关于非线性边值问题几个存在性定理的新结果

New Results of Some Existence Theorems on Nonlinear Boundary Value Problems

摘要: 本文运用同胚理论,研究微分方程边值问题解的存在性与唯一性,得到两个基本定理,推广了Brown(in Annali.Mat.Pura.Appli.1975,106:205～214)的结论,取消了有界性假设,并把我们的结果用于有限维的情况,考虑了非线性守恒系统在扰动情况下(Newton类运动方程)周期解的存在性与唯一性问题。把同胚用于这类问题的研究目前还是新的。
- 同胚 /
- 周期解 /
- 大范围反函数定理
Abstract: With the use of the homeomorohism theory and fixed point theory, the existence and uniqueness of solutions to boundary value problems are investigated. Two basic theorems are obtained without the boundness condition, which generalizes results of Brown. When our results are applied to the existence and uniqueness of periodic solutions for nonlinear perturbed conservative systems (Newtonian equations of motion), the existence and uniqueness of the solution obtained. The results in this note seem less restrictive than those of the former papers we have seen. Mean while, as far as we know, it seems that applying the homeomorphism theory to the research of this kind of problem is new.
- homeomorphism /
- existence and uniqueness /
- global inverse theorem

[1]	Brown K J.Nonlinear boundary value problems and a global inverse function theorem[J].Annali Mat Pura Appli,1975,106:205～214
[2]	Plastock R.Homeomorphisms between Banach spaces[J].Trans Amer Math Soc,1974,200:169～183
[3]	Brown K J,Lin S S.Periodically perturbed conservative system and a global inverse function theorem[J].Nonlinear Analysis,1980,TMA,4:193～201
[4]	Mawhin J.Recent trends in nonlinear boundary value problems[Z].In:Proc Int Conf Nonlinear Oscillations[C],Berlin:Academic-Verlag,1975:75～90
[5]	Bates P W.Solutions of nonlinear elliptic systems with meshed spectra[J].Nonlinear Analysis,1979,TMA,4(6):1023～1030
[6]	Dunford N,Schwartz J.Linear Opeator[M],volume Ⅰ,New York:Interscience,1963
[7]	Dunford N,Schwartz J.Linear Operator[M],Volume Ⅱ,New York:interscience,1963
[8]	Adams R A.Sobolev Spaces[M].New York:Academic Press,1975
[9]	Shen Zuhe,Wolfe M A.On the existence of periodically perturbed conservative systems[J].J Math Anal and Appl,1989,151:78～83
[10]	Lamberto Cesari.Optimization-Theory and Applications,Problems with Ordinary Differential Equations[M].Springer-verlag,1983.