胡迎春

（濟(jì)寧學(xué)院初等教育學(xué)院，山東 曲阜 273100）

一類二階非線性微分方程的振動(dòng)性

胡迎春

（濟(jì)寧學(xué)院初等教育學(xué)院，山東 曲阜 273100）

利用函數(shù)類、算子方法和廣義Riccati變換研究了二階非線性微分方程的振動(dòng)性,給出了Kamenev型振動(dòng)準(zhǔn)則,改進(jìn)和推廣了現(xiàn)有的若干結(jié)論．

非線性微分方程；振動(dòng)性；Kamenev型振動(dòng)準(zhǔn)則；算子方法

1 引 言

本文主要研究二階非線性微分方程

的振動(dòng)性問題. 假設(shè)以下條件成立

(A1)函數(shù)1/r∈Lloc([t0,∞),R+),在[t0,∞)上是局部可積函數(shù),且有r＞0.

(A2)當(dāng)t∈[t0,∞)時(shí),函數(shù)Q∶[t0,∞)×R2→R 是局部可積的,且對(duì)于y, z是連續(xù)的.

如果方程(1.1)的一個(gè)解有任意大的零點(diǎn),則稱這個(gè)解是振動(dòng)的,否則稱此解是非振動(dòng)的.如果方程(1.1)的所有解都是振動(dòng)的,則稱方程(1.1)是振動(dòng)的.

受文獻(xiàn)[9,10]的啟發(fā),本章利用和[10]中等價(jià)的定義,得到方程(1.1)的若干新的振動(dòng)準(zhǔn)則,所得結(jié)果改進(jìn)和推廣了已有的相應(yīng)結(jié)論.

稱函數(shù)Φ=Φ(t, s, l)屬于函數(shù)類X,如果Φ∈C( E, R),當(dāng)l＜s＜t時(shí),滿足Φ(t, t, l )=0, Φ(t, l, l)=0,Φ(t, s, l)＞0且在E上存在連續(xù)偏導(dǎo)數(shù)?Φ/?s,?Φ/?s在E上關(guān)于S是局部可積的,其中 E={(t, s, l)∶t0≤l≤s≤t ＜∞}.

如果函數(shù)f∈C1( R, R)且存在常數(shù)K＞0,對(duì)所有的y≠0有yf( y)＞0,f′(y)≥K＞0,則稱函數(shù)f屬于函數(shù)類P.

2 Kamenev型振動(dòng)準(zhǔn)則

下面給出方程(1.1)的Kamenev型振動(dòng)準(zhǔn)則.

定理2.1 假設(shè)對(duì)于每一個(gè)l≥t0,存在函數(shù)

算子T如(1.2)定義,函數(shù)φ=φ(t, s, l)如(1.3)定義.則方程(1.1)是振動(dòng)的.

證明 設(shè)y( t)是方程(1.1)的一個(gè)非振動(dòng)的解.不失一般性,不妨假設(shè)存在t1≥t0,對(duì)所有的t≥t1有y( t)＞0.

對(duì)(2.2)式兩端關(guān)于S求導(dǎo),由(1.1)得

這與(2.1)矛盾.證明完畢.

定理2.2 假設(shè)對(duì)于每一個(gè)l≥t0,存在函數(shù)

算子T如(1.2)定義,函數(shù)φ=φ(t, s, l)如(1.3)定義.則方程(1.1)是振動(dòng)的.

證明設(shè)y( t)是方程(1.1)的一個(gè)非振動(dòng)的解.不失一般性,不妨假設(shè)存在t1≥t0,對(duì)所有的t≥t1有y( t)＞0.由定理2.1的證明,可以得到

這與(2.4)矛盾.證明完畢.

在定理2.1中選擇?(t)≡1,ρ(t)≡0,可以得到下面的振動(dòng)性結(jié)果.

定理2.3假如對(duì)于每一個(gè)l≥t0,都存在函數(shù)υ∈C1([t0,∞),R+),f ∈Ρ以及兩個(gè)常數(shù)α, β＞1,在[t0,∞)上對(duì)任意的y∈C1([t0,∞),R)當(dāng)y(t)≠0時(shí)有

則方程(1.1)是振動(dòng)的.

在定理2.3中取r( t)≡1,υ(t)≡1,可以得到下面的振動(dòng)性結(jié)果.

定理2.4 假如對(duì)于每一個(gè)l≥t0,存在函數(shù)f∈Ρ和兩個(gè)常數(shù)α, β＞1,在[t0,∞)上對(duì)任意的y∈C1([t0,∞),R)當(dāng)y( t)≠0時(shí)有

證明 注意到

由定理2.3知方程(1.1)r(t)≡1是振動(dòng)的.證明完畢.

由定理2.4,可得下面兩個(gè)推論.

則方程(1.1)r(t)≡1是振動(dòng)的.

推論2.2[10,定理2.4]假如對(duì)于每一個(gè)l≥t0，存在函數(shù)f∈Ρ和常數(shù)μ＞，在 上對(duì)任意的[t,∞)當(dāng)

0y∈C1([t,∞),R)時(shí)有y( t)≠0

0

則方程(1.1)r(t)≡1是振動(dòng)的.

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（責(zé)任編輯 龐新琴）Oscillation Criteria for Second Order Nonlinear Differential Equations

HU Yingchun

(The Elementary Education College of Jining University,Qufu 273100,China)

This paper mainly studies the oscillation of second order nonlinear differential equationby applying the function class, operator method and generalized Riccati transformation, we gives Kamenev-type oscillation criteria. The obtained results extend and improve the existing results.

nonlinear differential equations; oscillation; Kamenev-type oscillation criteria; operator method

O175．12

A

1004—1877（2014）03—066—03

2014-03-06

胡迎春（1982-），女，河北秦皇島人，濟(jì)寧學(xué)院初等教育學(xué)院教師 ，碩士，研究方向：微積分.