非线性二阶中立型时滞微分方程的振动和非振动准则

26(3)(2006,6),325–334

(030008)

(571158)

(100080)

(E)

MR(2000)34K11

1

[r(t)(x(t)+P(t)x(t−τ)) ] +Q1(t)f(x(t−σ1))−Q2(t)g(x(t−σ2))=0(E)τ>0,σi≥0,p(t)=0,p,Q i∈C([t0,+∞),R),i=1,2.,r∈C([t0,+∞),(0,+∞)), f,g∈C(r,R)xf(x)>0,xg(x)>0,(x=0).

[1,2]

Kulenovi´c[3]

[3](E).2(E)

[3][1][4]3(E)

[5–8](E)

(E)

2003-09-28,2004-09-22.32626

1.1(E)[T x,+∞),T x≥t0x(t)

sup{|x(t)|:t≥T}>0,∀T≥T x

1.2(E)

1.3(E)

2

(E)

(c1)f g Lipchitz Lipchitz L f(A),L g(A),A

(c2)R(t)= t

t0

1

r(s)

d s,t≥t0,Q i(t)≥0,

+∞

R(t)Q i(t)dt<+∞,i=1,2.

(c3)a>0,aQ1(t)−Q2(t)≥0.

2.1(c1)–(c3)P0

|P(t)|≤P0<1

2

(1)

(E)

Banach X={x|x∈C([t0,+∞),R) x =sup

t≥t0

|x(t)|},

N1≥M1>0

1

1−P01−M1

P0

<

1

P0

(2)

A1={x∈X:M1≤x(t)≤N1,t≥t0}.

A1X

L1=max{L f(A1),L g(A1)},α1=max

x∈A1

{f(x)},

β1=min

x∈A1{f(x)},α2=max

x∈A1

{g(x)},β2=min

x∈A1

{g(x)}

(2)t≥t0,

+∞

t1R(s)[Q1(s)+Q2(s)]d s<

1−P0

L1

,(3)

0≤

+∞

t1

R(s)[α1Q1(s)−β2Q2(s)]d s≤(1−P0)N1−1(4)

+∞

t1

R(s)[β1Q1(s)−α2Q2(s)]d s≥0,(5)

3

327

T 1:A 1→X

(T 1x )(t )=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1−P (t )x (t −τ)+R (t ) +∞t [Q 1(s )f (x (s −σ1))−Q 2(s )g (x (s −σ2))]d s +

t

t 1

R (s )[Q 1(s )f (x (s −σ1))−Q 2(s )g (x (s −σ2))]d s,

t ≥t 1

(T 1x (t 1),

t 0≤t ≤t 1.T 1x

(1)

(4),

(T 1x )(t )≤1+P 0N 1+

+∞

t 1

R (s )[α1Q 1(s )−β2Q 2(s )]d s ≤N 1,t ≥t 1,

(1),(2)

(5),

(T 1x )(t )≥1−P 0N 1≥M 1

T 1x ⊂A 1.

T 1

A 1

∀x 1,x 2∈A 1

t ≥t 1,

(3)

|(T 1x 1)(t )−(T 1x 2)(t )|≤P 0|x 1(t −τ)−x 2(t −τ)|

+R (t ) +∞

t

Q 1(s )|f (x 1(s −σ1))−f (x 2(s −σ1))|d s

+R (t ) +∞

t

Q 2(s )|g (x 1(s −σ2))−g (x 2(s −σ2))|d s +

t

t 1R (s )Q 1(s )|f (x 1(s −σ1))−f (x 2(s −σ1))|d s +

t

t 1R (s )Q 2(s )|g (x 1(s −σ2))−g (x 2(s −σ2))|d s

≤P 0 x 1−x 2 +L 1 x 1−x 2

+∞

t R (s )[Q 1(s )+Q 2(s )]d s

=q 0 x 1−x 2 ,

q 0=P 0+

+∞

t

R (s )[Q 1(s )+Q 2(s )]d s <1,T 1A 1

Banach

T 1A 1x 1,

x 1(t )

(E)

2.1

P 1=lim sup t →+∞

{P (t )},P 2=lim inf t →+∞

{P (t )},

2.2(c 1)–(c 3)(i)0

P (t )≥0;

(6)(ii)

−1

P (t )≤0;

(7)

(E)

2.2

2.1,

A 2={x ∈X :M 2≤x (t )≤N 2,t ≥t 0}.

1−P14

3P1+1

[(1−P1)−M2].

(6)∃t2≥t0t≥t2

0≤P(t)<1+3P1

4

.

+∞

t2R(s)[Q1(s)+Q2(s)]d s<

3(1−P1)

4L2

,

0≤

+∞

t2R(s)[α1Q1(s)−β2Q2(s)]d s≤N2+P1−1,

+∞

t2

R(s)[β1Q1(s)−α2Q2(s)]d s≥0, L2=max{L f(A2),L g(A2)},α1,β1,α2,β2

T2:A2→X

(T2x)(t)=⎧

⎪⎪⎪

⎪⎨

⎪⎪⎪

⎪⎩

1−P1−P(t)x(t−τ)+R(t)

+∞

t

[Q1(s)f(x(s−σ1))−Q2(s)g(x(s−σ2))]d s +

t

t2

R(s)[Q1(s)f(x(s−σ1))−Q2(s)g(x(s−σ2))]d s,t≥t2 (T2x)(t2),t0≤t≤t2.

