Existence Of Positive Solutions Of Nonlinear Differential Equations In Abstract Spaces

Along with science's and technology's development, various non-linear problem 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 methods, and plays an important role in dealing with all kinds of nonlinear integral equations, nonlinear differential equations and partial differential equations. 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 several kinds of application discipline. It is one of most active domains of functional analysis at present. The nonlinear differential equation boundary value problem in Banach space 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.In this paper, we use the cone theory, the fixed point theory and the fixed point index theory, to study some kinds of boundary value problems for nonlinear impulsive difterential equation and we apply the main results to the boundary value problem for the differential equation.The thesis is divided into three chapters.In Chapter 1, we investigate the positive solutions of boundary problems for first order nonlinear impulsive differential equations on the half-line in Banach space. where(?) and (?) (?). We use the cone theory and cone expansion, compression fixed point theorem and fixed-point index theory to obtain the positive solutions for boundary value problem (1.1.1). This paper generalize and improve the results of [10, 13, 15](sec Remarks 1.3.1-1.3.4), and apply the main results to the infinite system of scalar first order impulsive equations.In Chapter 2, we talk about the positive solutions of two-point boundary problems for second order impulsive differential equationswhereα_i(t), b_i(t) : Jâ†'[0, +∞) arc continuous and I_i,k∈C[E. E], f_i,g_i∈C(J×E×E,E) (i=1,2),(?),k=1,2,…,m. J=[0,1].The fixed point index theory in cone are used to obtain the existence of positive solutions for boundary value problem (2.1.1) and generalize and improve the results in [18, 19, 20](see Remarks 2.3.1-2.3.4).In Chapter 3, we use the cone theory and cone expansion and compression fixed point theorem to investigate the positive solutions of nth-order m-point boundary value problem in Banach space:where n≥2. m > 2, 0 <η₁ <η₂ <…η_m-2 < 1,α_i > 0 (i = 1, 2,…, m - 2) and (?),J=[0,1].(?). Under the condtionds of f : [0. +∞)â†'[0. +∞) without term t and E is a real space, [30] and [31] give a sufficient condition of one positive solution for multi-point boundary value problem (3.1.1) using fixed point index and cone expansion and compression fixed point theorem. This papaer is to investigate the existence of at least one or at least two positive solutions of general multi-points BVP (3.1.1) in Banach space, we generalize and improve the results in [30] and [31].