一类非线性中立型时滞微分方程周期解的存在性

∗

(450052)

(455000)

(100081)

x (t)=f(t,x(t),x(t−τ),x (t−τ))+p(t)τ

Fredholm k-

1

[1−6],[1–5] Mawhin[6]Krasnoselskii

x (t)=−a(t)y(t)+λh(t)f

y

t−τ(t)

,(1)

x

x (t)=f

t,x(t),x(t−τ),x (t−τ)

+p(t)(2)

f(t,x,y,z)∈C(R×R3,R),f(t+T,x,y,z)≡f(t,x,y,z),∀(x,y,z)∈

R3;τ∈R p(t)∈C(R,R),p(t+T)≡p(t),T>0

f f(t)=x(t)

a(t)−β(t)x(t)−

b(t)x(t−τ)−c(t)x (t−τ)

,p(t)≡0

200113120011018

∗(19371006)(1999000722)

90

27

(2)

f

x

f

[1–6]

k -

(2)

τ

2

E

Banach

S ⊂E

αE (S )=inf

δ>0

S

:S =

n i =1

S i ,

S i

diam E (S i )≤δ

,

(3)

αE S Kuratowski [8−10]

.E 1,E 2

Banach

D ⊂

E 1,A :D →E 2

k ≥0,

S ⊂D

αE 2 A (S )

≤kαE 1(S ),

(4)

A

D k -[8−10]

.L :dom L ⊂E 1→E 2Fredholm

[10]

B ⊂dom L ,sup r >0|rαE 1(B )≤αE 2 L (B )

l (L )=sup r >0|rαE 1(B )≤αE 2

L (B ) ,

B ⊂dom (L )

.

L :X →Y Fredholm X,Y Banach

Ω⊂X

N :Ω→Y

k -

k 1

[7]

L :X →Y

Fredholm

y ∈Y

N :Ω→Y

k -

k 0∈Ω

(R 1)

Lx =λNx +λy,∀x ∈∂Ω,∀λ∈(0,1),(R 2)

QN (x )+Qy,x · QN (−x )+Qy,x

<0,

∀x ∈Ker L ∩∂Ω,

[·,·]Y ×X Q :Y →coker L x ∈Ω

Lx =Nx +y .

C T = x |x (t )∈C (R,R ),x (t +T )=x (t )

, x 0=max t ∈[0,T ]

x (t ) ,C T

· 0Banach C 1

T = x |x (t )∈C 1(R,R ),x (t )=x (t +T ) ,

∀x ∈C 1T , x 1=max x 0, x 0 ,C 1

T · 1Banach

X =C 1

T ,Y =C T ,L :dom L ⊂X →Y Lx =x ,N :X →Y Nx (t )=

f t,x (t ),x (t −τ),x (t −τ) .L

Fredholm

Q :Y →Y/Im L Q (y )=1

T T 0

y (t )d t ,∀y ∈Y ,ker L = x |x ∈X,x =c,c ∈R

,

1

91

Im L = y ∈Y T 0y (s )d s =0

Y

dim ker L =codim Im L =dim Im Q =

1,

L

Fredholm

(1)

x (t )

Lx =Nx +P x .

(H 1)

k ∈[0,1),

f (t,x,y,z )−f (t,x,y,z ) ≤k |z −z |.

(H 2)

M >0,x ≥y >M

∀(t,z )∈R ×R ,f (t,x,y,z )+p (t )>0(<0);x ≤y <−M

∀(t,z )∈R ×R ,

f (t,x,y,z )+p (t )<0(>0).

2(H 1)1)l (L )≥1;2)N :X →Y k -l (L )

N

1)∀B ⊂X

η=αY L (B )

≥0.

C 1

T

Y =C T ,B

B

C T

· 0

αY (B )=0.

