【摘要】：本篇論文研究了幾類微分方程、差分方程的振動性與非振動性,主要圍繞二階時滯動力方程、二階Emden-Fowler微分方程、二階差分方程、二維差分方程組、四階差分方程的振動性與非振動性展開研究,修正并完善了文獻中的一些已有結(jié)果,并建立了若干新的判定準(zhǔn)則,推廣了微分方程、差分方程的振動性與非振動性理論的一些已有結(jié)果.第一章介紹了本文研究問題的背景以及相關(guān)進展,并簡要敘述了本文所做的工作.第二章運用廣義Riccati變換等方法研究了一類時間尺度上二階非線性時滯動力方程的振動性,推廣了一些文獻中的已有結(jié)果.本章所研究方程如下.其中γ ≥ 1是兩個正奇數(shù)的商,并且滿足下面條件:本章,我們作如下假設(shè):定理0.0.1.假設(shè)條件(2),(3)及γ≥1成立.如果存在某個正常數(shù)k∈(0,1)使得成立,那么方程(1)在[t0,∞)T是振動的.推論0.0.1.假設(shè)(2),(3)和γ≥1成立.如果存在某個正常數(shù)k∈(0,1)使得成立,那么方程(1)在[t0,∞)T是振動的.定理0.0.2.假設(shè)條件(2)和(3)成立,且存在函數(shù)σ(t)≥t使得當(dāng)γ ≥ 1時,有成立,或當(dāng)0γ1時,有成立,那么方程(1)在[t0,∞)T是振動的.我們引入下列記號:定理0.0.3.假設(shè)條件(2)和(3)成立,并且Toσ = σoT.假設(shè)存在某個正常數(shù)k∈(0,1)和一個函數(shù)使得其中那么方程(1)在[t0,∞)T是振動的.定理0.0.4.假設(shè)條件(2)和(3)成立,并且τoσ = σoτ.假設(shè)存在函數(shù)H,h ∈ Crdr(D,R),D≡{(t,s)∈ T2:t≥s≥ t0}和一個函數(shù)δ(t)∈Crd([(t0,∞)T,R),同時,H有一個關(guān)于s的非正的連續(xù)偏導(dǎo)數(shù)H△s(t,s),并且滿足且成立,其中 那么方程(1)在[t0,∞)T是振動的.第三章利用Riccati變換法研究了 一類中立時滯微分方程的振動準(zhǔn)則,推廣了 一些已有的Emden-Fowler微分方程的振動結(jié)果.文章對α,β的大小分情況討論,并建立不同的權(quán)函數(shù),權(quán)函數(shù)的構(gòu)造基于所研究方程的特點.文章得到了如下的二階Emden-Fowler微分方程的振動準(zhǔn)則,并給出相應(yīng)的例子.其中,是兩個奇數(shù)的商.本章,我們假設(shè)下列條件成立:主要結(jié)果如下.定理0.0.5.如果0α≤β,并且存在函數(shù)p ∈C(t0,∞))使得那么方程(4)振動.定理0.0.6.如果αβ0,并且存在一個函數(shù)p ∈(C([t0,+∞))使得那么方程(4)振動.第四章包括兩部分內(nèi)容.利用差分的定義和相關(guān)公式以及不等式技巧研究了二階線性差分方程和一階二維線性差分方程組的振動準(zhǔn)則.改進并推廣了文獻中的已有結(jié)果,并給出相應(yīng)的例子.本章研究如下二階差分方程的振動準(zhǔn)則.本章我們使用如下記號:定理0.0.7.令q ≤ 1/4.如果存在常數(shù)α ∈[0,1)使得那么方程(5)是振動的.定理0.0.8.令px(0)≤1/4和q≤1/4.如果存在常數(shù)α ∈[M2,1)使得那么方程(5)是振動的.本章我們總是假設(shè)總是成立的.定理0.0.9.假設(shè)g。成立,并且存在λ ∈[2,+∞),使得那么方程組(6)振動.推論0.0.2.令 成立,假設(shè)那么方程組(6)振動.第五章我們給出了四階差分方程非振動的若干充分條件.本章研究如下四階差分方程的非振動準(zhǔn)則,假設(shè)z(n)= △y(n),則上面方程等價于即等價于下列方程假設(shè)有下列條件:(A8)1+P1(n)-p3(n)0;(A9)1+2P1(n)-p2(n+1)0;(A10)1+p1(n)+-p2(n)0;(A11)P2(n)0;(A12)p1(n)-2.定理0.0.10.如果p1(n),p2(n)和P3(n)滿足下面條件中的一個,則方程(8)是非振動的.(ⅰ)(A1),(A2),(A3);(ⅱ)(A11),(A2),(A4),(A8),(A10);或(A1),(A2),(A4),(A8),(A10);(ⅲ)(A12),(A4),(A5),(A6),(A10);(ⅳ)(A1),(A2),(A3),(A7),(A9);(ⅴ)(A1),(A2),(A7),(A8),(A9),(A10);(ⅵ)(A1),(A2),(A10);(ⅶ)(A1)(A1),(A10),(A11);(ⅷ)(A1),(A5),(A11);(ⅸ)(A1)(A5),(A6),(A10),(A11).注 0.0.1.將關(guān)系式P1(n)= a(n + 1)+ b(n + 1)-3,P2(n)= 3-2a(n + 1)-b(n +1)+c(n + 1),p3(n)= a(n+ 1)-1 代入條件(A1)-(A12)中,可以導(dǎo)出a(n),b(n),c(n)的不等關(guān)系式,作為方程(7)的非振動性的判定條件.第六章介紹了本文結(jié)論的意義、創(chuàng)新點及研究前景.
