Existence Of Positive Solutions Of Nonlinear Impulsive Differential Equations And Applications

Along with science's and technology's development, various non-linear problem has aroused people's widespread interest day by day, and so the nonlinear analysis has become one important research directions in modern mathematics. The nonlinear functional analysis is an important branch in nonlinear analysis, because it can explain well various the natural phenomenon. The boundary value problem of nonlinear differential equation stems from the applied mathematics, the physics, the cybernetics and each kind of application discipline. Nonlinerar analysis offers effective tools for solving those probles. It is one of most active domains of functional analysis studiesin at present. The nonlinear differential equation boundary value problem 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. The existence of positive solutions is one of the most imporant thesis of these problems.In this paper, we use the cone theory, the fixed point theory. the topological degree theory as well as the fixed point index theory and combined with a iterative technique, to study several kinds of boundary value problems for nonlinear differential equation and we give out the corresponding appplications.apply the main results to the boundary value problem for the singular integral differential equation.The thesis is divided into four chapters.In chapter 1, we use the cone expansion and compression theorem and cone theories to talk about positive solutions of a class of second order boundary value problem with integral boundary condition in Banach space:where g(s),h(s)∈L¹[0,1], are nonnegativc.J=[0,1], J¹=J\{t₁.t₂,...,t_m}. Under the condition of f_i having different linearity, we not only talk about the existence of positive solutions but also talk about the nonexistence of positive solutions of BVP(1.1.1). The equation in [4]only has one integral boundary condition(the case when h(s)=0 in BVP(1.1.1)), and it doesn't contain the impulsive terms(the case when I_k=0 in BVP(1.1.1)) ,and the author doesn't consider about the situation of f having different linearity. In this paper, the main results is theorem 1.3.1-theorem 1.3.5, comparing to in [4, 6], the problems in this paper are more general and comprehensive, the method we used is different. and the results we get are more profound(Remark 1.3.1). At the same time, we apply the main results to second order impulsive differential equation with integral boundary conditions.In chapter 2, we use fixed point index and some cone theories to discuss the existence of positive solutions of singular second order three -pint impulsive differential equation:both u and u¹ have jumping points. We obtained the at least one or at least two positive solutions of (2.1.1) using the cone expansion and compression theorem and some theories in cone. The main results in this paper is theorem 2.3.1-theorem 2.3.3, we improve and generalize the main results in [18, 19] in essence (Remarks 2.3.1-2.3.4).In chapter 3, we use the cone expansion and compression theorem, discuss the existence of positive solution of the following three-point third order impulsive differential equation:abstract space and under the conditions of f_i have different linearity, we talk about the existence of positive solutions of BVP(3.1.1) using the cone expansion and compression theorem. The main results in this paper is theorem 3.3.1-theorcm 3.3.2. We generalize and improve the main results in [33, 36](Remarks 3.3.1-3.3.3).