(7)

A3={x∈X:M3≤x(t)≤N3,t≥t0}. N3≥M3>0

04 3

(7)∃t3≥t0t≥t3

−1<3P2−1

4

≤P(t)≤0, +∞

t3R(s)[Q1(s)+Q2(s)]d s<

3(1+P2)

4L3

,

0≤

+∞

t3R(s)[α1Q1(s)−β2Q2(s)]d s≤(1+P2)

3

4

N3−1

, +∞

t3

R(s)[β1Q1(s)−α2Q2(s)]d s≥0,3329 L3=max{L f(A3),L g(A3)},α1,β1,α2,β2

T3:A3→X

(T3x)(t)=⎧

⎪⎪⎪

⎪⎨

⎪⎪⎪

⎪⎩

1+P2−P(t)x(t−τ)+R(t)

+∞

t

[Q1(s)f(x(s−σ1))−Q2(s)g(x(s−σ2))]d s +

t

t3

R(s)[Q1(s)f(x(s−σ1))−Q2(s)g(x(s−σ2))]d s,t≥t3 (T3x)(t3),t0≤t≤t3.

2.2

2.3(c1)–(c3)

(i)0P(t)>1;(8) (ii)−∞P(t)<−1.(9) P1,P2(E)

2.3 2.1,

A4={x∈X:M4≤x(t)≤N4,t≥t0}.

N4>M4>0

01

P1+ε

−1

P2−ε

N4,

1

P2−ε

P2+ε

P1+ε

<1 0<ε<

P2−1

2

.

(8)∃t4≥t0t≥t4

P2−ε≤P(t)≤P1+ε,(10) +∞

t4R(s)[Q1(s)+Q2(s)]d s<

(P2−ε)−1

L4

,

0≤

+∞

t4R(s)[α1Q1(s)−β2Q2(s)]d s≤(P2−ε)N4−1,

+∞

t4

R(s)[β1Q1(s)−α2Q2(s)]d s≥0, L4=max{L f(A4),L g(A4)},α1,β1,α2,β233026 T4:A4→X

(T4x)(t)=⎧

⎪⎪⎪

⎪⎨

⎪⎪⎪

⎪⎩

1

P(t+τ)

−x(t+τ)

P(t+τ)

+R(t+τ)

P(t+τ)

+∞

t+τ

[Q1(s)f(x(s−σ1))−Q2(s)g(x(s−σ2))]d s

+1

P(t+τ)

t+τ

t4

R(s)[Q1(s)f(x(s−σ1))−Q2(s)g(x(s−σ2))]d s,t≥t4 (T4x)(t4),t0≤t≤t4.

t+τ≥t0+max{σ1,σ2}.

(9)∃t5≥t0t≥t5

P2−δA5={x∈X:M5≤x(t)≤N5,t≥t0}.

N5>M5>0

M5<

−1

1+P2−δ

<

−1

1+P1+δ

≤N5,

+∞

t5R(s)[Q1(s)+Q2(s)]d s<−

1+P1+δ

L5

,

0≤

+∞

t5R(s)[α1Q1(s)−β2Q2(s)]d s≤

P1+δ

P2−δ

[1+(1+P2−δ)M5] +∞

t5

R(s)[β1Q1(s)−α2Q2(s)]d s≥0,

L5=max{L f(A5),L g(A5)},α1,β1,α2,β2 T5:A5→X

(T5x)(t)=⎧

⎪⎪⎪

⎪⎨

⎪⎪⎪

⎪⎩

−1

P(t+τ)

−x(t+τ)

P(t+τ)

+R(t+τ)

P(t+τ)

+∞

t+τ

[Q1(s)f(x(s−σ1))−Q2(s)g(x(s−σ2))]d s

+1

P(t+τ)

t+τ

t5

R(s)[Q1(s)f(x(s−σ1))−Q2(s)g(x(s−σ2))]d s,t≥t5, (T5x)(t5),t0≤t≤t5.

T5:A5→A5(E) 2.3

1[3]Q2(t)≡0f(x)≡x[9] (E)Q2(t)≡00≤P(t)≤1

(E)

(r(t)x (t)) +Q(t)x(t)=0(E )

[1] 2.2[4]2 +∞

t

Q(t)d t<+∞,

+∞t (

+∞

s

Q(u)d u)2

r(s)

≤1

4

+∞

t

Q(s)d s

3331

(E )

1

[t(x(t)+P(t)x(t−τ)) ] + 1

t2

+

1

t3

x35(t−σ1)−

2

t3

x3(t−σ2)=0(12)

P(t) 2.1-2.3R(t)=ln t,Q1(t)=1t2+1t3>0,Q2(t)=2t3,

+∞1R(t)Q i(t)d t<+∞,i=1,2.aQ1(t)−Q2(t)≥0,a>0,t≥2−a

a

(12)[1,3,4,9]

3

(E)(E)

+∞

1

r(t)

d t=+∞(13)

3.1(13)0≤P(t)≤1,Q1(t)≥0,Q2(t)f(x)x≥α>

0,(x=0),

+∞

Q1(s)[1−P(s−σ1)]d s=+∞

(E)

(E)x(t).

