引 言
高阶非线性微分方程在数学物理、工程数学、生态数学、物理化学等学科中都有很广泛的应用．例如,描述控制论中的电气系统、流体力学中的流体流动以及电磁波的传播等一些实际应用领域中都有其重要的研究成果[1-19]．非线性三阶微分方程边值问题已成为近年来的研究热点[4-17]．
文献[6]研究了非线性两点边值问题:
通过使用 Krasnosel’skii 锥拉伸与压缩不动点定理，确定了该边值问题正解的存在性和非存在性的一些结论．
文献[7]考虑了
通过使用Krasnosel’skii锥拉伸与压缩不动点定理，得到了该边值问题单独一个正解和多个正解的存在性．
文献[17]研究了非线性三点边值问题：
应用不动点指数理论得到了该边值问题正解的存在性准则．
Mo和Yang等也应用泛函分析、奇摄动、微分不等式讨论了一些大气物理、生物数学、理论物理、传染病传播等实际问题中的非线性微分方程的定性理论和定量方法的数学模型[20-33]．
本文讨论如下三阶非线性微分方程：
w‴(t)+f(t,w(t))=0, t∈(0,1)．
(1)
其满足边值条件
w(0)=w′(0)=0, w″(1)=g(w(1))
(2)
的非线性边界值问题，其中f∈C((0,1)×R→R)，g是关于其变量的光滑函数．利用Banach不动点定理证明了边值问题(1)和(2)解的存在唯一性．
1 引 理
设E=C[0,1]是一个Banach空间，定义其范数是‖w‖=maxt∈[0,1]|w(t)|．
引理1 设w,h∈E,α∈R，则边值问题
(3)
有唯一解：
其中
证明 设
(4)
其中A表示一个常数．对式(4)两边分别求导, 得到
w′(t)=-
(t-s)h(s)ds+2At．
(5)
由式(4)和(5)可知，显然w(0)=w′(0)=0成立．对式(5)两边求导得
于是
因此得到
(6)
引理证毕．
在式(6)中，用f(s,w(s))代替h(s),用g(w(1))代替α,可以得到
定义积分算子T：
(7)
其中
显然求边值问题(1)和(2)的解等价于寻求算子T在空间E中的不动点w0, 使得
Tw0=w0, w0∈E．
现有如下引理：
引理2[19](Banach 不动点定理) 设X是一个非空的度量空间，若X是完备的度量空间，且X→X是一个压缩映像，则T在X中有一个不动点．
2 主 要 结 论
定理1 设函数f(t,w)是一个满足 Lipschitz 条件的函数, 即存在正常数L1，使得
|f(t,w)-f(t,v)|≤L1|w-v|, w,v∈E,
(8)
且函数g也是一个Lipschitz 函数，即存在正常数L2，使得
|g(w)-g(v)|≤L2|w-v|, w,v∈E．
(9)
设
β
(10)
则边值问题(1)和(2)存在唯一解．
证明 显然Banach空间E是一个完备的度量空间．由式(7)和引理1 知,T:E→E．仅需证明T是一个压缩映像．事实上，不难得到
(11)
对于w,v∈E,t∈[0,1], 由式(7)～(9)和(11)可得
|(Tw-Tv)(t)|≤
再由假设(10)有
‖Tw-Tv‖≤β‖w-v‖,
其中0<β<1．于是算子T是一个压缩映像．由引理2可知，算子T在空间E中有唯一不动点，即边值问题(1)和(2)在空间E中有唯一解．
由式(7)和定理1, 可确定一个迭代程序：
从而可求非线性边值问题(1)和(2)的唯一解．
定理2 设函数f(s,w)是一个连续函数，关于w的偏导存在且有界，且函数g满足Lipschitz 条件，即存在常数L3使得
|g(w)-g(v)|≤L3|w-v|,
取
γ
(12)
其中
Df
(13)
则边值问题(1)和(2)存在唯一解．
证明 类似于定理1的证明,仅需证明算子T是一个压缩映像．根据式(7)、(11)～(13)，对于w,v∈E,t∈[0,1],可得
|(Tw-Tv)(t)|≤
故
‖Tw-Tv‖≤γ‖w-v‖,
其中0<γ<1．即算子T是一个压缩映像．由引理2可知，算子T在空间E中有唯一不动点．即边值问题(1)和(2)在空间E中有唯一解．
3 举 例
例 设w(t)表示流体随时间变化的流速，w′(t)，w″(t)和w‴(t)分别表示流体在时间t时的隅角、弯矩和剪切力．函数w(t)在t=1处的弯矩为sin w．这时边值问题(1)和(2)为如下三阶非线性微分方程边值问题：
(14)
由定理1，边值问题(14)存在唯一解．通过迭代方法, 可以得到唯一流速解满足
4 总 结
本文研究了非线性三阶微分方程边值问题，提出了适当的假设条件，利用Banach不动点定理确定了边值问题存在唯一解的两个充分条件，并给出了求解的迭代程序式．对于更弱的条件将有待进一步深入探讨．
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