几类微分方程、差分方程的振动性研究
发布时间：2018-07-29 19:59
【摘要】：本篇论文研究了几类微分方程、差分方程的振动性与非振动性,主要围绕二阶时滞动力方程、二阶Emden-Fowler微分方程、二阶差分方程、二维差分方程组、四阶差分方程的振动性与非振动性展开研究,修正并完善了文献中的一些已有结果,并建立了若干新的判定准则,推广了微分方程、差分方程的振动性与非振动性理论的一些已有结果.第一章介绍了本文研究问题的背景以及相关进展,并简要叙述了本文所做的工作.第二章运用广义Riccati变换等方法研究了一类时间尺度上二阶非线性时滞动力方程的振动性,推广了一些文献中的已有结果.本章所研究方程如下.其中γ ≥ 1是两个正奇数的商,并且满足下面条件:本章,我们作如下假设:定理0.0.1.假设条件(2),(3)及γ≥1成立.如果存在某个正常数k∈(0,1)使得成立,那么方程(1)在[t0,∞)T是振动的.推论0.0.1.假设(2),(3)和γ≥1成立.如果存在某个正常数k∈(0,1)使得成立,那么方程(1)在[t0,∞)T是振动的.定理0.0.2.假设条件(2)和(3)成立,且存在函数σ(t)≥t使得当γ ≥ 1时,有成立,或当0γ1时,有成立,那么方程(1)在[t0,∞)T是振动的.我们引入下列记号:定理0.0.3.假设条件(2)和(3)成立,并且Toσ = σoT.假设存在某个正常数k∈(0,1)和一个函数使得其中那么方程(1)在[t0,∞)T是振动的.定理0.0.4.假设条件(2)和(3)成立,并且τoσ = σoτ.假设存在函数H,h ∈ Crdr(D,R),D≡{(t,s)∈ T2:t≥s≥ t0}和一个函数δ(t)∈Crd([(t0,∞)T,R),同时,H有一个关于s的非正的连续偏导数H△s(t,s),并且满足且成立,其中 那么方程(1)在[t0,∞)T是振动的.第三章利用Riccati变换法研究了 一类中立时滞微分方程的振动准则,推广了 一些已有的Emden-Fowler微分方程的振动结果.文章对α,β的大小分情况讨论,并建立不同的权函数,权函数的构造基于所研究方程的特点.文章得到了如下的二阶Emden-Fowler微分方程的振动准则,并给出相应的例子.其中,是两个奇数的商.本章,我们假设下列条件成立:主要结果如下.定理0.0.5.如果0α≤β,并且存在函数p ∈C(t0,∞))使得那么方程(4)振动.定理0.0.6.如果αβ0,并且存在一个函数p ∈(C([t0,+∞))使得那么方程(4)振动.第四章包括两部分内容.利用差分的定义和相关公式以及不等式技巧研究了二阶线性差分方程和一阶二维线性差分方程组的振动准则.改进并推广了文献中的已有结果,并给出相应的例子.本章研究如下二阶差分方程的振动准则.本章我们使用如下记号:定理0.0.7.令q ≤ 1/4.如果存在常数α ∈[0,1)使得那么方程(5)是振动的.定理0.0.8.令px(0)≤1/4和q≤1/4.如果存在常数α ∈[M2,1)使得那么方程(5)是振动的.本章我们总是假设总是成立的.定理0.0.9.假设g。成立,并且存在λ ∈[2,+∞),使得那么方程组(6)振动.推论0.0.2.令 成立,假设那么方程组(6)振动.第五章我们给出了四阶差分方程非振动的若干充分条件.本章研究如下四阶差分方程的非振动准则,假设z(n)= △y(n),则上面方程等价于即等价于下列方程假设有下列条件:(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)满足下面条件中的一个,则方程(8)是非振动的.(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.将关系式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 代入条件(A1)-(A12)中,可以导出a(n),b(n),c(n)的不等关系式,作为方程(7)的非振动性的判定条件.第六章介绍了本文结论的意义、创新点及研究前景.
[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. Obtains Marxist societies traditions societies traditions incomes traditions incomes incomes traditions incomes traditions traditions incomes traditions incomes incomes traditions incomes incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes incomes incomes incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples traditions incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes peoples incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes traditions incomes incomes incomes incomes peoples incomes incomes peoples incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes traditions incomes incomes incomes incomes peoples incomes incomes incomes peoples incomes peoples) (A2)), A7))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) it can be derived from the condition. As a criterion for the Nonoscillation of equation (7), the sixth chapter introduces the significance, innovation and research prospect of the conclusion.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175
[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. Obtains Marxist societies traditions societies traditions incomes traditions incomes incomes traditions incomes traditions traditions incomes traditions incomes incomes traditions incomes incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes traditions incomes incomes incomes incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes traditions incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples traditions incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes incomes incomes peoples incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes traditions incomes incomes incomes incomes peoples incomes incomes peoples incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes peoples incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes incomes peoples incomes traditions incomes incomes incomes incomes peoples incomes incomes incomes peoples incomes peoples) (A2)), A7))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) it can be derived from the condition. As a criterion for the Nonoscillation of equation (7), the sixth chapter introduces the significance, innovation and research prospect of the conclusion.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175
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