一类带次线性中立项和分布时滞的三阶阻尼微分方程的振动性

第62卷 第1期吉林大学学报(理学版)
V o l .62 N o .1
2024年1月
J o u r n a l o f J i l i nU n i v e r s i t y (
S c i e n c eE d i t i o n )J a n 2024
d o i :10.13413/j .c n k i .j
d x b l x b .2023226一类带次线性中立项和分布时滞的
三阶阻尼微分方程的振动性
林 文 贤
(韩山师范学院数学与统计学院,广东潮州521041
)摘要:利用处理次线性中立项的技术㊁广义R i c c a t i 变换和积分平均技巧,首先,给出一种估计R i c c a t i 变换不等式的解析方法;其次,考虑一类具有次线性中立项和分布时滞的三阶阻尼微分方程,给出解振动或收敛于零的一些充分条件;最后,通过实例验证所得结果.关键词:振动性;分布时滞;次线性中立项
中图分类号:O 175.1 文献标志码:A 文章编号:1671-5489(2024)01-0055-08
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l e s .K e y
w o r d s :o s c i l l a t i o n ;d i s t r i b u t e dd e l a y ;s u b l i n e a r n e u t r a l t e r m 收稿日期:2023-06-06.
作者简介:林文贤(1966 ),男,汉族,教授,从事泛函微分方程理论及应用的研究,E -m a i l :2450512677@q q .c o m.基金项目:国家自然科学基金(批准号:12026253)㊁广东省普通高校特色创新类项目(批准号:2023K T S C X 083
)㊁广东省一流课程‘数学分析“建设项目(批准号:Z 21011)和2022年度韩山师范学院质量工程建设项目(批准号:E 22033).
0 引 言
考虑具有分布时滞和阻尼项的三阶中立型微分方程:
p (x )z (x )y (x )+ʏβα
r (x ,t )y
μ(τ(x ,t ))d ()t []ᶄ(
)κ
[
]ᶄᶄ+
q (x )z (x )y (x )+ʏβα
r (x ,t )y μ
(τ(x ,t ))d ()t []ᶄ()κᶄ+ʏb
a
G (x ,ξ,y (σ(x ,ξ)))d ξ=0(1
)的振动性,且满足下列条件:
(H 1)0<μɤ1,κȡ1,μ,
κ均为两个正奇数之商;(H 2)p (x )ɪC 1
([x 0,ɕ),(0,ɕ)),q (
x )ɪC ([x 0,ɕ),(0,ɕ)),ʏ
ɕ
x 01
p (x
)e x p -ʏ
x
x 0q (s
)p (s
)d {}
s d x =
ɕ;
(H 3)z (x )ɪC 1
([x 0,
ɕ),(0,ɕ)),ʏ
ɕ
x 01
z
1/κ
(x )d x =ɕ;(H 4)r (x ,t )ɪC ([x 0,ɕ)ˑ[α,β]
,(-ɕ,+ɕ)),0ɤr (x )=ʏ
β
α
r (x ,t )d t ɤr <1,l i m x ңɕ
r (x )=0;(H 5)τ(x ,t )ɪC ([x 0,ɕ)ˑ[α,β],(-ɕ,+ɕ)),∂τ(x ,t )∂t ȡ0,τ(x ,t )ɤx ,l i m x ңɕm i n t ɪ[α,β
]τ(x ,t )=+ɕ;
(H 6)G (x ,ξ,v )ɪC ([x 0,ɕ)ˑ[a ,b ]ˑ(0,ɕ),(0,ɕ)),m (x ,ξ)
ɪC ([x 0,ɕ)ˑ[a ,b ],(0,ɕ)),G (x ,ξ,v )/v κȡm (x ,ξ
);(H 7)σ(x ,ξ)ɪC ([x 0,ɕ)ˑ[a ,b ],(-ɕ,+ɕ)),∂σ(x ,ξ)∂ξ
ȡ0,σ(x ,ξ)ɤx ,l i m x ңɕm i n ξɪ[a ,b ]σ(x ,ξ)=+ɕ.
