Solutions Of Some Differential Equations And Applications

The theory of impulsive differential is one of important branches of differential equations. In the field of modern applied mathematics, it has made considerable headway in recent years, because all the structure of its emergence has deep physical background and realistic mathematical model.The singular ordinary differential equations first arise in the fields of gas dynamics, Newtonian fluid mechanics,nuclear physics, the theory of boundary layer, nonlinear optics and so on. It is also one of important areas in the study.This thesis is divided into four chapters. In the first chapter and the second chapter, we study the existence of solutions of impulsive differential equations in Banach spaces, the main results derive from paper [36] and [37]. In the third chapter, and the fourth chapter , we study the existence of positive solutions of singular differential equations, and the main results derive from paper [38] and [39].In the first chapter, we study the initial value problems (IVP) of mixed monotone impulsive itegro-differential equations in Banach spaceswhere , E is real Banach spaces . denote the right and left limits of ,and the operators T. S are gives byIn the special case where in. the existence and uniqueness of solutions has been obtained by means of the lower and supper solutions and comparison results by [3] . But, it is easy to see that the comparison result, in [3] is not applicable inthe impulsive case. With the impulsive conditions, the paper [2] obtained by means of equivalent norm the existence of solutions and coupled mininal and maximal solutions of IVP(l.l). The paper [4] studied the existence of solutions by means of lower and supper solutions and compactness with f non-monotone and without Tu, Su, which the conditions are waker than the paper [5]. But the conditions of the paper [4] and [4] are also comparatively strong. Therefore, in the first chapter, we shall construct two new comparison results for IVP(1.1), and then we obtain an existence theorem of IVP(l.l) by means of the Monch fixed theorem, which conditions are wider than the paper [2] and [4].Now, we state the main results as follows:Lemma 1.5(comparison results) Letsatisfywhere M(t),N(t) are positive bounded and integrable functions. are constants. Satisfyingwhere Lebesgue integral function. are constants. Then m(t) = Let us list the following conditions for convenience.(H1) there exist which are the coupled quasi-solution of the quation (1.1).(H2) there are positive bounded and integrable functions M(t), N(t),,such thatwhere(H3) when such thatthere exist constants and Lebesgue integrable functions gi(t)(i=1,2,3,4),such that the following hold for all countable monotone sequences Theorem 1.1 Let E be a Bauach space and P be a regular cone in E. Assume that (H1)- (H5) are satisfied. Then IVP(l.l) possess solution x* and coupled minimal and maximal quasi-solutions Moreover existing the sequence un,vn converge in the ordered sector [u0,v0] to the coupled minimal and maximal quasi-solutions satisfyingRemark 1.1 In the paper[2]. (H4) require gi(t)(i = 1.2.3.4) being bounded and integrable functions. Yet. in this paper, we only need gi(t)(i=1.2.3,4) nonnegativeLebesgue integrable functions. When gi(t) are unbounded, the paper[2] can not answer. At the back of this paper, we give an example that can explain above.Remark 1.2 In [2] where also requireBut, when , [2] can do nothing. .In this paper, we completely delete the requirements. Moreover, if let N(t) = 0, Lk = 0, the theorem 1.1 is as same as the main theorem 1 of [2], which can also be given explanations by an example. Therefore, this paper has improved and generalized [2], and the method is different form [2].In the second chapter, we consider the initial value problem for nonlinear second order impulsive integro-differential equations in Banach space .where E is a real Banach space, , R denote real number Guo Dajun [14] construct the existence of solutions of IVP(2.1). Moreove...