近年来, 国内外许多学者关于微分算子和微分方程边值问题特征值的依赖性问题进行了大量研究, 并且取得了一系列研究成果. 对于二阶微分算子, 1987年, P$ \rm{\ddot{o}} $eschel和Trubowitz在文献[1] 中证明了Dirichlet边界条件下第$ n $个特征值关于势函数$ q(x) $是可微的. 1993年, Dauge和Helffer在文献[2, 3] 中讨论了Neumann边界条件下的特征值是关于区间端点的可微函数. 1996年, Kong和Zettl在更弱的条件下进行了研究, 得到了类似的结论, 并且考虑了特征值关于区间端点、边界条件、方程系数和权函数的可微性, 以及当区间长度趋于零时, Dirichlet特征值和Neumann特征值的性质^{[4, 5]}. 后来, 学者们将这些成果推广到了四阶及更高阶情形, 并得到类似的结果^[6-10]. 随着具有转移条件Sturm-Liouville边值问题在自伴性、特征函数的完备性、Green函数等方面研究工作的进展^[11-13], 学者们进一步研究了具有转移条件的二阶Sturm-Liouville问题^[14]及四阶和高阶Sturm-Liouville微分算子特征值的依赖性^[15-18], 证明了特征值和特征函数不仅连续依赖于问题(即问题的有关参数), 而且给出特征值关于问题中参数可微依赖的微分表达式. 此外, 还有奇数阶, 左定, 奇异等其它情形的一些推广工作可以参考文献[19-22].

在以上研究成果中, 特征参数都只出现在微分算式或微分方程中, 其实还可以出现在边界条件中^{[12, 13, 22-25]}. 许多力学及工程技术领域中的一些问题, 如滑竿上的弦振动问题, 再如热传导方程或波动方程以及一些偏微分方程经过分离变量法之后可以得到边界条件中含有特征参数的微分算子问题^[26]. 这类微分方程及边界条件中都带有特征参数的微分算子问题是算子谱理论中的重要研究内容之一, 也是近年来数学物理及科技工作者们研究的热点, 并从不同角度对此问题进行了大量研究^{[23, 24, 27-30]}. 但是, 关于这类问题的特征值依赖性的研究成果目前甚少, 然而, 2020年, Zhang和Li考虑了边界条件一端点含有特征参数的二阶正则Sturm-Liouville算子特征值的依赖性^[31].

另外, 具有转移条件(即内部具有不连续性)的Sturm-Liouville问题有着重要的应用前景, 例如热传导和质量转移问题, 衍射问题, 中间有结点的弦振动问题等, 都可以转化为内部具有不连续性的微分算子问题. 其中有许多实际问题往往需要转化为具有转移条件的高阶微分算子问题, 而四阶微分算子又是偶数阶高阶微分算子的一个典型形式, 因此, 我们对一类边界条件一端带有谱参数且具有转移条件的四阶微分算子的自伴性及特征值的依赖性研究也是很有必要的.

本文虽然是将文献[31]的结论推广到了四阶情形, 但是, 因为我们考虑的微分算式更具有一般性且带有特征参数的边界条件与前人工作中带谱参数的边界条件不同, 所以首先要在适当的Hilbert空间中证明问题的自共轭性并给出其特征值存在的条件, 进一步考虑问题特征值的依赖性. 本文不仅给出了特征值关于方程系数函数, 权函数, 转移条件相关矩阵和边界点的微分表达式, 而且也给出了特征值关于特征参数依赖的边界条件系数矩阵的Fr$ \acute{\rm{e}} $chet导数及不连续点左右两侧的一阶导数表达式.

本文安排如下: 第1节为引言; 第2节给出了本文要研究的基本问题; 第3节证明算子$ T $是自共轭的; 第4节给出了特征值和特征函数的连续性定理; 第5节得出了特征值关于问题相关参数的微分表达式.

考虑一般形式的四阶微分方程

(2.1) $ \begin{equation} l(y): = (p_{2}(x)y'')''-(p_{1}(x)y')'+q(x)y = \lambda w(x)y, x\in J' = (a', c)\cup(c, b'). \end{equation} $

一端具有特征参数的分离型边界条件

(2.2) $ \begin{eqnarray} & &l_{1}y: = (\alpha_{1}\lambda+\beta_{1})y(a)-(\alpha_{2}\lambda+\beta_{2})((p_{2}y'')'-p_{1}y')(a) = 0, \end{eqnarray} $

(2.3) $ \begin{eqnarray} & &l_{2}y: = (\alpha_{3}\lambda+\beta_{3})y'(a)-(\alpha_{4}\lambda+\beta_{4})(p_{2}y'')(a) = 0, \end{eqnarray} $

(2.4) $ \begin{eqnarray} &&l_{3}y: = \cos\gamma y(b)-\sin\gamma ((p_{2}y'')'-p_{1}y')(b) = 0, \end{eqnarray} $

(2.5) $ \begin{eqnarray} &&l_{4}y: = \cos\gamma y'(b)-\sin\gamma (p_{2}y'')(b) = 0 \end{eqnarray} $