(3)

∀ε>0,

B 1,B 2,···,B m ⊂X

B =

m

i =1

B i ,

diam Y (B i )<ε(i =1,2,···,m ).αY (L (B ))=η

L (B i )⊂L (B ),(3)αY L (B i )

≤η.ε>0,

∀i ∈{1,2,···,m },

A 1i ,A 2

i ,···,A m i i ⊂Y ,L (B j i )=m i j =1

A j i ,diam Y (A j i )≤η+ε.

B j

i

=

x |x ∈B i ,Lx ∈A j i

,

diam Y

L (B j i

≤η+ε,

B =

m i =1m i j =1

B j

i .

∀x,y ∈B j

i ,

x −y 0= Lx −Ly 0≤η+ε,

(5)

diam Y (B i )<ε,diam Y (B j

i )<ε.(5)

diam X (B j

i )≤η+ε,

αX (B )≤η=αY L (B )

,l (L )≥1.

2)

f :[0,T ]×R 3→R B ⊂X

N u :B →Y

N u (x )(t )=f t,x (t ),x (t −τ),u (t −τ)

,

f

{N u |u ∈X }B u · 0∀ε>0,∃δ(ε)>0,

u ∈X ,x,x ∈B x −x 0<δ(ε)

N u (x )−N u (x ) 0

<ε.

δ (ε)=min δ(ε),ε

,

∀Ω⊂B ,diam Y (Ω)<δ (ε)

diam Y (N u (Ω))<ε.

η=αX (B ),ε>0

(3)

B 1,B 2,···,B m ⊂B

m j =1

B j =B

diam X (B j )<η+ε.(6)

X =C 1

T Y =C T ,B

X

B Y

∀j ∈{1,2,···,m },B j Y

αY (B j )=0.

(3)

∀j ∈{1,2,···,m }

B 1j ,B 2j ,···,B n (j )

j

⊂B j B j =

n (j ) i =1

B i

j ,

92

27

diam Y (B i

j )<δ (ε)(1≤j ≤m,1≤i ≤n (j )).

diam Y N u (B i

j ) ≤ε,

(7)

∀x,u ∈B i

j

1≤j ≤m,1≤i ≤n (j ) , Nx −Nu 0≤sup 0≤t ≤T f t,x (t ),x (t −τ),x (t −τ) −f t,u (t ),u (t −τ),u (t −τ)

≤ Nx −N u (x ) 0+ N u (x )−N u (u ) 0.(8)

Nx −N u (x ) 0

=sup 0≤t ≤T

f t,x (t ),x (t −τ),x (t −τ) −f t,x (t ),x (t −τ),u (t −τ)

≤k sup 0≤t ≤T

x (t −τ)−u (t −τ)

=k x (s )−u (s ) 0≤k x −u 1≤k diam X B i

j .

(9)

(6),(9)

Nx −N u (x ) 0

≤kη+kε.

(10)

(7)

∀x,u ∈B i

j

1≤j ≤m,1≤i ≤n (j ) ,

N u (x )−Nu 0= N u (x )−N u (u ) 0

≤ε,

(11)

(8),(10)

(11)

∀x,u ∈B i

j

1≤j ≤m j ,1≤i ≤n (j ) , Nx −Nu 0≤

kη+(k +1)ε,

εαY (NB )≤kαX (B ),

N k -3

τ∈ −

T 2,0 ∪(0,T 2)

∀x ∈C 1

T ,

T

x (t )−x (t − τ) 2d t ≤ τ

2 1+| τ|T

T 0

x (t ) 2

d t.