[Abstract]:In this paper, the vibrational and non oscillatory properties of several differential equations are studied. The vibration and non oscillations of the two order delay dynamic equations, the two order Emden-Fowler differential equations, the two order difference equations, the two dimensional difference equations, the four order difference equations are studied, and some existing results in the literature are modified and perfected. A number of new criteria are established to extend some existing results of the theory of oscillation and non oscillation of differential equations and differential equations. The first chapter introduces the background and related progress of this paper, and briefly describes the work done in this paper. In the second chapter, a class of time scale is studied by using the generalized Riccati transform. The oscillations of the upper two order nonlinear time delay dynamic equations have been generalized. The equations studied in this chapter are as follows. In this chapter, gamma > 1 is a quotient of two positive odd numbers and satisfies the following conditions: in this chapter, we assume the following hypothesis: theorem 0.0.1. hypothesis conditions (2), (3) and gamma > 1 if there is a certain normal number k 0,1 ( Set up, then the equation (1) T is vibrational in [t0, infinity. Inference the 0.0.1. hypothesis (2), (3) and gamma > 1. If a normal number k (0,1) is established, then the equation (1) in [t0, infinity is vibrational. Theorem 0.0.2. assumes that conditions (2) and (3) are established, and the existence of a function (T) > t makes it set up when gamma > 1, or when 0 gamma 1 is established, and that Me Fangcheng (1) is vibrational in [t0, infinity T. We introduce the following notation: theorem 0.0.3. hypothesis conditions (2) and (3), and To sigma = oT. hypothesis that there is a normal number k (0,1) and a function that make the equation (1) T in [t0, infinity are vibrational. Theorem 0.0.4. hypothesis (2) and (3) are established, and tau o sigma = sigma o tau. Suppose existence function H Crdr (D, R), D D {(T, s), T2:t > s > t0}, and a non positive continuous partial derivative, which is satisfied and established, in which the equation (1) is vibrational. In Chapter third, the oscillation of a class of neutral delay differential equations is studied. This paper generalizes the vibration results of some existing Emden-Fowler differential equations. The paper discusses the size of alpha and beta, and sets up different weight functions. The construction of the weight function is based on the characteristics of the equation. The following two order Emden-Fowler differential equations are obtained, and the corresponding examples are given. Among them, two This chapter, in this chapter, we assume the following conditions to be established: the main results are as follows. Theorem 0.0.5. if 0 alpha < < beta, and there is a function P C (T0, infinity)), then the equation (4) vibrate. Theorem 0.0.6. if alpha beta 0, and there is a function p (C ([t0, + infinity)) so that the equation (4) vibrate. The fourth chapter includes two parts. Use the difference. The definition, related formulas and inequality techniques are used to study the two order linear difference equations and the first order two-dimensional linear difference equations. The existing results in the literature are improved and extended, and the corresponding examples are given. This chapter studies the following two order difference equations. We use the following notation in this chapter: theorem 0.0.7. order q The equation (5) is vibrational. Theorem 0.0.8. makes PX (0) less than 1/4 and Q < 1/4. if there is a constant alpha [M2,1) so that the equation (5) is vibrational. In this chapter, we always assume that the equation is always established. Theorem 0.0.9. assumes that G. is established, and there is a [2, + infinity, so that then the equation group (6) vibrate. The inference 0.0.2. order is set up, assuming then the equation group (6) vibration. In Chapter fifth, we give some sufficient conditions for the non vibration of the four order difference equation. This chapter studies the non vibration criterion of the four order difference equation, assuming that Z (n) = delta y (n) is equivalent to the following equation that is equivalent to the following equation: (A8) 1+P1 (n) -p3 (n) 0; (A9) 1+2P. 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As a criterion for the Nonoscillation of equation (7), the sixth chapter introduces the significance, innovation and research prospect of the conclusion.
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O175

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