具有局部弱稳定退化解二阶非线性方程的奇摄动问题

秦赵娜,姚静荪

(安徽师范大学 数学计算机科学学院,安徽 芜湖 241003)

0 引 言

解的存在性及其渐近估计.

(3)

则称退化方程F(t,u,u′)=0的解u=u(t)在[a,b]上是(Iq)稳定的.

且在D0(u)∩([a,a+δ1]×)中,在D0(u)∩([b-δ1,b]×)中,则称退化方程F(t,u,u′)=0的解u=u(t)在[a,b]上是稳定的.

定义3如果存在常数δ2>0,使得当u(a)≠A时,在D(u)∩([a,a+δ2/2]×2)中,Fy′(t,y,y′)≤0,且当u(b)≠B时,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)给出.则称退化方程F(t,u,u′)=0的解u=u(t)是局部弱稳定的.

假设:

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

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

1 主要结果

|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为待定正常数.

证明: 不妨设u(a)≠A,u(b)≠B.用EST表示指数型小项.

1) 当u=u(t)在[a,b]上是(Iq)稳定时,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).显然

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分别由定义1和定义3给出.下证

εβ″-F(t,β,β′)≤0,

(5)

εα″-F(t,α,α′)≥0,

(6)

当t∈[a,b-δ0/2]时,

(7)

当t∈[a+δ0/2,b]时,

(8)

r=2C1+‖u″‖.

(9)

Γ(ε)≤δ1/2.

(10)

WR≤δ1/2.

(11)

当t∈[b-δ0/2,b]时,同理存在ε2>0,使得对式(9)给定的r>0,当0<ε≤ε2时,式(5)成立.

令ε0=min{ε1,ε2,ε3},对式(9)给定的r>0,当0<ε≤ε0时,对一切t∈[a,b]都有式(5)成立.

同理对式(9)给定的r>0,当0<ε≤ε0时,对一切t∈[a,b]式(6)也成立.于是,由(H2)根据文献[9]中引理2.2.1知,在(Iq)稳定的条件下结论成立.

当t∈[a,b-δ0/2]时,式(7)成立,且

(12)

当t∈[a+δ0/2,b]时,式(8)成立,且

(13)

取

r=2(C2+‖u″‖).

(14)

当t∈[a,a+δ0/2]时,

当t∈[b-δ0/2,b]时,同理存在ε5>0,使得对式(14)式给定的r>0,当0<ε≤ε5时,式(5)成立.当t∈[a+δ0/2,b-δ0/2]时,

Γ(ε)≤δ3/2.

(15)

WL+WR≤δ3/2.

(16)

故对式(14)给定的r>0,当0<ε≤min{ε4,ε5,ε6}时,对一切t∈[a,b]都有式(5)成立.

另一方面,当t∈[a+δ0/2,b-δ0/2]时,

当t∈[a,a+δ0/2]∪[b-δ0/2,b]时,

综上所述,令ε0=min{ε4,ε5,ε6,ε7},对式(14)给定的r>0,当0<ε≤ε0时,对一切t∈[a,b]都有式(4)～(6)成立.于是由(H2),根据文献[9]中引理2.2.1知情形①成立.

② 当A>u(a),B

(17)

(18)

(19)

(20)

(21)

(22)

r=2[C3(1+M2)+‖u″‖].

(23)

当t∈[a+δ0/2,b-δ0/2]时,

根据式(18)同理可证,对式(23)给定的r>0,当0<ε≤ε8时,式(6)成立.

当t∈[a,a+δ0/2]时,

同理可证当t∈[b-δ0/2,b]时,对上述的ε9>0及式(23)给定的r>0,当0<ε≤ε9时,式(6)成立.

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

根据式(21),(22)同理可证,对式(23)给定的r>0,存在ε11>0,使得当0<ε≤ε11时,对t∈[a,a+δ0/2]都有式(6)成立.

综上所述,令ε0=min{ε8,ε9,ε10,ε11},对上述取定的r>0,当0<ε≤ε0时,对一切t∈[a,b]都有式(4)～(6)成立.于是由(H2)根据文献[9]中引理2.2.1知,情形②得证.

③ 当Au(b)时,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的证明过程可证.

④ 当A

作为问题(1)-(2)的特例,对二阶半线性问题:

和二阶拟线性问题:

应用定理1得到的结果与文献[6]的结论一致.

2 应用实例

例1

例2

|y(t,ε)-u(t)|≤WL(t,ε)+WR(t,ε)+Γ(ε),

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