(12)

1)

τ∈

0,T 2 ,

∀r ∈[0,T ],

T

x (t )−x (t − τ) 2

d t =

r +T

r

x (t )−x (t − τ) 2

d t =

r +T

r

t

t − τ

x

(σ)d σ

2

d t

≤ τ r +T

r

t

t − τ x (σ) 2

d σd t ≤

τ r +T

r − τ

σ+ τσ

x (σ)

2d t d σ=

τ2 r +T

r − τ

x (σ) 2

d σ= τ2

r

r − τ

x (σ) 2

d σ+ τ2

r +T

r x (σ) 2

d σ

= τ2

r r − τ

x (σ)

2d σ+ τ2

T 0

x

(σ) 2

d σ.

r

[0,T]

T

0 x (t )−x (t − τ) 2

d t ≤ τ

2

T

x (σ) 2

d σ+min

r ∈[0,T ]

r

r −

τ x (σ) 2d σ .

(13)

1

93

min

r ∈[0,T ]

r

r − τ

x (σ) 2

d σ

τ≤ T 0

x (σ) 2

d σT r

r −

τ x (σ) 2d σ≤ τT T 0

x (σ) 2

d σ,(4)

T

0 x (t )−x (t − τ) 2d t ≤ τ

2 1+ τT

T

x (σ) 2

d σ.

2) τ∈ −

T

2,0

,

T

0 x (t )−x (t − τ) 2d t =

T − τ− τ

x (s )−x (s + τ) 2

st =

T

x (s )−x (s −| τ|) 2

d s.(14)

1)

(14)

T

x (t )−x (t − τ) 2d t ≤ τ

2 1+| τ|T

T

x (t ) 2

d t.

1

(H 1),(H 2)

(A 1)

a 1,a 2,a 3,a 4,∀(t,x,y,z )∈R ×R 3

f (t,x,y,z )

≤f (t,x,y,z )+a 1|x |+a 2|y |+a 3|z |+a 4

f (t,x,y,z )

≤−f (t,x,y,z )+a 1|x |+a 2|y |+a 3|z |+a 4;

(A 2)

δ=1−a 1T −a 2T −a 3>0

(2)

Lx =λNx +λp ,λ∈(0,1),x ∈X ,

x (t )=λf t,x (t ),x (t −τ),x (t −τ)

+λp (t ).

(15)

ξ∈[0,T ], x (ξ)

≤M 1,

(16)

M 1

λ

x (t )t 0,t 1,x (t 0)=x (t 1)=0,

x (t 0)≥x (t 0−τ),x (t 1)≤x (t 1−τ).x (t 0−τ)>M f

t 0,x (t 0),x (t 0−τ),x (t 0−τ)

+p (t 0)=0,

H 2x (t 0−τ)≤M ;x (t 1−τ)<−M f t 1,x (t 1),x (t 1−τ),x (t 1−τ)

+p (t 1)=0,(H 2)x (t 1−τ)≥−M ,

(16)

(16),∀t ∈[0,T ],x (t )=x (ξ)+

t

ξ

x (t )d t ,

x (t ) ≤ x (ξ)

+

t

ξ x (t )

d t ≤M 1+

T

x (t ) d t.

x 0≤M1+ T

x (t)

d t.(17)

T 0

x (t)

d t

(15)0T

0=

T

0f

t,x(t),x(t−τ),x (t−τ)

d t+

T

p(t)d t,

T

0f

t,x(t),x(t−τ),x (t−τ)

d t=−

T

p(t)d t≤

T

p(t)

d t.(18)

(15),(17),(18)(A1)

T 0

x (t)

d t

≤ T

f

t,x(t),x(t−τ),x (t−τ)

d t+

T

p(t)

d t

≤ T

f

t,x(t),x(t−τ),x (t−τ)

d t+a1

T

x(t)

d t+a2

T

x(t−τ)

d t +a3

T

x (t−τ)

d t+a4T+

T

p(t)

d t

≤2 T

p(t)

d t+a1

T

x(t)

d t+a2

T−τ

−τ

x(t)

d t+a3

T−τ

−τ

x (t)

d t+a4T

=2

T

p(t)

d t+a1

T

x(t)

d t+a2

T

x(t)

d t+a3

T

x (t)

d t+a4T

≤2 T

p(t)

d t+(a1+a2)T x 0+a3

T

x (t)

d t+a4T

≤2 T

p(t)

d t+(a1+a2)T

M1+

T

x (t)

d t

+a3

T

x (t)

d t+a4T

=2

T

p(t)

dt+(a1+a2)T M1+

(a1+a2)T+a3

T

x (t)

d t+a4T,

[1−a1T−a2T−a3]

