具有局部弱穩(wěn)定退化解二階非線性方程的奇攝動問題

秦趙娜,姚靜蓀

(安徽師范大學(xué) 數(shù)學(xué)計算機科學(xué)學(xué)院,安徽 蕪湖 241003)

0 引 言

解的存在性及其漸近估計.

(3)

則稱退化方程F(t,u,u′)=0的解u=u(t)在[a,b]上是(Iq)穩(wěn)定的.

且在D0(u)∩([a,a+δ1]×)中,在D0(u)∩([b-δ1,b]×)中,則稱退化方程F(t,u,u′)=0的解u=u(t)在[a,b]上是穩(wěn)定的.

定義3如果存在常數(shù)δ2>0,使得當(dāng)u(a)≠A時,在D(u)∩([a,a+δ2/2]×2)中,Fy′(t,y,y′)≤0,且當(dāng)u(b)≠B時,在D(u)∩([b-δ2/2,b]×2)中,Fy′(t,y,y′)≥0,其中D(u)∶={(t,y,y′)|a≤t≤b,|y-u(t)|≤d(t),|y′|<+∞},d(t)由式(3)給出.則稱退化方程F(t,u,u′)=0的解u=u(t)是局部弱穩(wěn)定的.

假設(shè):

(H1) 退化方程F(t,u,u′)=0具有解u=u(t)∈C2[a,b],且u=u(t)是局部弱穩(wěn)定的;

(H2)F,Fy,Fy′∈C(D(u)),Fy′(t,y,y′)在[a,b]×2上是有界函數(shù).

1 主要結(jié)果

|y(t,ε)-u(t)|≤WL(t,ε)+WR(t,ε)+Γ(ε),

其中:

σ=m1/2q[(q+1)(2q+1)!]-1/2;r>0為待定正常數(shù).

證明: 不妨設(shè)u(a)≠A,u(b)≠B.用EST表示指數(shù)型小項.

1) 當(dāng)u=u(t)在[a,b]上是(Iq)穩(wěn)定時,令α(t,ε)=u(t)-WL(t,ε)-WR(t,ε)-Γ(ε),β(t,ε)=u(t)+WL(t,ε)+WR(t,ε)+Γ(ε).顯然

α(t,ε)≤β(t,ε),α(a,ε)≤A≤β(a,ε),α(b,ε)≤B≤β(b,ε).

(4)

令δ0=min{δ1,δ2,(b-a)/2},其中δ1,δ2分別由定義1和定義3給出.下證

εβ″-F(t,β,β′)≤0,

(5)

εα″-F(t,α,α′)≥0,

(6)

當(dāng)t∈[a,b-δ0/2]時,

(7)

當(dāng)t∈[a+δ0/2,b]時,

(8)

r=2C1+‖u″‖.

(9)

Γ(ε)≤δ1/2.

(10)

WR≤δ1/2.

(11)

當(dāng)t∈[b-δ0/2,b]時,同理存在ε2>0,使得對式(9)給定的r>0,當(dāng)0<ε≤ε2時,式(5)成立.

令ε0=min{ε1,ε2,ε3},對式(9)給定的r>0,當(dāng)0<ε≤ε0時,對一切t∈[a,b]都有式(5)成立.

同理對式(9)給定的r>0,當(dāng)0<ε≤ε0時,對一切t∈[a,b]式(6)也成立.于是,由(H2)根據(jù)文獻(xiàn)[9]中引理2.2.1知,在(Iq)穩(wěn)定的條件下結(jié)論成立.

當(dāng)t∈[a,b-δ0/2]時,式(7)成立,且

(12)

當(dāng)t∈[a+δ0/2,b]時,式(8)成立,且

(13)

取

r=2(C2+‖u″‖).

(14)

當(dāng)t∈[a,a+δ0/2]時,

當(dāng)t∈[b-δ0/2,b]時,同理存在ε5>0,使得對式(14)式給定的r>0,當(dāng)0<ε≤ε5時,式(5)成立.當(dāng)t∈[a+δ0/2,b-δ0/2]時,

Γ(ε)≤δ3/2.

(15)

WL+WR≤δ3/2.

(16)

故對式(14)給定的r>0,當(dāng)0<ε≤min{ε4,ε5,ε6}時,對一切t∈[a,b]都有式(5)成立.

另一方面,當(dāng)t∈[a+δ0/2,b-δ0/2]時,

當(dāng)t∈[a,a+δ0/2]∪[b-δ0/2,b]時,

綜上所述,令ε0=min{ε4,ε5,ε6,ε7},對式(14)給定的r>0,當(dāng)0<ε≤ε0時,對一切t∈[a,b]都有式(4)～(6)成立.于是由(H2),根據(jù)文獻(xiàn)[9]中引理2.2.1知情形①成立.

② 當(dāng)A>u(a),B

(17)

(18)

(19)

(20)

(21)

(22)

r=2[C3(1+M2)+‖u″‖].

(23)

當(dāng)t∈[a+δ0/2,b-δ0/2]時,

根據(jù)式(18)同理可證,對式(23)給定的r>0,當(dāng)0<ε≤ε8時,式(6)成立.

當(dāng)t∈[a,a+δ0/2]時,

同理可證當(dāng)t∈[b-δ0/2,b]時,對上述的ε9>0及式(23)給定的r>0,當(dāng)0<ε≤ε9時,式(6)成立.

εβ″-F(t,β,β′)≤ε(u″+C3+M2C3)-Γ(m-M2Γ)≤0;

根據(jù)式(21),(22)同理可證,對式(23)給定的r>0,存在ε11>0,使得當(dāng)0<ε≤ε11時,對t∈[a,a+δ0/2]都有式(6)成立.

綜上所述,令ε0=min{ε8,ε9,ε10,ε11},對上述取定的r>0,當(dāng)0<ε≤ε0時,對一切t∈[a,b]都有式(4)～(6)成立.于是由(H2)根據(jù)文獻(xiàn)[9]中引理2.2.1知,情形②得證.

③ 當(dāng)Au(b)時,令α(t,ε)=u(t)-WL(t,ε)-Γ(ε),β(t,ε)=u(t)+WR(t,ε)+Γ(ε).仿情形②的證明過程可證.

④ 當(dāng)A

作為問題(1)-(2)的特例,對二階半線性問題:

和二階擬線性問題:

應(yīng)用定理1得到的結(jié)果與文獻(xiàn)[6]的結(jié)論一致.

2 應(yīng)用實例

例1

例2

|y(t,ε)-u(t)|≤WL(t,ε)+WR(t,ε)+Γ(ε),

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