摘要
利用不动点指数理论和上下解方法,本文主要研究了二阶奇异微分方程周期边值问题正解的存在性以及二阶脉冲微分方程三点边值问题极值解的存在性,给出了解存
在的充分条件.全文共分三章:
第一章简要介绍了微分方程和脉冲微分方程多点边值问题研究的背景和研究现状以及本文的主要工作.
第二章研究了具变号非线性项的二阶奇异微分方程周期边值问题正解的存在性.利用锥上的不动点指数理论,并借助于其相应的线性周期边值问题解的一种新的积分表示以及特殊的技巧,得到了此类问题存在正解的充分条件,接着又给出了正周期解存在的结果,所得结果改进了已有的部分结论.作为应用,我们给出了具体的例子.
第三章研究了具偏差变元的脉冲微分方程在非线性的三点边界条件下解的存在性.首先,我们给出了几个新的比较结果,并基于此引入了一种更广泛的上下解定义.然后,利用上下解方法和单调迭代技巧,得到了此类问题存在极值解的充分条件.将已有的常微分方程三点边值问题的相应结果推广到脉冲微分方程.为说明结论的可行性,我们给出了具非线性边界条件的实例.
This thesis is mainly concerned with the existence of positive solutions of singular sec-ond order periodic boundary value problems with sign-changing nonlinearities and extremalsolutions of three-point boundary value problems for second order impulsive differentialequations with deviating arguments. Some results are obtained by virtue of the fixed pointindex theory and the method of lower and upper solutions coupled with monotone itera-tive technique, respectively, which either extend or complement some previously obtainedresults.
It consists of three chapters.
As the introduction, in the first chapter, the background and history of two-point andthree-point boundary value problems of ordinary and impulsive differential equations arebrie?y addressed and the main work of this paper are given.
In Chapter 2, by using the fixed point index theory, we discuss the existence of positivesolutions of singular second order periodic boundary value problems with sign-changingnonlinearities. With the help of a new integral representation of the solution of the associatedlinear periodic boundary value problems, we obtain some sufficient conditions guaranteeingthe existence of at least one positive solution followed by one existence result of positiveperiodic solutions. Finally, we give some examples to illustrate the validity of our results.
Chapter 3 deals with the existence of extremal solutions of three-point boundary valueproblem for second order impulsive differential equations with deviating arguments andnonlinear boundary conditions. Some new comparison results under weaker conditions aregiven, which allow us to introduce new concept of lower and upper solutions. Meanwhile,the monotone iterative technique is employed to establish some sufficient conditions on theexistence of solutions of this problem. Our results extend part of previous results of thecorresponding ordinary differential equations. Moreover, some examples with nonlinearboundary conditions are presented.
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