T

x (t)

d t≤2

T

p(t)

d t+(a1+a2)T M1+a4T.(19)

(A2) T

x (t)

d t≤M2(M2λ).

x 0≤M1+

T

x (t)

d t≤M1+M2=M3.

Ω=

x(t): x 0,M=max{M,M3}+1,Ω1

(R1),1(R2)Y×X[·,·]

1

95

[y,x ]=

T

y (t )x (t )d t Q :Y →coker L

Q (y )=

1T

T

y (t )d t ,

∀x ∈ker L ∩∂Ω,

x

|x |=M ,

QN (x )+Qy,x QN (−x )+Qy,x

=M

2T 2

T

0 f (t,M,M,0)+p (t ) d t · T 0

f (t,−M,−M,0)+p (t ) d t.(H 2)

(R 2)211

x ∈Ω,

Lx =Nx +p ,

(2)

T -

f t,x (t ),x (t −τ),x (t −τ) =f 1 x (t ) +f 2 x (t −τ) +f 3 t,x (t −τ)

,

x (t )=f 1 x (t ) +f 2 x (t −τ) +f 3 t,x (t −τ)

+p (t ).

(20)

2(H 1),(H 2)(B 1) f 2(x 1)−f 2(x 2)

≤L |x 1−x 2|,∀x 1,x 2∈R ,L

(B 2)τ∈ k ∈Z

kT −δ,kT )∪(kT,kT +δ ,δ

0<δ<

T

2

0<δ<

231−k L

(20)x (t )

(15)

(H 2)

1 x 0≤M 1+ T 0

x (t ) d t , T 0

x (t ) d t

τ∈(kT,kT +δ],k ∈Z ,

τ =τ−kT ∈(0,δ].(20)

x (t )

[0,T ]

T

0 x (t ) 2d t = T

0f 1

x (t )

x (t )d t +

T

0f 2 x (t −τ)

x

(t )d t +

T

f 3

x (t −τ)

x (t )d t +

T

p (t )x (t )dt

=

T 0 f 2 x (t −τ) −f 2 x (t )

x (t )d t +

T

0f 2 x (t )

x (t )d t

+

T

f 3 t,x

(t −τ) −f 3(t,0) x (t )d t +

T

0f 3(t,0)x

(t )d t +

T

p (t )x (t )d t

≤L

T 0

x (t −τ)−x (t )

x (t ) d t +k

T

0 x

(t −τ) x (t ) d t

+M 4

T

x (t )

d t + p 0

T

x (t ) d t

≤L

T

x (t −τ )−x (t ) 2d t

12

T

|x (t )|2

d t

1

2

+k

T

x (t −τ ) 2d t ]1

2

T

x (t ) 2d t 1

2

+M4

T

x (t)

d t+ p 0

T

x (t)

d t,(21)

M4=max

t∈[0,T]

f3(t,0)

,3(21)

T 0

x (t)

2

d t

≤L

τ 2

1+

τ

T

T

x (t)

2

d t

1

2

T

x (t)

2

d t

1

2

+k

T

x (t)

2

d t+(M4+ p 0)

T

x (t)

d t

≤Lτ

1+

τ

T

T

x (t)

2

d t+k

T

x (t)

2

d t+(M4+ p 0)

T

x (t)

d t

≤Lδ

1+

δ

T

T

x (t)

2

d t+k

T

x (t)

2

d t+(M4+ p 0)

T

x (t)

d t

≤

3

2

Lδ

T

x (t)

2

d t+k

T

x (t)