令
w (x )=y (x )+ʏ
β
α
r (x ,t )y
μ(τ(x ,t ))d t .(2
)若函数y (x )满足方程(1)及y (x )ɪC 1([T y ,ɕ),(-ɕ,+ɕ)),p (
x )[z (x )(w ᶄ(x ))κ]ᶄɪC 1
[T y ,ɕ),T y ȡt 0,则称y
(x )是方程(1)的一个解.若y (x )既非最终为正,也非最终为负,则称y (x )在[T y ,ɕ)上振动;否则,称y (x )
是非振动的.由于中立型微分方程广泛应用于自然科学和应用技术等领域,因此关于三阶中立型微分方程解的
振动性和渐近性研究备受关注[1-12]
.文献[3
]研究了三阶方程(a (x )(b (x )(y (x )+p y (x -τ))ᶄ)ᶄ)ᶄ+q (x )f (y (x -σ))=0的振动性;文献[7
]研究了具连续分布滞量的三阶微分方程r (x )y (x )+ʏ
β
αp (x ,s )y (τ(x ,s ))d s [
])[
]ᵡᶄ+ʏ
b a
q (x ,ξ)f (y (σ(x ,ξ))d ξ=0
的振动性;文献[8]研究了方程(1)当μ=κ=1及Z (x )恒为1时特例的振动性;文献[10
]研究了三阶阻尼微分方程
[r (t )([x (t )+p (t )x (τ(t ))]ᵡ)α]ᶄ+m (t )([x (t )+p (t )x (τ(t ))]ᵡ)α
+q (t )f (x (σ(t )))g (
x ᶄ(t ))=0的振动性;文献[11]研究了方程(1)当μ=κ=1时特例的振动性,得到了若干解的振动性定理.
本文受文献[7-10]的启发,考虑中立型方程(1)当0<μɤ1和κȡ1时更广泛的情形,给出该方程
的振动性条件及实例.
1 引 理
引理1 若y (x )是方程(1)的正解,则当x ȡx 1ȡx 0时有下列两种可能:
1)w (x )>0,w ᶄ(x )>0,(z (x )[w ᶄ(x )]κ
)ᶄ>0;
2)w (x )>0,w ᶄ(x )<0,(z (x )[w ᶄ(x )]κ
)ᶄ>0.
证明:首先,设y (x )是方程(1)在[x 0,ɕ)上的一个正解,由条件(H 4)和(H 5)知,存在x 1>x 0,
使得当x ȡx 1时,有y (τ(x ,t ))>0,y (σ(x ,ξ
))>0.于是由式(2)和条件(H 3)有w (x )>y (x )>0.其次,由方程(1)和条件(H 6)
有(p (x )(z (x )[w ᶄ(x )]κ)ᶄ)ᶄ+q (x )(z (x )[w ᶄ(x )]κ
)ᶄ=-ʏ
b a G (x ,ξ,y (σ(x ,ξ)))d ξɤ
-ʏb
a
m (x ,
ξ)y (σ(x ,ξ))d ξ<0,因此可得
e x p
ʏ
x
x 1q (s )p (s
)d {}
s p (x )(z (x )[w ᶄ(x )]κ
)éë
êùûúᶄᶄɤ0,65 吉林大学学报(理学版) 第62卷
从而e x p
ʏ
x
x 1
q (s )p (s )d {}
s p (x )(z (x )[w ᶄ(x )]κ
)ᶄ递减且最终定号.由条件(H 1)知,当x ȡx 2ȡx 1时,有(z (x )[w ᶄ(x )]κ)ᶄ<0或(z (x )[w ᶄ(x )]κ
)ᶄ>0.
