【摘要】：在科學(xué)研究的應(yīng)用中,微分方程作為一種重要工具在物理,化學(xué),生物中扮演著重要的角色,其中脈沖微分方程是微分方程的一個重要分支.本文主要討論三類脈沖微分方程正解的存在性,其中有兩類脈沖微分方程帶有時滯現(xiàn)象.脈沖現(xiàn)象和時滯現(xiàn)象在現(xiàn)代科技各領(lǐng)域的實際問題中是普遍存在的.相比沒有脈沖和時滯的微分方程而言,脈沖微分方程和時滯微分方程能更真實地反映事物的發(fā)展過程,其最突出的特點是能夠充分考慮到瞬時突變和時間延滯現(xiàn)象對狀態(tài)的影響,能夠更精確、更深刻地反映事物的變化規(guī)律.本文主要分為四章,具體內(nèi)容安排如下:第一章,概述脈沖微分方程的歷史背景和研究現(xiàn)狀.第二章,研究二階脈沖微分方程周期邊值問題正解的存在性,本章主要通過假設(shè)f和IK是遞增的條件下,通過Leggett-Williams不動點定理,證明所給系統(tǒng)至少存在三個正周期解.第三章,研究一類帶有時滯的一階脈沖微分方程,主要通過轉(zhuǎn)化技術(shù)把脈沖微分方程轉(zhuǎn)化為非脈沖微分方程,再運用不動點定理去建立時滯脈沖微分方程正周期解的存在性.在一定的條件下證明所給系統(tǒng)至少存在兩個正周期解,使得研究結(jié)果得到進(jìn)一步完善.第四章,研究一類二階時滯脈沖微分方程邊值問題,二階時滯脈沖微分方程往往在求解過程中由于其脈沖性而較為繁瑣,本章借助轉(zhuǎn)化技術(shù)將二階時滯脈沖微分方程轉(zhuǎn)化為二階時滯微分方程,并通過錐拉伸與錐壓縮不動點定理,研究所給系統(tǒng)正解的存在性.
[Abstract]:In the application of scientific research, differential equations play an important role in physics, chemistry and biology as an important tool, among which impulsive differential equations are an important branch of differential equations. In this paper, we discuss the existence of positive solutions for three kinds of impulsive differential equations, in which two kinds of impulsive differential equations have delay phenomena. Pulse phenomenon and delay phenomenon are common in various fields of modern science and technology. Compared with differential equations without impulses and delays, impulsive differential equations and delay differential equations can more truly reflect the development of things. Its most outstanding feature is that it can fully consider the influence of transient abrupt change and time delay on the state, and can more accurately and profoundly reflect the changing law of things. This paper is divided into four chapters. The main contents are as follows: chapter 1, the historical background and research status of impulsive differential equations. In chapter 2, we study the existence of positive solutions for periodic boundary value problems of second order impulsive differential equations. In this chapter, we prove that there are at least three positive periodic solutions for the system under the assumption that f and IK are increasing, and by Leggett-Williams fixed point theorem. In chapter 3, a class of first order impulsive differential equations with time delay is studied. The impulsive differential equations are transformed into nonimpulsive differential equations by transformation technique, and the existence of positive periodic solutions of delay impulsive differential equations is established by using fixed point theorem. Under certain conditions, it is proved that there are at least two positive periodic solutions for the given system, which further improves the research results. In chapter 4, the boundary value problems of a class of second order delay impulsive differential equations are studied. In this chapter, the second order delay impulsive differential equation is transformed into the second order delay differential equation by means of transformation technique, and the existence of positive solutions of the system is studied by using the fixed point theorem of cone stretching and cone contraction.
【學(xué)位授予單位】：廣西師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175.8

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相關(guān)期刊論文 前1條

1 ;Periodic boundary value problem for the first order functional differential equations with impulses[J];Applied Mathematics:A Journal of Chinese Universities(Series B);2009年01期

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本文編號：2199317