Boundary Value Problems Of Nonlinear Differential Equations In Abstract Space

Along with science's and technology's development, various non-linear prob-lem has come up from the fields of physics, chemisty, mathematics, biology, medicine, economics, engineering, cybernetics, and these problems has aroused people's widespread attention day by day. However, the nonlinear functional analysis offers effective theoretic tools for these problems, and it is a subject of profound theories and broad applications. The nonlinear functional analysis bases on nonlinear problems of math and science, constructs general theories and meth-ods, and plays an important role in dealing with all kinds of nonlinear integral or differential equations and partial differential equations. So the nonlinear analy-sis has become one important research directions in modern mathematics. With the development of science and technology, there appears all kinds of nonlinear problems in the fields of physics, chemistry, biology, economics, engineering etc. Numerous nonlinear problems involving gas dynamics, fluid mechanics, boundary layer theory etc, can be described by ordinary differential equations with singular-ity and in abstract space it is also the hot spot which has been discussed in recent years. So it become a very important domain of differential equation research at present. With the P-Laplace operator boundary value problem have a wide range of applications, in the non-Newtonian mechanics, space physics, plasma problem, elasticity, gas vortex, astrophysics and the P-Laplacian of the radial symmetric solution of practical problems and theoretical studies. And there are many re-search results about the P-Laplace operator boundary value problem, however, we find few results in abstract spaces. Therefore, it is necesary to do some research on diferential equation boundary problem with P-laplace operator.In this paper, we use the cone theory and the fixed point theory to study some kinds of boundary value problems for nonlinear differential equation and we apply the main results to the boundary value problem for the differential equation.The thesis is divided into two chapters.In Chapter 1, we obtained a class of m-point boundary value problem exis-tence results of the nonlinear second order singular differential equations con-taining first order derivative in abstract spaces, using the properties of non- compactness measure and Sadovskii fixed point theorem of the generalized con-densing mapping. whereθis zero element in E, is continuous and h(t) is singular at t=0,1.In Chapter 2, we talk about the solutions of boundary problems for P-Laplace differential equations whereθis zero element in E, andφp(s)=‖s‖p-2 s for p> 1.