及转移条件

(2.6) $ \begin{equation} CY(c-)+DY(c+) = 0, \end{equation} $

其中$ -\infty<a'<a<c<b<b'<+\infty $, 且令$ J = [a, c)\cup(c, b], \lambda\in{\mathbb C} $是特征参数,

(2.7) $ \begin{equation} p_{2}, p_{1}, q, w: J'\rightarrow {\mathbb R}, \frac{1}{p_{2}}, p_{1}, q, w\, \in L^{1}_{loc}(J'), w>0{\quad} a.e. {\quad} J', Y = \left(\begin{array}{c}y\\y'\\p_{2}y''\\(p_{2}y'')'-p_{1}y'\end{array}\right), \end{equation} $

(2.8) $ \begin{equation} \alpha_{i}, \beta_{i}\in {\mathbb R}, i = 1, 2, 3, 4. \theta_{1} = \alpha_{1}\beta_{2}-\beta_{1}\alpha_{2}>0, \theta_{2} = \alpha_{4}\beta_{3}-\beta_{4}\alpha_{3}>0, \gamma\in (0, \pi]. \end{equation} $

$ Y(c\pm) = \lim\limits_{x\rightarrow c\pm}Y(x). C = (c_{ij}), D = (d_{ij}) $是$ 4\times 4 $实矩阵, det$ C = \eta^{2}, $ det$ D = \xi^{2}, $$ \eta>0, \xi>0 $, 且满足

(2.9) $ \begin{eqnarray} \xi CJ_{0}C^{T} = \eta DJ_{0}D^{T}, \, J_{0} = \left(\begin{array}{cccc}0&\ 0\ &0&\ 1\\0&0&-1&\ 0\\ 0&1&0&\ 0\\-1&0&0&\ 0\end{array}\right), \end{eqnarray} $

$ C^{T} $表示矩阵$ C $的转置, $ J_{0}^{T} = J_{0}^{-1} = -J_{0}, J_{0}\cdot J_{0} = -E $, 其中$ E $为四阶单位矩阵.

定义2.1   令微分算式$ l(y): = (p_{2}(x)y'')''-(p_{1}(x)y')'+q(x)y, $由微分算式$ l(y) $生成的最大算子$ T_{M} $的定义为

$ D_{M} = \{y\in L^{2}_{w}(J)|y, y', p_{2}y'', (p_{2}y'')'-p_{1}y'\in AC_{loc}(J), l(y)\in L^{2}_{w}(J)\}, $

$ T_{M}(y) = l(y), y\in D_{M} $

其中$ D_{M} $称为算式$ l(y) $的最大算子域.

引理2.1^[32]   (Naimark Patching Lemma) 对任意的复数组$ \sigma_{1}, \sigma_{2}, \sigma_{3}, \sigma_{4} $及$ \zeta_{1}, \zeta_{2}, \zeta_{3}, \zeta_{4} $, 存在$ y\in D_{M} $, 满足

$ y(a) = \sigma_{1}, y'(a) = \sigma_{2}, p_{2}y''(a) = \sigma_{3}, ((p_{2}y'')'-p_{1}y')(a) = \sigma_{4}; $

$ y(b) = \zeta_{1}, y'(b) = \zeta_{2}, p_{2}y''(b) = \zeta_{3}, ((p_{2}y'')'-p_{1}y')(b) = \zeta_{4}. $

定义2.2   若微分方程(2.1) 在端点$ a $是正则的, 函数$ u\in D_{M}, $则$ u, u', p_{2}u'', (p_{2}u'')'-p_{1}u' $在$ a $点是连续的, 即

$ u(a) = \lim\limits_{x\rightarrow a^{+}}u(x), u'(a) = \lim\limits_{x\rightarrow a^{+}}u'(x), $

$ p_{2}u''(a) = \lim\limits_{x\rightarrow a^{+}}p_{2}u''(x), ((p_{2}u'')'-p_{1}u')(a) = \lim\limits_{x\rightarrow a^{+}}((p_{2}u'')'-p_{1}u')(x), $

存在且有限, 类似可以定义$ b $点是正则的.

用$ L^{2}_{w}(J) $表示区间$ J $上权函数为$ w $平方可积的复值可微函数全体组成的空间. 在此空间中定义如下内积

$ \langle f, g\rangle _{1} = \eta \int^{c}_{a}f(x)\overline{g}(x)w(x){\rm d}x+\xi \int^{b}_{c}f(x)\overline{g}(x)w(x){\rm d}x, \forall f, g\in L^{2}_{w}(J). $

易知$ H_{1} = (L^{2}_{w}(J), \langle\cdot, \cdot\rangle_{1}) $是Hilbert空间. 在直和空间$ H = H_{1}\oplus{\mathbb C}\oplus{\mathbb C} $中定义内积, 令$ \rho_{1} = \frac{1}{\theta_{1}}, \rho_{2} = \frac{1}{\theta_{2}}, $