下面证明(z (x )[w ᶄ(x )]κ)ᶄ>0.假设(z (x )[w ᶄ(x )]κ
)ᶄɤ0,由e x p
ʏ
x
x 1
q (s
)p (s )d {}
s p (x )ˑ(z (x )[w ᶄ(x )]κ
)ᶄ的单调性知,存在常数T >0,使得
e x p ʏ
x
x 1
q (s )p (s )
d {}
s p (x )(z (x )[w ᶄ(x )]κ
)ᶄɤ-T ,(3
)将式(3)从x 2到x 积分得
z (x )[w ᶄ(x )]κɤz (x 2)[w ᶄ(x 2)
]κ
-T ʏx
x 21
p (s
)e x p -ʏ
s
x 1q (v )p (v )d {}
v d s .令x ңɕ,由条件(H 1)得z (x )[w ᶄ(x )]κң-ɕ.因此由(z (t )[w ᶄ(x )]κ
)ᶄɤ0,可知当x ȡx 3ȡx 2时,
z (x )[w ᶄ(x )]κɤz (x 3)[w ᶄ(x 3)
]κ
<0,从而有
w ᶄ(x )ɤz 1/κ
(x 3)w ᶄ(x 3)1z 1/κ
(x )
.(4
)对式(4)从x 3到x 积分,有
w (x )-w (x 3)ɤz 1/κ(x 3)w ᶄ(x 3)
ʏ
x x 31
z
1/κ
(s )d s .令x ңɕ,由条件(H 2)可得w (x )ң-ɕ,与w (x )>0矛盾,进而有(z (x )[w ᶄ(x )]κ
)ᶄ>0.证毕.
引理2 设y (x )是方程(1)的最终正解,w (x )
满足引理1中结论2),若ʏɕ
x 01z (v )ʏɕv 1p (u )ʏ
ɕu m (s )d s d æè
çöø÷u 1/
κ
d v =ɕ,(5
)则l i m x ңɕ
y (x )=0,其中m (x )=ʏb
a
m (x ,
ξ)d ξ.证明:由于w (t )满足引理1的结论2),即w (x )>0,w ᶄ(x )<0,又由单调有界定理可知
l i m x ңɕ
w (x )存在,记l i m x ңɕ
w (x )=j ,则j ȡ0.假设j >0,由l i m x ңɕ
r (x )=0,则∀0<ε<m i n {j ,2-μj 1-μ}
,存在j <w (x )<j +
ε,0ɤr (x )<ε,使得y (x )=w (x )-ʏ
βα
r (x ,t )y
μ(τ(x ,t ))d t ȡj -ʏ
β
α
r (x ,t )w μ(τ(x ,t ))d t >j -(j +ε)μεȡj -(2j )με=K (j +ε
)>K w (x ),(6
)其中K =j -(2j )μεj +
ε>0,利用条件(H 6),(H 7)及引理1中结论2)和式(6
),可得(p (x )(z (x )[w ᶄ(x )]κ)ᶄ)ᶄ+q (x )(z (x )[w ᶄ(x )]κ)ᶄɤ-ʏ
b a
m (x ,ξ)y κ
(σ(x ,ξ))d ξɤ
-K κw κ
(
σ(x ,b ))m (x ). 令α(x )=e x p
ʏ
x
x 1q (s
)p (s
)d {}
s ,则有(α(x )p (
x )(z (x )[w ᶄ(x )]κ
)ᶄ)ᶄɤ-K κα(x )w κ(σ(x ,b ))m (x ).(7
)对式(7)在(x ,+ɕ)
上积分,有-α(x )p (
x )(z (x )[w ᶄ(x )]κ
)ᶄ+K κʏ
ɕ
x
α(s )w κ(σ(s ,b ))m (s )d s ɤ0.再注意到w (σ(x ,b ))>j 和w ᶄ(x )>0,有-(
z (x )[w ᶄ(x )]κ
)ᶄ+K κj κ
