Nonlinear Differential Equations

In recent years, the rapid development of Applied Mathematics, especially the research and analysis in-depth application of mathematics,which makes use of various disciplines of the mathematical content is more speci?c, actionable. At present, the non-linear problem to obtain different using different degrees in various disciplines, such as economics,control theory,chemistry, mathematics, physics. People especially attention to these nonlinear problem and its application, and even become the research direction of the discipline of modern mathematics. However, the nonlinear differential equations boundary value problems is widely discussed by the mathematical researchers. We mainly used the theory of?xed point and cone theory to analyzes the existence of positive solutions for some nonlinear differential equations.The thesis is divided into four sections according to contents:Chapter 1 Preference, we introduce the overall background and the calculus on time scales of this paper.Chapter 2 In this chapter, we consider the following boundary value problem?-u′′(t) = f(t, v), 0 < t < 1,v(4)(t) = g(t, u), 0 < t < 1,u(0) = u(1) = 0,v(0) = v(1) = v′′(0) = v′′(1) = 0.(2.1.4)Where f, g ∈ C([0, 1] × [0, +∞], [0, +∞]), f(t, 0) ≡ 0, g(t, 0) ≡ 0We use the cone theory and cone expansion and compression ?xed point to obtain the positive solutions for boundary value problem(2.1.4). This chapter generalizes and improves the results of [7].Chapter 3 In this chapter, we mainly consider the two-point boundary value problems of following mixed type singular semipositone differential equations- u′′(t) = f(t, v), 0 < t < 1,v(4)(t) = g(t, u), 0 < t < 1,u(0) = u(1) = 0,v(0) = v(1) = v′′(0) = v′′(1) = 0,(3.1.1)Where f ∈ C((0, 1)×[0, +∞), R), g ∈ C((0, 1)×[0, +∞), R+), with f 在 t =0, t = 1 singular.We get the positive solution to the equations depended on the compression ?xed point and cone theory. This chapter generalizes and improves the results of the last chapter.Chapter 4 In this part, we mainly consider the two-point boundary value problem of the following mixed singular semipositone system of differential equations.- u′′′(t) = f(t, u, v), 0 < t < 1,v′′(t) = g(t, u, v), 0 < t < 1,u(0) = u′(0) = u′(1),v(0) = v(1) = 0,(4.1.3)Where f, g ∈ C((0, 1)×[0, +∞)×[0, +∞), R+), 且f(t, 0, 0) = 0, g(t, 0, 0) =0.