$ \langle {\cal F}, {\cal G}\rangle = \langle f, g\rangle _{1}+\eta\rho_{1}h_{1} \overline{k}_{1}+\eta\rho_{2}h_{2}\overline{k}_{2}, \forall {\cal F} = (f(x), h_{1}, h_{2})^{T}, {\cal G} = (g(x), k_{1}, k_{2})^{T}\in H. $

在Hilbert空间$ H $中定义算子$ T $如下

$ T{\cal F} = (\frac{l(f)}{w}, \beta_{1}f(a)-\beta_{2}((p_{2}f'')'-p_{1}f')(a), -(\beta_{3}f'(a)-\beta_{4}(p_{2}f'')(a)))^{T}, $

$ \forall {\cal F} = (f, -(\alpha_{1}f(a)-\alpha_{2}((p_{2}f'')'-p_{1}f')(a)), \alpha_{3}f'(a)-\alpha_{4}(p_{2}f'')(a))^{T}\in D(T), $

其中

$ \begin{eqnarray*} D(T)& = &\Big\{(f(x), h_{1}, h_{2})^{T}\in H|f, f', p_{2}f'', (p_{2}f'')'-p_{1}f'\in AC_{loc}(J), \\ &&{\quad} l(f)\in H_{1}, l_{j}f = 0, j = 3, 4, CF(c-)+DF(c+) = 0, \\ &&{\quad} h_{1} = -(\alpha_{1}f(a)-\alpha_{2}((p_{2}f'')'-p_{1}f')(a)), h_{2} = \alpha_{3}f'(a)-\alpha_{4}(p_{2}f'')(a)\Big\}. \end{eqnarray*} $

由定义2.2可知, 对于任意的$ (f(x), h_{1}, h_{2})^{T}\in D(T), $函数$ f(x), f'(x), p_{2}f''(x), $$ ((p_{2}f'')'-p_{1}f')(x) $在$ [a, c] $和$ [c, b] $上是连续的.

为了简便, 令

(2.10) $ \begin{eqnarray} M(f) = \beta_{1}f(a)-\beta_{2}((p_{2}f'')'-p_{1}f')(a), M'(f) = \alpha_{1}f(a)-\alpha_{2}((p_{2}f'')'-p_{1}f')(a), \end{eqnarray} $

(2.11) $ \begin{eqnarray} N(f) = \beta_{3}f'(a)-\beta_{4}(p_{2}f'')(a), N'(f) = \alpha_{3}f'(a)-\alpha_{4}(p_{2}f'')(a), \end{eqnarray} $

即$ {\cal F} = (f, -M'(f), N'(f))^{T}, $

(2.12) $ \begin{eqnarray} T{\cal F} = (\frac{l(f)}{w}, M(f), -N(f))^{T} = (\lambda f, -\lambda M'(f), \lambda N'(f))^{T} = \lambda {\cal F}. \end{eqnarray} $

因此, 我们可以通过在$ H $中讨论算子方程$ T{\cal F} = \lambda {\cal F} $来研究问题(2.1)–(2.9).

由算子$ T $的定义可知:

引理3.1   边值问题(2.1)–(2.9) 的特征值与$ T $的特征值相同, 特征函数是算子$ T $的相应向量特征函数的第一个分量.

引理3.2  算子$ T $的定义域$ D(T) $在$ H $中是稠密的.

证  设$ {\cal F} = (f(x), h_{1}, h_{2})^{T}\in H, $且$ {\cal F}\perp D(T), $并且令$ \widetilde{C^{\infty}_{0}} $表示下列函数的集合

$ \phi(x) = \left\{ \begin{array}{ll} \varphi_{1}(x), &x\in[a, c), \\ \varphi_{2}(x), &x\in(c, b], \end{array} \right. $

其中, $ \varphi_{1}(x)\in C^{\infty}_{0}[a, c), \varphi_{2}(x)\in C^{\infty}_{0}(c, b]. $于是, $ \widetilde{C^{\infty}_{0}}\oplus\{0\}\oplus\{0\} \subset D(T), (0\in {\mathbb C}). $设$ {\cal U} = (u(x), 0, 0)^{T}\in \widetilde{C^{\infty}_{0}}\oplus\{0\}\oplus\{0\}, $则$ {\cal F}\perp {\cal U}. $由

$ \langle {\cal F}, {\cal U}\rangle = \langle f, u\rangle _{1}+\eta\rho_{1}h_{1} \overline{k}_{1}+\eta\rho_{2}h_{2}\overline{k}_{2} = \langle f, u\rangle _{1} = 0, $