p (x )ʏ
ɕ
x
m (s )d s <0,从而
z (x )[w ᶄ(x )]κ+K κj
κ
ʏɕ
x
1p (
u )ʏ
ɕu m (s )
d s d u <0,7
5 第1期 林文贤:一类带次线性中立项和分布时滞的三阶阻尼微分方程的振动性
进而有
-w ᶄ(x )ȡK j 1z (x )ʏɕx 1p (u
)ʏ
ɕu m (s )d s d æèçöø÷u 1/
κ
.(8
)对式(8)在(x 1,
ɕ)上积分,得ʏɕ
x 1
1z (v )ʏɕv 1p (u )ʏ
ɕu m (s )d s d æèçöø÷u 1/κd v <w (x 1)K j ,与条件(5)矛盾.因此j =0.又因为w (x )>y (
x )>0,所以有l i m x ңɕ
y (x )=0.证毕.引理3
[13
] 设0<λɤ1,则:
1)X λ+Y λɤ21-λ(X +Y )λ
,X ,Y 为非负实数;
2)(1+X )λ
ɤ1+λX ,其中1+X >0.2 主要结果
下面利用R i c c a t i 变换和文献[14]中的一些估计,证明方程(1
)的一些新的振动性结果.记E ={(x ,s )x ȡs ȡx 0}, E 0={(x ,s )x >s ȡx 0}
.如果函数J (x ,s )ɪC (E ,R )具有下列性质,则称J (x ,s )属于Y 类,记作J ɪY :
(i )J (x ,x )=0,x ȡx 0,
J (x ,s )>0,(x ,s )ɪE 0;(i i )∂J (x ,s )∂s
ɤ0,(x ,s )ɪE ;
(i i i )存在函数f (x ,s )ɪC (E ,R )
,使得∂J (x ,s )∂s
=-f (
x ,s )J (x ,s ), (x ,s )ɪE . 定理1 设式(5)成立,且存在函数J ɪY 和ψɪC 1
([x 0,
ɕ),(-ɕ,+ɕ)),使得0<i n f s ȡx 0l i mi n f x ңɕJ (x ,s )J (x ,x 0éëê
ùûú)ɤɕ,(9)ʏ
ɕ
x 0
ψ2+(x )g (x )p (x )d x =ɕ,(10)l i ms u p x ңɕ1J (x ,X )ʏ
x
X J (x ,s )β(s )-14ηg (s )p (s )f 2(x ,s éë
ê
ùûú)d s ȡψ(X ),(11
)其中
φ+(s )=m a x {φ(
s ),0},g (x )=e x p ʏ
x
x 0q (s )p (s
)d s ,β(x )=g (x )1-μ21-μ+(21-μ-1)k éëêùûú1
{}
r κγm (x )z (x ), η>1, γ>0.(12
)则方程(1)的解y (x )振动或l i m x ңɕ
y (x )=0.
证明:设y (x )是方程(1)的一个非振动解,不失一般性,设y (x )>0,x ɪ[x 1,
ɕ).由条件(H 4),(H 5)
有y (τ(x ,t ))>0, (x ,t )ɪ[x 1,ɕ)ˑ[α,β], y (σ(x ,ξ))>0, (x ,ξ
)ɪ[x 1,ɕ)ˑ[a ,b ]. 当w (x )满足引理1中结论1)时,由条件(H 4),(H 5),有y (
x )ȡw (x )-(1+w μ(x ))r +r ȡw (x )-21
-μ(1+w (x ))μr +r ȡ(1-μ21-μr )w (x )-(21-μ-1
)r .(13)由引理1中结论1)知,w (x )>0,w ᶄ(x )>0,所以
w (σ(x ,a ))ȡw (σ(x 1,a ))=k 1, x ȡx 1.
