Some Problems On Nonlinear Singular Ordinary Differential Equations

With the great development of science and technology, all kinds of nonlinear problems have been brought out in many fields such as physics , chemistry , mathematics , biology , medicine , economics , cybernetics and other system. People have paid more and more close attention to these nonlinear problems for their deep applied background and mathematical significance. At present, nonlinear functional analysis has been one of the most important branch of learning in modern mathematics. It provides an effect theoretical tool for studying many nonlinear problems, mainly including topological degree method , variational method , partial ordering method and analytic method and so on. There are many excellent monographs on nonlinear functional analysis (see [1.5.6]).Many nonlinear problems can be studied by converting them to corresponding differential equation, integral equation, integral-differential equation, difference equation, operator equation and other type of equation. The singular ordinary differential equations arise in the fields of gas dynamics. Newtonian fluid mechanics, nuclear physics, the theory of boundary layer. nonlinear optics and so on. The existences of positive solutions and multiple solutions for singular differential equations have been considered extensively in scalar case. In recent ten years, the theory of ordinaiy equations in abstract space has become a new and important branch (see [4.11.12]). Up to now, only a few papers consider singular differential equations in abstract space. The theory of impulsive differential equations has been emerging as an another new important area of investigation in recent years (see [8,71]),which has been developed rapidly in modern applied mathematics. This theory is a lot richer than the theory of differential equations since it represents a natural framework for mathematical model of several real world phenomena. However, the corresponding theory for impulsive integro-differential equations in Banach spaces is yet to be developed. One of the goals of this thesis is to devote ourselves to the differential equations which permit singularities and impulsive in abstract space.The fixed point theory of increasing operator and the method of upper and lower solutions based on it are one of the basic methods in the study of various nonlinear problems. Traditionally. the corresponding nonlinear operator are requested to be increasing when we use these method. It's worth deep researching not only in the theory but also in the application for how to use the method of upper and lower solutions. The definition of limit-increasing operator is put forward to delete increasing condition of the operator in [77] and the existence results of solutions for operator equation satisfying some conditions are also obtained by using the method of upper and lower solutions. One of the goals in this thesis is to give a further study on limit-increasing...