知$ f(x) $在$ H_{1} $中正交于$ \widetilde{C^{\infty}_{0}}, $故$ f(x) $为零. 设$ {\cal G} = (g(x), k_{1}, 0)^{T}\in \widetilde{C^{\infty}_{0}}\oplus\{0\} (\widetilde{C^{\infty}_{0}}\oplus\{0\}\subset\widetilde{C^{\infty}_{0}}\oplus\{0\}\oplus\{0\}), $则$ \langle {\cal F}, {\cal G}\rangle = \langle f, g\rangle _{1}+\eta\rho_{1}h_{1}\overline{k}_{1} = 0, $由于$ k_{1} = -M'(g) $是任意选取的, 故$ h_{1} = 0. $又设$ {\cal G} = (g(x), 0, k_{2})^{T}\in D(T), $则$ \langle {\cal F}, {\cal G}\rangle = \langle f, g\rangle_{1}+\eta\rho_{2}h_{2}\overline{k}_{2} = 0, $由于$ k_{2} = N'(g) $是任意选取的, 故$ h_{2} = 0. $于是$ {\cal F} = (0, 0, 0)^{T}. $所以, 与$ D(T) $正交的只有零元素, 从而证得$ D(T) $在$ H $中是稠密的.

定理3.1   线性算子$ T $是定义在$ H $上的自共轭算子.

证  对任意的$ {\cal F}, {\cal G}\in D(T), $由分部积分法可得

$ \begin{eqnarray*} \langle T{\cal F}, {\cal G}\rangle& = &\langle \frac{l(f)}{w}, g\rangle_{1}+\eta\rho_{1}M(f)(-M'(\overline{g}))-\eta\rho_{2}N(f)N'(\overline{g})\\ & = &\eta\int_{a}^{c}[(p_{2}f'')''-(p_{1}f')'+qf]\overline{g}{\rm d}x+\xi\int_{c}^{b}[(p_{2}f'')''-(p_{1}f')'+qf]\overline{g}{\rm d}x\\ &&-\eta\rho_{1}M(f)M'(\overline{g})-\eta\rho_{2}N(f)N'(\overline{g})\\ & = &\langle {\cal F}, T{\cal G}\rangle-\eta[f, g](c-)+\eta[f, g](a)-\xi[f, g](b)+\xi[f, g](c+)\\ &&+\eta\rho_{1}(M'(f)M(\overline{g})-M(f)M'(\overline{g}))+\eta\rho_{2}(N'(f)N(\overline{g})-N(f)N'(\overline{g})), \end{eqnarray*} $

其中

(3.1) $ \begin{eqnarray} [f, g](x)& = &f(x)((p_{2}\overline{g}'')'-p_{1}\overline{g}')(x)-f'(x)(p_{2}\overline{g}'')(x){}\\ &&+(p_{2}f'')(x)\overline{g}'(x)-((p_{2}f'')'-p_{1}f')(x)\overline{g}(x) {} \\ & = &-G^{\ast}(x)J_{0}F(x), \end{eqnarray} $

且$ G^{\ast} $表示向量$ G $的共轭转置, $ G = (g, g', p_{2}g'', (p_{2}g'')'-p_{1}g')^{T}. $

由$ l_{3}f = 0, l_{3}\overline{g} = 0, $可知$ f(b)((p_{2}\overline{g}'')'-p_{1}\overline{g}')(b)-((p_{2}f'')'-p_{1}f') (b)\overline{g}(b) = 0. $再由$ l_{4}f = 0, l_{4}\overline{g} = 0, $可知$ f'(b)(p_{2}\overline{g}'')(b) -(p_{2}f'')(b)\overline{g}'(b) = 0. $因此, $ [f, g](b) = 0. $由(2.10)式和(2.11) 式分别可以得到如下的(3.2) 式和(3.3) 式

(3.2) $ \begin{equation} \rho_{1}(M'(f)M(\overline{g})-M(f)M'(\overline{g})) = -f(a)((p_{2}\overline{g}'')'-p_{1}\overline{g}')(a) +((p_{2}f'')'-p_{1}f')(a)\overline{g}(a), \end{equation} $

(3.3) $ \begin{equation} \rho_{2}(N'(f)N(\overline{g})-N(f)N'(\overline{g})) = f'(a)(p_{2}\overline{g}'')(a)-(p_{2}f'')(a)\overline{g}'(a). \end{equation} $

再由(3.2)和(3.3) 式得

$ \eta\rho_{1}(M'(f)M(\overline{g})-M(f)M'(\overline{g}))+\eta\rho_{2}(N'(f)N(\overline{g})-N(f)N'(\overline{g})) = -\eta[f, g](a). $

由转移条件(2.6) 知

(3.4) $ \begin{eqnarray} F(c-) = -C^{-1}D F(c+), {\quad} G^{\ast}(c-) = -G^{\ast}(c+)D^{\ast}(C^{\ast})^{-1}, \end{eqnarray} $

将(3.4) 式代入(3.1) 式并结合(2.9) 式可得

(3.5) $ \begin{eqnarray} \eta[f, g](c-) = -\eta G^{\ast}(c+)D^{\ast}(C^{\ast})^{-1}J_{0}C^{-1}DF(c+) = \xi[f, g](c+), \end{eqnarray} $

因此, $ \langle T{\cal F}, {\cal G}\rangle = \langle {\cal F}, T{\cal G}\rangle, $故算子$ T $是对称的. 接下来根据自伴算子的定义证明算子$ T $是自伴的.

