【作者】 孙肖丽；

【导师】 闫宝强；

【作者基本信息】 山东师范大学 ， 应用数学， 2006， 硕士

【摘要】 在自然界中,许多事物的变化规律不仅依赖于当时的状态,还依赖于过去或将来某时刻或某时间段的状态,并且往往伴有瞬时突变现象,这些现象的数学模型可以用脉冲泛函微分系统来描述（[1]-[14]）。非共振泛函微分系统是其中一类常见的系统,在物理、生物、医学、控制论等领域都有着广泛的实际应用背景,因此对该系统的研究逐渐成为一个热点（[14]-[18]）。本文即利用非线性泛函分析理论研究了两类非共振脉冲泛函边值问题解的存在性。全文分为两章。 第一章中我们利用不动点指数理论讨论了非共振的含参数脉冲泛函边值问题 （?）多个正解的存在性。 相比于文[14],本文在加脉冲的同时将右端项f（t,u（t））推广到f（t,u（w（t）））,使[14]成为本章的特殊情况。在本章中我们利用上下解方法以及不动点指数理论得到如下结论:存在λ^*,λ^**>0,使当0<λ<λ^*时,方程（1）至少存在两个正解,而当λ>λ^**时,方程（2）无正解。其次我们又考虑了在w（t）=t情况下所得的更优的结论,包括f（t,u）在u=0处奇异和非奇异两种情况。在非奇异情况下,我们得到:存在0<λ^*≤λ^***,使当λ∈（0,λ^*）时,（1）至少有两个正解;当λ∈[0,λ^***]时,（1）至少有一个正解;而当λ>λ^***时,（1）无解。这时由于脉冲的影响,需要建立新的上下解方法, 第二章中我们通过建立相应的比较定理并运用单调迭代技巧研究了如下非共振的脉冲泛函边值问题解的存在性。更多还原

【Abstract】 In the nature, the law of development of many things relies on not only the state of the time, but also at some state that has gone or will come, and there usually are impulses. The mathematical model of these phenomenons can be described by functional differential systems（[1]-[14]） with impulses. Nonresonant differential systems have widely practical background in many fields, such as physics, biology, medicine and other areas and there were many results on this aspect in late years（[14]-[18]）. Nonlinear Functional analysis became an important mathematical branch from 1930s, and it provides useful tools for nonlinear problems coming from science and technology field（[1]-[9]）. In this paper, we study existence of solutions for boundary value problems of nonresonant impulsive functional differential equations using the theory of nonlinear functional analysis.In chapter one, we consider the following nonresonant impulsive functional equationsusing fixed point index.In this chapter, using upper and lower technique and fixed point index theory, we first prove that there is a λ^* > 0 and λ^** > 0 such that the above problem has at least two positive solutions if 0 < λ < λ^* and there are no solutuons if λ > λ^**. Our results improve the conclusions in [14]In chapter two, we present the existance of solutions for the following non-resonant impulsive functional boundary value problems’ -u" + f3u{t) = f（t, u{t）, ut), t e （0,1）, u（t） = <f>（t）, t€[-r,0];Aw|t=tl = Lu（ti）;Au’\t=tl = L*u’{tx);（2）The key difference between this chapter and chapter one is that the right end is not f（t,u（iu（t））） but f（t,u（t）^ut）. Namely, the nature of the system at the moment of t associated not only with the state of w（t）, but also the overall state from t — r to t. Monotone iterative technique has broad applications in periodical boundary value problems（[19]-[22]）, but it is rarely used to discuss impulsive functional boundary value problems（[23j）. In this chapter, we first establish the corresponding comparison theorem, and then we get the existence of solutions of （2）. At last, an example is worked out to indicate that our conditions is reasonable.更多还原