(14
)从而由条件(H 6),(H 7)
和上述不等式,有(p (x )(z (x )[w ᶄ(x )]κ)ᶄ)ᶄ+q (x )(z (x )[w ᶄ(x )]κ)ᶄɤ-ʏ
b
a
m (x ,ξ)y κ
(σ(x ,ξ))d ξɤ
85 吉林大学学报(理学版) 第62卷
-ʏ
b a m (x ,ξ)[(1-μ21-μr )w (σ(x ,a ))-(21-μ-1)r ]κd ξɤ
-w κ
(σ(x ,a ))1-μ
21-μ+(21-μ-1)k éëêùûú1
{}
r κm (x ).令H (x )=g (x )p (x )(z (x )[w ᶄ(x )]κ
)ᶄz (x )[w ᶄ(x )
]κ
,x ȡx 1,其中g (x )由式(12)定义,于是H ᶄ(x )ɤg ᶄ(x )g (x )H (x )+g (x )(p (x )(z (x )[w ᶄ(x )]κ)ᶄ)ᶄz (x )[w ᶄ(x )]κ
-H 2
(x )
g (x )p (
x )ɤg ᶄ(x )g (
x )H (x )-g (x )1-μ21-μ
+(21-μ-1)k éëêêùûúú1{}
r κw κ(σ(x ,a ))m (x )z (x )[w ᶄ(x )]κ
éëêê+q (x )(z (x )[w ᶄ(x )]κ
)ᶄz (x )[w ᶄ(x )]ùûúúκ
-H 2(x )g (x )p (
x )=-g (x )1-μ21-μ+(21-μ-1)k éëêùûú1{}
r κw κ(σ(x ,a ))m (x )z (x )[w ᶄ(x )]κ
-H 2(x )g (x )p (
x ).(15
) 由引理1中结论1)知,存在极限l i m x ңɕ1[w ᶄ(x )]κ,记为l i m x ңɕ1[w ᶄ(x )
]κ=θ.选取ε=θ2,则1[w ᶄ(x )
]κ>θ2,x ȡx 2ȡx 1.再取γ=w κ(λ)θ2,则由条件(H 6)有σ(x ,a )ȡλ,从而H ᶄ(x )=-β(x )-H 2(x )g (x )p (x ).(16)将式(16)两边乘以J (x ,s ),并在[x 2,
x ]上积分,得ʏ
x
x 2
J (x ,s )β(
s )d s ɤJ (x ,x 2)H (x 2)-ʏ
x
x 2
f (
x ,s )J (x ,s )H (s )d s -ʏ
x
x 2
J (x ,s )H 2(s
)g (s )p (s )d s =J (x ,x 2)H (x 2)-ʏ
x
x 2f (x ,s )ηg (s )p (s )2+H (s )J (x ,s )η
g (s )p (s æèçöø÷)2d s +ʏ
x x 2ηg (s )p (s )f 2(x ,s )4d s -ʏ
x
x 2
(η-1)J (x ,s )H 2(s )ηg (s )p (s )d s ,从而
l i ms u p x ңɕ1J (x ,x 2)ʏ
x x 2
J (x ,s )β(s )-ηg (s )p (s )f 2(x ,s )æèçöø÷4d s ɤ H (x 2)-l i mi n f x ңɕ1J (x ,x 2)ʏ
x
x 2
(η-1)J (x ,s )H 2(s )ηg (s )p (s )d s ,(17)于是
l i mi n f x ңɕ1J (x ,x 2)ʏ
x
x 2
(η-1)J (x ,s )H 2(s )ηg (s )p (s )d s ɤH (x 2)-ψ(x 2)<ɕ, x ȡx 2.(18
)断言ʏ
ɕx 2H 2(x )g (x )p (x )d x <ɕ.若ʏ
ɕ
x 2
H 2(x )g (x )p (x )d x =ɕ,则由式(9),对某个δ>0有i n f s ȡx 0l i mi n f x ңɕJ (x ,s )J (x ,x 0é
ë
ê
ùûú)>δ.表明J (x ,x 3)J (x ,x 0)
>δ,x ȡx 3ȡx 2,从而对任意N >0,有ʏ
x
x 3H 2(x )g (x )p (x
)d x ȡN
δ,进而
1J (x ,x 0)ʏ
x x 3J (x ,s )H 2(s )g (s )p (s )d s =1J (x ,x 0)ʏx x 3
-∂J (x ,s )∂æèçöø÷
s ʏ
s
x 3H 2(v )
g (v )p (v
)d v d s ȡ1J (x ,x 0)N δʏx
x 3
-∂J (x ,s )∂æèç
öø÷
s d s =N δJ (
x ,x 3)J (x ,x 0
)
ȡN .