下面只需证明: 若对任意的$ {\cal F} = (f(x), -M'(f), N'(f))^{T}\in D(T), $有$ \langle T{\cal F}, {\cal Z}\rangle = \langle {\cal F}, {\cal U}\rangle $成立, 则$ {\cal Z}\in D(T) $且$ T{\cal Z} = {\cal U}. $其中, $ {\cal Z} = (z(x), m_{1}, m_{2})^{T}, {\cal U} = (u(x), n_{1}, n_{2})^{T}, $即

($ \rm{i} $) $ z(x), z'(x), (p_{2}z'')(x), ((p_{2}z'')'-p_{1}z')(x)\in AC_{loc}(J), \frac{l(z)}{w}\in H_{1}; $

($ \rm{ii} $) $ m_{1} = -M'(z) = -[\alpha_{1}z(a)-\alpha_{2}((p_{2}z'')'-p_{1}z')(a)], m_{2} = N'(z) = \alpha_{3}z'(a)-\alpha_{4}(p_{2}z'')(a); $

($ \rm{iii} $) $ l_{i}z = 0, i = 3, 4; $

($ \rm{iv} $) $ u(x) = \frac{l(z)(x)}{w}; $

($ \rm{v} $) $ CZ(c-)+DZ(c+) = 0; $

($ \rm{vi} $) $ n_{1} = M(z) = \beta_{1}z(a)-\beta_{2}((p_{2}z'')'-p_{1}z')(a), n_{2} = -N(z) = -(\beta_{3}z'(a)-\beta_{4}(p_{2}z'')(a)). $

对任意的$ {\cal F}\in \widetilde{C^{\infty}_{0}}\oplus\{0\}\oplus\{0\}\subset D(T), $由$ \langle T{\cal F}, {\cal Z}\rangle = \langle {\cal F}, {\cal U}\rangle $得

$ \eta\int_{a}^{c}l(f)\overline{z}{\rm d}x+\xi\int_{c}^{b}l(f)\overline{z}{\rm d}x = \eta\int_{a}^{c}f\overline{u}w{\rm d}x+\xi\int_{c}^{b}f\overline{u}w{\rm d}x, $

即

(3.6) $ \begin{equation} \langle \frac{l(f)}{w}, z\rangle_{1} = \langle f, u\rangle_{1}, \end{equation} $

由标准的Sturm-Liouville理论可知$ z(x)\in D(T), $故($ \rm{i} $) 成立.

因为算子$ T $是对称的, 所以有$ \langle T{\cal F}, {\cal Z}\rangle = \langle {\cal F}, T{\cal Z}\rangle, $再由上述$ {\cal F} $的取法可得$ \langle \frac{l(f)}{w}, z\rangle_{1} = \langle f, \frac{l(z)}{w}\rangle_{1}, $因此结合(3.6) 式可得$ \langle f, \frac{l(z)}{w}\rangle_{1} = \langle f, u\rangle_{1}, $即($ \rm{iv} $) 成立.

再由($ \rm{iv} $) 可知, 对任意的$ {\cal F}\in D(T) $方程$ \langle T{\cal F}, {\cal Z}\rangle = \langle {\cal F}, {\cal U}\rangle $即为

$ \begin{eqnarray*} & &\eta\int_{a}^{c}l(f)\overline{z}{\rm d}x+\xi\int_{c}^{b}l(f)\overline{z}{\rm d}x+\eta\rho_{1}M(f)\overline{m}_{1}-\eta\rho_{2}N(f)\overline{m}_{2}\\ & = &\eta\int_{a}^{c}f\overline{u}w{\rm d}x+\xi\int_{c}^{b}f\overline{u}w{\rm d}x-\eta\rho_{1}M'(f)\overline{n}_{1}+\eta\rho_{2}N'(f)\overline{n}_{2}\\ & = &\eta\int_{a}^{c}l(\overline{z})f{\rm d}x+\xi\int_{c}^{b}l(\overline{z})f{\rm d}x-\eta\rho_{1}M'(f)\overline{n}_{1}+\eta\rho_{2}N'(f)\overline{n}_{2}, \end{eqnarray*} $

于是

$ \langle \frac{l(f)}{w}, z\rangle_{1} = \langle f, \frac{l(z)}{w}\rangle_{1}-\eta\rho_{1}(M(f)\overline{m}_{1}+M'(f)\overline{n}_{1})+\eta\rho_{2}(N(f)\overline{m}_{2} +N'(f)\overline{n}_{2}), $

而

$ \langle \frac{l(f)}{w}, z\rangle_{1} = \langle f, \frac{l(z)}{w}\rangle_{1}-\eta[f, z](c-)+\eta[f, z](a)-\xi[f, z](b)+\xi[f, z](c+). $

