Boundary Value Problems Of Nonlinear Differential Equations And Applications

The ordinary differential equation has a widespread application in science and technology, many mathematical, physical, chemical and biological problems can be converted into the nonlinear problems, so various non-linear problems have aroused people's widespread attention. Among these problems, the impulsive and semipositone problems are the hot spot which have been discussed in recent years, and become two very important domains of differential equation research at present. There is a lot of research about the two domains, but the research of the impulsive and semipositone differential equations is very few. In this paper, we use the fixed point theory and the fixed point index theory to study several kinds of boundary value problems for nonlinear differential equations, and we apply the main results to the boundary value problems for the impulsive and semipositone differential equations.The thesis is divided into two chapters according to contents.In Chapter 1, we study the existence and multiplicity of positive solutions for systems of nonlinear second-order singular and impulsive differential equations.denote the right limit (left limit) of u_i~1(t) andu_i(t) at t = t_k, respectively. By constructing a cone K~n in the space C[0,1], the existence of one or two positive solutions is established by using the fixed point index theorem. This paper expands and improves some results in [11-14]. In Chapter 2, we will study the following fourth-order nonlinear singular semipositone system:where f,g∈C((0,1)×R~+×R~+×R~-×R~-,R),f, g may be singular at t = 0 and/or t = 1 and may take negative values, in which R~+=[0,+∞), R~-= (-∞,0]. By approximating the above singular semipositone system to a second-order singular system and using the well-known fixed point index theorem, we can get the existence of positive solutions of the two-point boundary value problems for the nonlinear fourth-order singular semipositone system. Finally, an example is given to demonstrate the application of our main results.