2

d t+

M4+ p 0

T

x (t)

d t,

1−

3

2

Lδ−k

T

x (t)

2

d t≤

M4+ p 0

T

x (t)

d t.(22)

0<δ<

2

3

1−k

L

,1−

2

3

Lδ−k>0,(22)λ

M2>0,

T

x (t)

d t≤M2.1

3(H1)-(H2)τ∈{jT,j∈Z}(20)

x(t)(15)1 x 0≤M1+

T

x (t)

d t,

T

x (t)

d t

(20)x (t)0T

T 0

x (t)

2

d t

=

T

0f1

x(t)

x (t)d t+

T

f2

x(t)

x (t)d t+

T

f3

t,x (t−τ)

x (t)d t+

T

p(t)x (t)d t

=

T

0f3

t,x (t)

x (t)d t+

T

p(t)x (t)d t

=

T

f3

t,x (t)

−f3(t,0)

x (t)d t+

T

f3(t,0)x (t)d t+

T

p(t)x (t)d t

≤k T

x (t)

2

d t+M4

T

x (t)

d t+ p 0

T

x (t)

d t,

(1−k)

T

x (t)

2

d t≤

M4+ p 0

T

x (t)

d t.(23)

1

97

k <1,

T 0

x (t ) 2

d t ≤ M 4+ p 01−k

2.

1

x (t )=

1

8πx (t )±112π

x (t −τ)+g t,x (t ),x (t −τ),x (t −τ) +sin t,g (t,x,y,z )∈C (R ×R 3,R )z t 2π−g

|∂g ∂z |<1.a 1=14π,a 2=16π,a 3=0,a 4=2g M g M =max g (t,x,y,z ) ,

f (t,x,y,z )+a 1|x |+a 2|y |+a 3|z |+a 4

=1

8πx ±

112πy +g (t,x,y,z )+14π|x |+16π

|y |+2g M ≥1

8π|x |+

112π

|y |+g M ≥ f (t,x,y,z ) ,δ=1−

1

4π

·2π−

16π

·2π=1−

12

−

13

=

16

>0.

1

A 1

(H 1),(H 2)

1

n -

Li´e nard

,1990,11(A):297–307

(Ge Weigao.Harmonic Solution of

n -dimension Li´e nard Equation.Chinese Annals of Mathematics ,

1990,11A(3):297–307)

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1991,16(2):183–1903

Duffing

x (t )+g (t,x (t −τ(t )))=p (t )

2π

,1994,39(3):201–203

(Huang Xiankai,Xiang Zigui.On the Existence of

2π-periodic Solutions of Duffing Type Equation

x

(t )+g (t,x (t −τ(t )))=p (t ).Chinese Science Bulletin ,1994,39(3):201–203)

4

Li´e nard

,1998,18(4):565–570

(Li Yongkun.Periodic Solutions of the Li´e nard Equation with Deviating Arguments.J.Mathematical Research and Exposition ,1998,18(4):565–570)5Ma Shiwang,Wang Zhicheng,Yu Jianshe.

Coincidence Degree and Periodic Solutions of Duffing

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Math.,No.568,Berlin:Springer-Verlag,1977

FOR NONLINEAR NEUTRAL DELAY

DIFFERENTIAL EQUATION

Ren Jingli

(Department of Mathematics,Zhengzhou University,Zhengzhou450052)

Ren Baoxian

(Department of Computer Science,Anyang Teachers College,Anyang455000)

Ge Weigao

(Department of Mathematics,Beijing Institute of Technology,Beijing100081)

Abstract By using the abstract continuity theorem,we study the problem of periodic solution of the nonlinear neutral differential equation x (t)=f(t,x(t),x(t−τ),x (t−τ))+p(t) and obtain sufficient conditions for the existence of periodic solutions related with delayτ. Key words Neutral differential equation,periodic solution,Fredholm operator,

k-set contractive operator