因此结合(3.1) 式有

(3.7) $ \begin{eqnarray} &&-\eta\rho_{1}(M(f)\overline{m}_{1}+M'(f)\overline{n}_{1})+\eta\rho_{2}(N(f)\overline{m}_{2} +N'(f)\overline{n}_{2}){}\\ & = &-\eta[f, z](c-)+\eta[f, z](a)-\xi[f, z](b)+\xi[f, z](c+){}\\ & = &\eta Z^{\ast}(c-)J_{0}F(c-)-\eta Z^{\ast}(a)J_{0}F(a)+\xi Z^{\ast}(b)J_{0}F(b)-\xi Z^{\ast}(c+)J_{0}F(c+). \end{eqnarray} $

根据引理2.1, 存在函数$ {\cal F}\in D(T), $使得

(3.8) $ \begin{eqnarray} F(b) = F(c-) = F(c+) = 0, \end{eqnarray} $

$ f(a) = \alpha_{2}, f'(a) = (p_{2}f'')(a) = 0, ((p_{2}f'')'-p_{1}f')(a) = \alpha_{1}, $

此时, $ M'(f) = N(f) = N'(f) = 0, M(f) = -\theta_{1}, $于是, 由(3.7) 式可知

$ m_{1} = -M'(z) = -[\alpha_{1}z(a)-\alpha_{2}((p_{2}z'')'-p_{1}z')(a)]. $

同理可证$ m_{2} = N'(z). $事实上, 存在函数$ {\cal F}\in D(T), $使得(3.8) 成立且满足

$ f(a) = 0, f'(a) = \alpha_{4}, (p_{2}f'')(a) = \alpha_{3}, ((p_{2}f'')'-p_{1}f')(a) = 0, $

由(3.7) 式可知$ m_{2} = N'(z) = \alpha_{3}z'(a)-\alpha_{4}(p_{2}z'')(a). $则($ \rm{ii} $) 成立. (vi) 证明类似.

下面证明($ \rm{iii} $) 成立, 选取函数$ {\cal F}\in D(T), $使得

$ F(a) = F(c-) = F(c+) = 0, $

$ f(b) = \sin\gamma, f'(b) = (p_{2}f'')(b) = 0, ((p_{2}f'')'-p_{1}f')(b) = \cos\gamma, $

此时, 由(3.7) 式可得$ l_{3}z = 0. $同理可得$ l_{4}z = 0. $则($ \rm{iii} $) 成立.

下面证明($ \rm{v} $) 成立, 选取函数$ {\cal F}\in D(T), $使得

$ F(a) = F(b) = 0, $

(3.9) $ \begin{equation} f'(c-) = (p_{2}f'')(c-) = ((p_{2}f'')'-p_{1}f')(c-) = 0, f(c-)\neq 0, \end{equation} $

则由(3.7) 式有

(3.10) $ \begin{eqnarray} &&-\eta f(c-)((p_{2}\overline{z}'')'-p_{1}\overline{z}')(c-)+\xi[f(c+)((p_{2}\overline{z}'')'-p_{1}\overline{z}')(c+){}\\ &&-f'(c+)(p_{2}\overline{z}'')(c+)+(p_{2}f'')(c+)\overline{z}'(c+)-((p_{2}f'')'-p_{1}f')(c+)\overline{z}(c+)] = 0, \end{eqnarray} $

因为$ f\in D(T), $故$ f $满足转移条件$ C\cdot F(c-)+D\cdot F(c+) = 0, $从而有

(3.11) $ \begin{eqnarray} F(c+) = -D^{-1}CF(c-) = -\frac{1}{\xi^{2}}D^{\star}C\cdot F(c-) , \end{eqnarray} $

其中

$ D^{\star} = \left(\begin{array}{cccc} N_{11}&\ -N_{21}\ &N_{31}&\ -N_{41}\\ -N_{12}&N_{22}&-N_{32}&\ N_{42}\\N_{13}&-N_{23}&N_{33}&\ -N_{43}\\ -N_{14}&N_{24}&-N_{34}&\ N_{44}\end{array}\right), $

$ N_{ij} $为$ d_{ij} $的余子式, $ D^{\star} $为矩阵$ D $的伴随矩阵. 将(3.11) 式中的$ f(c+), f'(c+), (p_{2}f'')(c+), $$ ((p_{2}f'')'-p_{1}f')(c+) $代入(3.10) 式, 并结合(3.9) 式整理得

(3.12) $ \begin{eqnarray} ((p_{2}\overline{z}'')'-p_{1}\overline{z}')(c-) & = &-\frac{1}{\xi\eta}[-(-c_{11}N_{14}+c_{21}N_{24}-c_{31}N_{34}+c_{41}N_{44})\overline{z}(c+){}\\ &&+(c_{11}N_{13}-c_{21}N_{23}+c_{31}N_{33}-c_{41}N_{43})\overline{z}'(c+){}\\ &&-(-c_{11}N_{12}+c_{21}N_{22}-c_{31}N_{32}+c_{41}N_{42})(p_{2}\overline{z}'')(c+){}\\ &&+(c_{11}N_{11}-c_{21}N_{21}+c_{31}N_{31}-c_{41}N_{41})((p_{2}\overline{z}'')'-p_{1}\overline{z}')(c+)].{\qquad} \end{eqnarray} $

