Impulsive Singular Second Order Differential Equations On The Half-line

The theory of impulsive di?erential equations is a new important branch of di?erentialequations. Because differential equations with impulses provide an adequate mathemat-ical model of many evolutionary processes that suddenly change their state at certainmoments, its theory can be applied to medicine and biology, optimal control models ineconomics, the dynamical system and other fields.The singular ordinary differential equations arise in the fields of gas dynamics, New-tonian Huid mechanics, nuclear physics, the theory of boundary layer, nonlinear opticsand so on. In the recent ten years the existence and uniqueness of positive solutions forsingular differential equations have been widely studied.The thesis is divided into two chapters. Applying upper and lower solution method,fixed point theorem,measure of noncompactness and fixed point index on a cone, theexistence, uniqueness and nonexistence for a class of second order impulsive singulardifferential equations on the half-line are discussed in this paper, and the impulsive effecton differential equations is also revealed . The contexts of this paper are presented asfollowing.In the first chapter, we consider the following initial value problem on the half linewith infinite impulses:Assuming the existence of the the upper and lower solutions, we transform the initialproblem into a kind of boundary value problems firstly. And then, using Schauder fixedpoint theorem, we establish the upper and lower solution method for boundary valueproblems with n fixed impulses in finite intervals . Finally, applying diagonalizationargument,we have proved the existence of bounded solutions with infinite impulses onhalf-line. The technique to deal with impulses is adequately exhibited in the proof ofcomparison principle .In view of the importance of the existence of upper solutions and lower solutions,both of such existence on half line are presented in §1.4 of this paper.Impulsive di?erential systems have a large di?erence from systems without impulses.In §1.5, in order to reveal impulsive e?ects on systems, we present the nonexistence ofsolutions of initial value problems for a class of second order impulsive di?erential equa-tions on the half line . In the virtue, there is not many investigations on the nonexistenceof impulsive di?erential equations so far.In the second chapter, we concern with the existence and uniqueness of solutions forthe following boundary value problem in a Banach space:By using compact conditions and Sadovskii theorem, the existence of solutions is firstlyestablished , in which the di?culties we should solve are the following: some properties ofoperator in continuous space should be extended to that in PC[R+,E] and the Corduneautheorem in PC[R+,E] should be established. Then, the uniqueness of solutions is provedby using Lipschitz conditions , as well as the existence of multiple solutions is presentedfixed point index theorem.In conclusion, by contrast with the ordinary di?erential systems without impulses,some existence theories of the impulsive systems come to be more complex because of thee?ect of infinite fixed impulses. Although the impulsive e?ect can be controlled by someconditions in some cases, it is possible that impulses lead to nonexistence of solutions,and multiple solutions .