于是
9
5 第1期 林文贤:一类带次线性中立项和分布时滞的三阶阻尼微分方程的振动性
l i mi n f x ңɕ1J (x ,x 0)ʏ
x
x 3
J (x ,s )H 2(s
)g (s )p (s )d s =ɕ,与式(18)矛盾.因此有ʏ
ɕ
x 2H 2
(x )g (x )p (x )d x <ɕ,与条件(10)矛盾.当w (x )满足引理1中结论2)时,由式(5)和引理2,有l i m x ңɕw (x )=l i m x ңɕ
y (x )=0.证毕.
定理2 设式(5)成立,若存在函数J ɪY 和B (x )ɪC ([x 0,ɕ),(-ɕ,+ɕ)),使得对x ȡX ȡx 0,
满足
l i ms u p x ңɕ1J (x ,X )ʏ
x X J (x ,s )A (s )-ηg (s )p (s )f 2(x ,s )4(η
-1æèçöø÷)d s =ɕ,(19)则方程(1)的解y (x )振动或l i m x ңɕ
y (x )=0,其中η>1,γ>0,
A (x )=M (x )-ηg (x )p (
x )B 2(x ), g (x )=e x p ʏ
x
x 0q (s )p
(s )d s ,(20
)M (x )=g (x )[p (x )B 2(x )-q (x )B (x )-(p (
x )B (x ))ᶄ+1-μ21-μ
+(21-μ-1)k éëêùûú1
{
}
r κγm (x )z (x )].
(21
) 证明:设y (x )是方程(1)的一个非振动解,不失一般性,设y (x )>0,x ɪ[x 1,ɕ).由条件(H 4),(H 5)
有y (τ(x ,t ))>0, (x ,t )ɪ[x 1,ɕ)ˑ[α,β], y (σ(x ,ξ))>0, (x ,ξ
)ɪ[x 1,ɕ)ˑ[a ,b ]. 当w (x )
满足引理1中结论1)时,令H (x )=g (x )p (x )(z (x )[w ᶄ(x )]κ)ᶄz (x )[w ᶄ(x )
]κ
+p (x )B (x éëêêùûúú), x ȡx 1,于是
H ᶄ(x )ɤg ᶄ(x )g (x
)H (x )+g (x )(p (x )B (x ))ᶄ-g (x )1-μ21-μ
+(21-μ-1)k éëêêùûúú1{}
r κw κ(σ(x ,a ))m (x )z (x )[w ᶄ(x )]éëêêκ+q (x )(z (x )[w ᶄ(x )]κ)ᶄz (x )[w ᶄ(x )
]κ+p (x )(z (x )[w ᶄ(x )]κ)ᶄz (x )[w ᶄ(x )]æèçöø÷κùûúú2.类似式(15),取l i m
x ңɕ1[w ᶄ(x )
]κ
=θ,ε=η2,γ=w κ(λ)θ2,则有H ᶄ(x )ɤg ᶄ(x )g (x
)H (x )+g (x )(p (x )B (x ))ᶄ-g (x )1-μ21-μ
+(21-μ-1)k éëêêùûúú1{}
r κγm (x )z (x éëêê)+q (x )H (x )g (x )p (
x )-q (x )B (x )+p (x )H (x )g (x )p (x )-B (x æèçöø÷)ùûúú2
=-M (x )+2B (x )H (x )-H 2(x )ηg (x )p (x )-(η-1)H 2(x )η
g (x )p (x ),其中M (x )由式(21)定义.利用不等式D u -C u 2
ɤD 24C
,C >0,u ɪℝ,得
H ᶄ(x )ɤ-A (x )-(η-1)H 2(x )η
g (x )p (x ),(22)其中A (x )来自式(20).将式(22)两边乘以J (x ,s ),并在[X ,x ]
上积分得ʏ
x X J (x ,s )A (s )d s ɤJ (x ,X )H (X )-ʏx x 2
f (x ,s )J (x ,s )H (s )+(η-1)J (x ,s )H 2(s )η
g (s )p (s æèçöø÷)d s =J (x ,X )H (X )-ʏ
x
X f (x ,s )ηg (s )p (s )4(η-1)+H (s )(η-1)J (x ,s )ηg (s )p (s æèçöø÷)2
d s +ʏ
x
X ηg (s )p (s )f 2(x ,s )4(η-1
)d s ,从而有
06 吉林大学学报(理学版)
第62卷
1J (x ,X )ʏ
x X J (x ,s )A (s )-ηg (s )p (s )f 2(x ,s )4(η-1æè
çöø÷)d s ɤH (X ),与式(16
)矛盾.当w (x )满足引理1中结论2)时,由式(5)和引理2有l i m x ңɕ
w (x )=l i m x ңɕ
y (x )=0.证毕.
