具有局部弱稳定退化解二阶非线性方程的奇摄动问题
2013-12-03秦赵娜姚静荪
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A 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. 例1 例2 |y(t,ε)-u(t)|≤WL(t,ε)+WR(t,ε)+Γ(ε), [1] YAO Jing-sun.Asymptotic Solution of the Three Points Boundary Value Problem for Nonlinear Third Order Equation [J].Mathematica Applicata,2009,22(2): 437-442. [2] YAO Jing-sun,MO Jia-qi,CHEN Xiu.The Solvability for Singularly Perturbed Quasi-linear Differential System of Second Order [J].J of Math,2010,30(6): 959-965. [3] WU You-ping,YAO Jing-sun,ZHUANG Hong-yan.A Class of Singularly Perturbed Problem with Two Parameters and Nonlinear Boundary Value Conditions [J].Applied Mathematics A Journal of Chinese Universities: Ser A,2011,26(3): 288-294.(吴有萍,姚静荪,庄红艳.一类具非线性边值条件的双参数奇摄动问题 [J].高校应用数学学报: A辑,2011,26(3): 288-294.) [4] CHEN Ting,YAO Jing-sun.The Shock Solution for a Class of Quasilinear Singularly Perturbed Boundary Value Problems [J].Chin Quart J of Math,2012,27(3): 317-322. [5] LIU Yan,YAO Jing-sun.A Class of Singular Perturbed Problems with Nonlinear Boundary Value Conditions for Higher Order Equations [J].Applied Mathematics A Journal of Chinese Universities: Ser A,2012,27(2): 175-181.(刘燕,姚静荪.一类高阶方程的非线性边界条件的奇摄动问题 [J].高校应用数学学报: A辑,2012,27(2): 175-181.) [6] Chang K W,Howes F A.Nonlinear Singular Perturbation Phenomena：Theory and Applications [M].New York: Springer-Verlag,1984: 6-66. [7] 林宗池,周明儒.应用数学中的摄动方法 [M].南京: 江苏教育出版社,1995: 205-222. [8] ZHOU Ming-ru.Boundary and Corner Layer Behavior in Singularly Perturbed Robin Boundary Value Problem [J].Ann of Diff Eqs,2005,21(4): 639-647. [9] 刘树德,鲁世平,姚静荪,等.奇异摄动边界层和内层理论 [M].北京: 科学出版社,2012: 15-40. [10] LI Qing,YAO Jing-sun.Singularly Perturbed Problem for the Second-Order Nonlinear Equation with a Locally Strongly Stable Reduce Solution [J].Journal of Anhui Normal University: Natural Science,2012,31(6): 514-518.(李青,姚静荪.具有局部强稳定退化解的二阶非线性方程奇摄动问题 [J].安徽师范大学学报: 自然科学版,2012,31(6): 514-518.)
2 应用实例
