems, semiconductor problems, hydrodynamic problems, so In chapter 4, we consider the equations with integral boundary conditions: where f1,f2∈C([0,1]×[0,+∞),[0,+∞)),g01(s),g11(s),g02(s),g12(s)∈C([0,1],(0,+∞α1,α2,β1,β2 are nonnegative real parameters. We obtained the existence of at least three positive solutions of the equations. The proof of our main results are based upon the Leggett-Williams fixed point theorem. Finally we give an example to demonstrate our result.