Singular Impulsive Differential Equations

The change of state abruptly is called impulse. Impulse is a common phenomenon in the technological field and in the area of mathematical modeling which can be transformed to impulsive differential system. The core character of this system is that it can both reflect how the change of state abruptly influences the state and reflect the basic discipline in the field. For example, this system can be used to design the absorption apparatus for space vehicle, the transfer device for satellite trajectory, the robot, neurotic network, Chaos Control, as well as the study of confidential communication. So the research on impulsive equations are of great value both in theory and application.In 1990s, there was some fundamental theory about impulsive differential equations [14]. After then, many researchers continued to study and develop the theory of these equations. Also there were some existence results under some conditions. For example, Dajun Guo[35], xilin Fu[36-37], Jiang Daqing[4],[6] in China and R.P.Agarwal,Donal O'Regan[18], Y.H.Lee. [19],[20],[22] abroad have done a lot of work. In these research work, some nonlinearities have singularities and some have not. Most methods they applied are fixed point theorem, fixed point index theory on cone, the method of upper and lower solutions and so on.The thesis contains three chapters. We will overcome the difficulties which lie in impulsive effect and singularities using the fixed point theory on cone and the method of upper and lower solutions and approximation technique. Thus we can get the existence of positive solutions of second order impulsive differential equations with singuar boundary value problems .In chapter one, we consider semipositone singular boundary value problems of second order impulsive differential equationwhere f∈C[(0,1)×(0,∞), (0,∞)], q∈C(0,1),I∈C[[0,∞), [0,∞)], and t₁∈(0,1) be given. Assume f may be singular at t = 0,1 or y = 0, I is continuous on [0,∞).In [4], Daqing Jiang, by upper and lower solutions, gave sufficient conditions for the existence of positive solutions to the semipositone singular boundary value problems of second order impulsive differential equationwhere f∈C[(0,∞), R], f(y) + M≥0, for any y∈(0,∞), q∈C(0,1)∩L¹[0,1],q(t)> 0, t∈(0,1), I_k∈C[[0,∞), [0,∞)], t_k∈(0,1).This paper gives new conditions , and gets positive solutions of problem (1) by using the theorem of fixed point index ([34]). We improve the conditions in [4] f(y) + M≥0 and find that q(t) may be unboundary function in problem (1).In chapter two, we consider the following singular boundary value problemwhereμ> 0, q(t) > 0, t∈J, f(t,x,y)≥0, (?)(t,x,y)∈D_A, and may be singular at x = 0, A, y = 0, and he function I∈C[[0,∞), [0,∞)] continuous and nondecreasing , and 0≤I(x) a,λis positive real parameter, f : R→[0,∞) is continuous, q : (0,1)→(0,∞) is continuous and singular at t = 0, and/or t = 1. Using fixed point theorem, Eun Kyoung Lee and Yong-Hoon Lee got sufficient conditions for the existence of positive solutions.The difference between the above equation and the one we consider is that there is some effect by the first derivative which may lead the equation (2) to have no solutions. In this chapter we will give sufficient conditions for the existence of positive solutions on the set (?)_A for the problem (2), whenμ, nonlinearity function f and q satisfied some given conditions. The existence theorem of positive solutions for problem (2) is based on nonsingular and sequential techniques. We construct 2-parameter family of nonsingular BVPs and obtain a priori bounds on solutions (Lemma 2.2.1). Using these bounds , we can apply the topological transversality theorem(see,e.g., [10,13]) for the existence of solutions of auxiliary BVPs(Lemma 2.2.2). In addition, we give lower bounds for these solutions (Lemma 2.2.3). The existence results on orignal BVP (2) follows by the Arzela - Ascoli Theorem.In chapter three , we study the following singular boundary value poroblem which may be change signwhere f(t, x, y) may be singular at t = 0, 1, x = 0, and may change sign, and the funtion I continuous and nondecreasing, and 0 < I(x) < x, x∈[0,∞), I∈C[[0,∞), [0,∞)]. This is different from the chapter two , when f may change sign here , and maybe singular only at t = 0, 1, x = 0. In the above two chapers, we get sufficient conditions for the existence of positive solutions by fixed point theorem . In this chapter we will spread the method of upper and lower solutions to study the problem (3). We get solutions of operators sequence by the method of upper and lower solutions, and using the Theorem of Arzela - Ascoli to get a positive solution of impulsive equation.