﻿ 线性泛函方程解的振动性 The Oscillation of the Linear Functional Equations

Pure Mathematics
Vol.06 No.04(2016), Article ID:18114,10 pages
10.12677/PM.2016.64048

The Oscillation of the Linear Functional Equations

Lina Dai, Yanfen Xu*, Quanwen Lin

Department of Mathematics, Science of School, Guangdong University of Petrochemical Technology, Maoming Guangdong

Received: Jul. 9th, 2016; accepted: Jul. 24th, 2016; published: Jul. 28th, 2016

ABSTRACT

In this paper, we study of oscillatory of all solutions to the high order equation

We get some new vibration conditions, and improve or promote some of the results of previous literature.

Keywords:Functional Equations, Solutions, Oscillation, Non-Oscillation

1. 引言

(1.1)

(1.2)

1995年，在文 [2] Nowakowska和Werbowski将方程(1.2)推广高阶线性泛函方程(1.1)的情形，得到方程(1.1)的所有解振动，如果

(1.3)

(1.4)

(1.5)

1998年以来，函数方程的振动性成为数学工作者研究研究的热门课题他们得到各类线性高级泛函方程和非线性高阶泛函方程解的振动准则(请参看文 [3] - [12] ) (从略)。

2003年，文 [7] 研究了方程(1.1)的一种特殊形式

(1.6)

(i) 如果下面条件之一成立：

(ii) 如果下面条件之一成立

(1)

(2) (1.7)

(iii)如果下面条件之一成立

(1),

(2) (1.8)

2. 引理

(2.1)

(2.2)

(2.3)

(2.4)

，注意到，从上式我们得到(2.2)。引理证毕。

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

(2.10)

(2.11)

，我们得(2.8)。利用引理2.1和(2.11)(令)，易得满足(2.9)。引理证毕。

(1) (2.12)

(2) (2.13)

3. 结果及证明

(1) (3.1)

(2)(3.2)

(3.3)

(1) (3.4)

(2)(3.5)

(1) (3,6)

(2) (3.7)

，有：

(3.8)

(1), (3.9)

(2), (3.10)

(3.11)

(3.12)

(1)(3.13)

(2) (3.14)

The Oscillation of the Linear Functional Equations[J]. 理论数学, 2016, 06(04): 327-336. http://dx.doi.org/10.12677/PM.2016.64048

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