注1 若方程(1)中取μ=κ=1,则定理1和定理2即为文献[10]
的振动结果,进而改进并推广了文献[7-9
]的相应结果.3 实 例
例1 考虑三阶阻尼微分方程
1x e -x y (x )+ʏ
2
1t 6x y 1/3(x -t )d æèçöø÷t éëêùûúᶄæè
çöø÷2éëêêùûúúᶄᶄ+2x 2e -x y (x )+ʏ
2
1t 6x y 1/3(x -t )d æèçöø÷t éëêùûúᶄæè
çöø
÷2ᶄ+ʏ11/2
2
x ξy 2
((x -3x )ξ)d ξ=0.(23
) 记p (x )=1x ,q (x )=1x
2,z (x )=e -x ,r (x ,t )=t 6x ,κ=2,α=1,β=2,a =12,b =1,τ(x ,t )=x -t ,σ(x ,ξ)=(x -3x )ξ,μ=13,则
r (x )=ʏβ
αr (
x ,t )d t =ʏ
2
1t 6x d t =14x ɤ12, l i m x ңɕr (x )=l i m x ңɕ1
4x
=0,
于是满足条件(H 1)~(H 7).进一步,令
J (x ,s )=(x -s )2
, η=2, γ=2, r =12
, m (x ,ξ)=2x ξ, ψ(x )=x , x 0=1,利用定理1,有
f (x ,s )=2,
g (x )=x 2-1, m (x )=38x , β(x )=38
x (x 2-1)e x
,l i ms u p x ңɕ1
J (x ,X )ʏ
x
X J (x ,s )β(s )-14ηg (s )p (s )f 2(x ,s éë
êêùûúú)d s ȡX =ψ(X ).因此满足定理1的所有条件.故由定理1知,方程(23
)的任意解振动或趋于0.例2 考虑三阶阻尼微分方程
2x x y (x )+ʏ
10e -x t 2y 1/313æèçöø÷x t d æèçöø÷t éëêêùûúúᶄæèçöø÷2éëêêùûúúᶄᶄ+1x 2x y (x )+ʏ
1
0e -x t 2y 1/313æèçöø÷x t d æèçöø÷t éëêùûúᶄæè
çöø÷2ᶄ+ʏ
1
1/2
ξx
y 2(x ξ)d ξ=0.
(24
) 记p (x )=2x ,q (x )=1x 2,z (x )=x ,r (x ,t )=e -x t 2
,κ=2,α=0,β=1,a =12,b =1,τ(x ,t )=13x t ,μ=13
,σ(x ,ξ)=x ξ,则满足条件(H 1)~(H 7).进一步,令J (x ,s )=(x -s )2
,η=2,γ=2,r =0.9,m (x ,ξ)=ξx
,B (x )=1x ,x 0=1,利用定理2,有f (
x ,s )=2, g (x )=x -1, m (x )=38x
, A (x )=(x -1)316x +1x æèçöø÷3,l i ms u p x ңɕ1J (x ,X )ʏ
x X J (x ,s )A (s )-ηg (s )p (s )f 2(x ,s )4(η-1æè
çöø÷)d s =ɕ.因此满足定理2的所有条件.故由定理2知,方程(24
)的任意解振动或趋于0.参
考
文
献
[1] A G A RWA LRP ,G R A C ESR ,O R E G A N D.T h eO s c i l l a t i o no fC e r t a i n H i g h e r -
O r d e rF u n c t i o n a lD i f f e r e n t i a l 1
6 第1期 林文贤:一类带次线性中立项和分布时滞的三阶阻尼微分方程的振动性
26吉林大学学报(理学版)第62卷
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