秦趙娜,姚靜蓀
(安徽師范大學 數學計算機科學學院,安徽 蕪湖 241003)
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]上是穩定的.
定義3如果存在常數δ2>0,使得當u(a)≠A時,在D(u)∩([a,a+δ2/2]×2)中,Fy′(t,y,y′)≤0,且當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)是局部弱穩定的.
假設:
(H1) 退化方程F(t,u,u′)=0具有解u=u(t)∈C2[a,b],且u=u(t)是局部弱穩定的;
(H2)F,Fy,Fy′∈C(D(u)),Fy′(t,y,y′)在[a,b]×2上是有界函數.
|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.
證明: 不妨設u(a)≠A,u(b)≠B.用EST表示指數型小項.
1) 當u=u(t)在[a,b]上是(Iq)穩定時,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t∈[a,b-δ0/2]時,
(7)
當t∈[a+δ0/2,b]時,
(8)
r=2C1+‖u″‖.
(9)
Γ(ε)≤δ1/2.
(10)
WR≤δ1/2.
(11)
當t∈[b-δ0/2,b]時,同理存在ε2>0,使得對式(9)給定的r>0,當0<ε≤ε2時,式(5)成立.
令ε0=min{ε1,ε2,ε3},對式(9)給定的r>0,當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0<ε≤ε0時,對一切t∈[a,b]式(6)也成立.于是,由(H2)根據文獻[9]中引理2.2.1知,在(Iq)穩定的條件下結論成立.
當t∈[a,b-δ0/2]時,式(7)成立,且
(12)
當t∈[a+δ0/2,b]時,式(8)成立,且
(13)
取
r=2(C2+‖u″‖).
(14)
當t∈[a,a+δ0/2]時,
當t∈[b-δ0/2,b]時,同理存在ε5>0,使得對式(14)式給定的r>0,當0<ε≤ε5時,式(5)成立.當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0<ε≤min{ε4,ε5,ε6}時,對一切t∈[a,b]都有式(5)成立.
另一方面,當t∈[a+δ0/2,b-δ0/2]時,
當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0<ε≤ε0時,對一切t∈[a,b]都有式(4)~(6)成立.于是由(H2),根據文獻[9]中引理2.2.1知情形①成立.
② 當A>u(a),B
(17)
(18)
(19)
(20)
(21)
(22)
r=2[C3(1+M2)+‖u″‖].
(23)
當t∈[a+δ0/2,b-δ0/2]時,
根據式(18)同理可證,對式(23)給定的r>0,當0<ε≤ε8時,式(6)成立.
當t∈[a,a+δ0/2]時,
同理可證當t∈[b-δ0/2,b]時,對上述的ε9>0及式(23)給定的r>0,當0<ε≤ε9時,式(6)成立.
εβ″-F(t,β,β′)≤ε(u″+C3+M2C3)-Γ(m-M2Γ)≤0;
根據式(21),(22)同理可證,對式(23)給定的r>0,存在ε11>0,使得當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0<ε≤ε0時,對一切t∈[a,b]都有式(4)~(6)成立.于是由(H2)根據文獻[9]中引理2.2.1知,情形②得證.
③ 當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:
應用定理1得到的結果與文獻[6]的結論一致.
例1
例2
|y(t,ε)-u(t)|≤WL(t,ε)+WR(t,ε)+Γ(ε),
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