同理, 与上述方法一样, 令$ f(c-) = (p_{2}f'')(c-) = ((p_{2}f'')'-p_{1}f')(c-) = 0, f'(c-)\neq 0, $我们可以得到

(3.13) $ \begin{eqnarray} (p_{2}\overline{z}'')(c-)& = &\frac{1}{\xi\eta}[-(-c_{12}N_{14}+c_{22}N_{24}-c_{32}N_{34}+c_{42}N_{44})\overline{z}(c+){}\\ &&+(c_{12}N_{13}-c_{22}N_{23}+c_{32}N_{33}-c_{42}N_{43})\overline{z}'(c+){}\\ &&-(-c_{12}N_{12}+c_{22}N_{22}-c_{32}N_{32}+c_{42}N_{42})(p_{2}\overline{z}'')(c+){}\\ &&+(c_{12}N_{11}-c_{22}N_{21}+c_{32}N_{31}-c_{42}N_{41})((p_{2}\overline{z}'')'-p_{1}\overline{z}')(c+)]. \end{eqnarray} $

令$ f(c-) = f'(c-) = ((p_{2}f'')'-p_{1}f')(c-) = 0, (p_{2}f'')(c-)\neq 0, $我们可以得到

(3.14) $ \begin{eqnarray} \overline{z}'(c-)& = &-\frac{1}{\xi\eta}[-(-c_{13}N_{14}+c_{23}N_{24}-c_{33}N_{34}+c_{43}N_{44})\overline{z}(c+){}\\ &&+(c_{13}N_{13}-c_{23}N_{23}+c_{33}N_{33}-c_{43}N_{43})\overline{z}'(c+){}\\ &&-(-c_{13}N_{12}+c_{23}N_{22}-c_{33}N_{32}+c_{43}N_{42})(p_{2}\overline{z}'')(c+){}\\ &&+(c_{13}N_{11}-c_{23}N_{21}+c_{33}N_{31}-c_{43}N_{41})((p_{2}\overline{z}'')'-p_{1}\overline{z}')(c+)]. \end{eqnarray} $

令$ f(c-) = f'(c-) = (p_{2}f'')(c-) = 0, ((p_{2}f'')'-p_{1}f')(c-)\neq 0, $我们可以得到

(3.15) $ \begin{eqnarray} \overline{z}(c-)& = &\frac{1}{\xi\eta}[-(-c_{14}N_{14}+c_{24}N_{24}-c_{34}N_{34}+c_{44}N_{44})\overline{z}(c+){}\\ &&+(c_{14}N_{13}-c_{24}N_{23}+c_{34}N_{33}-c_{44}N_{43})\overline{z}'(c+){}\\ &&-(-c_{14}N_{12}+c_{24}N_{22}-c_{34}N_{32}+c_{44}N_{42})(p_{2}\overline{z}'')(c+){}\\ &&+(c_{14}N_{11}-c_{24}N_{21}+c_{34}N_{31}-c_{44}N_{41})((p_{2}\overline{z}'')'-p_{1}\overline{z}')(c+)]. \end{eqnarray} $

结合(3.12)–(3.15) 式得

$ \begin{eqnarray*} Z(c-)& = &\frac{1}{\xi\eta} \left(\begin{array}{ccccc}c_{14}&c_{24}&c_{34}&\ c_{44}\\-c_{13}&\ -c_{23}\ &-c_{33}&\ -c_{43}\\ c_{12}&c_{22}&c_{32}&\ c_{42} \\-c_{11}&-c_{21}&-c_{31}&\ -c_{41} \end{array}\right) \left(\begin{array}{ccccc}N_{14}&N_{13}&N_{12}&\ N_{11}\\-N_{24}&\ -N_{23}\ &-N_{22}&\ -N_{21} \\N_{34}&N_{33}&N_{32}&\ N_{31}\\-N_{44}&-N_{43}&-N_{42}&\ -N_{41} \end{array}\right)Z(c+)\\ & = &-\frac{1}{\xi\eta}(CJ_{0})^{T}(D^{\star})^{T}J_{0}Z(c+). \end{eqnarray*} $

根据$ D^{\star} = \xi^{2}D^{-1}, J_{0}^{T} = -J_{0} $和(2.9) 式有

$ CZ(c-) = \frac{\xi}{\eta}(CJ_{0}C^{T})(D^{T})^{-1}J_{0}Z(c+) = -DZ(c+). $

综上所述, 线性算子$ T $在$ H $中是自共轭的.

由自共轭算子的性质可知:

推论3.1   边值问题(2.1)–(2.9) 的特征值是实的.

推论3.2   设$ \lambda_{1} $和$ \lambda_{2} $是算子$ T $的两个特征值, 且$ \lambda_{1}\neq\lambda_{2}, $则相应的特征函数$ f(x) $与$ g(x) $在下述意义下是正交的:

$ \eta\int^{c}_{a}f\overline{g}w{\rm d}x+\xi\int^{b}_{c}f\overline{g}w{\rm d}x+\eta\rho_{1}M'(f)M'(\overline{g}) +\eta\rho_{2}N'(f)N'(\overline{g}) = 0. $

但是, 在通常意义下, 算子$ T $对应的两个不同特征值的特征向量不是正交的.