Z(t)=x(t)+P(t)x(t−τ)(14) Z(t)(E)t1≥t0t≥t1(r(t)Z (t)) <0.

r(t)Z (t))t

Z (t)≥0,t≥t1.(15) Z (t)<0

Z(t)≤Z(t1)+r(t1)z (t1)

t

t1

1

r(s)

d s,t≥t1.

(13),lim

t→+∞

Z(t)=−∞,Z(t)

(E)

(r(t)Z (t)) +αQ1(t)x(t−σ1)≤0,t≥t1. (14)

(r(t)Z (t)) +αQ1(t)[Z(t−σ1)−P(t−σ1)x(t−σ1−τ)≤0 Z(t)≥x(t)(15),

(r(t)Z (t)) +αQ1(t)[1−P(t−σ1)]Z(t−σ1)≤0,t≥t1.33226

W(t)=r(t)Z (t) Z(t−σ1)

W (t)=(r(t)Z (t))

Z(t−σ1)

−r(t)Z

(t)Z (t−σ1)

Z2(t−σ1)

≤−αQ1(t)[1−P(t−σ1)] 0≤W(t)≤W(t1)−α

t

t1

Q1(t)[1−P(s−σ1)]d s

3.1

2

1 t (x(t)+

1

1+t

(x(t−τ))

+

k

t

x3

t−

π

2

−(1−3t)x5

t−

π

3

=0,(16)

k>0, 3.1(16)[5–10]

(16).

(E)H(t)

[r(t)(x(t)+P(t)x(t−τ)) ] +Q1(t)f(x(t−σ1))−Q2(t)g(x(t−σ2))=H(t),(E1)

(E).

3.2σ1=0,σ2=τ>0,P(t)≥0,

f(x) x ≥α>0,g(x)

x

≥β>0(x=0)(17)

T≥t0T≤s1−βQ2(t)≥αQ1(t)P(t)≥0,t∈[s1,t1]∪[s2,t2],(18)

H(t)≤0,t∈[s1,t1]H(t)≥0,t∈[s2,t2],(19)

B i={u∈C1[s i,t i]:u(t)=0,u(s i)=u(t i)=0},i=1,2.

u∈B i,

K i(u)= t

i

s i

[αQ1(t)u2(t)−r(t)u 2(t)]d t≥0,i=1,2.(20)

(E1)

(E1)x(t),

Z(t)=x(t)+P(t)x(t−τ) Z(t)

V(t)=−r(t)Z (t)

Z(t)

"c":

V (t)=V2(t)

r(t)

+

Q1(t)f(x(t))−Q2(t)g(x(t−τ))

Z(t)

−H(t)

Z(t)

(21)

(18)(19),T≤s1H(t)≤0,t∈[s1,t1],

−βQ2(t)≥αQ1(t)P(t)≥0,t∈[s1,t1].

(21),

V (t)≥V2(t)

r(t)

+

αQ1(t)(x(t)−βQ2(t)(x(t−τ)

Z(t)

≥V 2(t)

r(t)

+αQ1(t),t∈[s1,t1].

u(t)∈B1(20).u2(t)(22)[s1,t1] t

1 s1u2(t)V (t)d t≥

t

1

s1

u2(t)V2(t)

r(t)

d t+

t

1

s1

αQ1(t)u2(t)d t

u(s1)=u(t1)=0)

− t

1

s1

2u(t)u (t)V(t)d t≥

t

1

s1

u2(t)V2(t)

r(t)

d t+

t

1

s1

αQ1(t)u2(t)d t

K1(u)+

t

1

s1

r(t)u (t)+

u(t)V(t)

r(t)

2

d t≤0(23)

(20)(23),

r(t)u (t)+u(t)V(t)=0,t∈[s1,t1].

(u(t)

Z(t)

) =0,t∈[s1,t1].

u(t)

Z(t)

=const,Z(t)>0,u(t)=0,t∈(s1,t1).c=0,u(s1)=u(t1)=0 (E1)[s2,t2]H(t)≥0B2

(E1) 3.2

2(E1)

(r(t)(x(t) ) +Q(t)f(x(t))=H(t),(E2)

3.2[11]1,[12][1][8]33426

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OSCILLATION AND NONOSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR NEUTRAL DELAY

DIFFERENTIAL EQUATIONS

Zhang Zhiyu Wang Xiaoxia

(North China Institute of Technology,Taiyuan030008)

Lin Shizhong

(Department of Mathematics,Hainan University,Haikou571158)

Yu Yuanhong

(Institute of Applied Mathematics,Chinese Academy of Sciences,Beijing100080) Abstract Some new oscillation and nonoscillation criteria for second order nonlinear neu-tral delay differential equations are established.Illustrative examples are given.Our theorems improve and generalize several known results.

Key words Positive and negative coefficient,second order neutral equation,oscillation, nonoscillation.