根据常微分理论中解的存在唯一性定理, 给出边值问题(2.1)–(2.9) 特征值存在的充分必要条件.

为了方便, 我们记边界条件(2.2) 和(2.3) 中的有关系数$ \alpha_{i}, \beta_{i}\ (i = 1, 2, 3, 4) $为矩阵

$ A = \left(\begin{array}{cccccc} \alpha_{1}&0&0&\ \beta_{1}\\ 0&\ \alpha_{3}\ &\beta_{3}&\ 0\\ 0&\alpha_{4}&\beta_{4}&\ 0\\ \alpha_{2}&0&0&\ \beta_{2} \end{array}\right), $

且记边界条件(2.2)–(2.5) 的矩阵形式为

(4.1) $ \begin{equation} A_{\lambda}Y(a)+BY(b) = 0, \end{equation} $

其中

$ A_{\lambda} = \left(\begin{array}{ccccccc}\alpha_{1}\lambda+\beta_{1}&0&0&\ -(\alpha_{2}\lambda+\beta_{2})\\ 0&\ \alpha_{3}\lambda+\beta_{3}\ &-(\alpha_{4}\lambda+\beta_{4})&\ 0\\ 0&0&0&\ 0\\ 0&0&0&\ 0\end{array}\right), $

$ B = \left(\begin{array}{cccccc} 0&0&0&\ 0\\ 0&0&0&\ 0\\ \cos\gamma&0&0&\ -\sin\gamma\\ 0&\ \cos\gamma\ &-\sin\gamma&\ 0 \end{array}\right). $

接下来我们给出特征值所满足的判别函数.

设$ \psi_{11}, \psi_{12} $和$ \psi_{13}, \psi_{14} $是方程(2.1) 在区间[$ a, c $) 满足如下初始条件的线性无关解

$ \begin{eqnarray*} (\Psi_{11}(a, \lambda), \Psi_{12}(a, \lambda), \Psi_{13}(a, \lambda), \Psi_{14}(a, \lambda)) = E, \end{eqnarray*} $

其中$ E $是四阶单位矩阵, $ \Psi_{1j}(x) = (\psi_{1j}(x), \psi_{1j}'(x), p_{2}\psi''_{1j}(x), ((p_{2}\psi_{1j}'')'-p_{1}\psi_{1j}')(x))^{T} $$ (j = 1, 2, 3, 4). $它们的Wronskian与变量$ x $无关, 且是关于特征参数$ \lambda $的整函数.

设$ \psi_{21}, \psi_{22} $和$ \psi_{23}, \psi_{24} $是方程(2.1) 在区间$ (c, b] $满足如下初始条件的线性无关解

$ C(\Psi_{11}(c, \lambda), \Psi_{12}(c, \lambda), \Psi_{13}(c, \lambda), \Psi_{14}(c, \lambda))+D(\Psi_{21}(c, \lambda), \Psi_{22}(c, \lambda), \Psi_{23}(c, \lambda), \Psi_{24}(c, \lambda)) = 0, $

其中$ \Psi_{2j}(x) = (\psi_{2j}(x), \psi_{2j}'(x), p_{2}\psi''_{2j}(x), ((p_{2}\psi_{2j}'')' -p_{1}\psi_{2j}')(x))^{T}, $$ (j = 1, 2, 3, 4). $且令

$ \Phi(x, \lambda) = \left\{ \begin{array}{ll} (\Psi_{11}(x, \lambda), \Psi_{12}(x, \lambda), \Psi_{13}(x, \lambda), \Psi_{14}(x, \lambda)), x\in[a, c), \\ (\Psi_{21}(x, \lambda), \Psi_{22}(x, \lambda), \Psi_{23}(x, \lambda), \Psi_{24}(x, \lambda)), x\in(c, b]. \end{array} \right. $

引理4.1   一个复数$ \lambda $是算子$ T $的特征值当且仅当$ \lambda $满足

$ \Delta(\lambda) = {\rm det}(A_{\lambda}+B\Phi(b, \lambda)) = 0, $

称$ \Delta(\lambda) $为判别函数.

证  这与文献[12, Theorem 3.1] 讨论的边界条件两端含有特征参数的证明过程类似, 因此省略.

接下来, 我们引入Banach空间及相应的范数. 比较任意参数发生微小改变时不同问题的谱的变化, 给出特征值和特征函数对边值问题(2.1)–(2.9) 的参数是连续依赖的. 为此我们令

$ \Omega = \{\omega = (1/p_{2}, p_{1}, q, w, a, b, \gamma, c-, c+, A, C, D)\}, $

且令$ \widetilde{\Omega} = \{\widetilde{\omega} = (\widetilde{1/p_{2}}, \widetilde{p_{1}}, \widetilde{q}, \widetilde{w}, a, b, \gamma, c-, c+, A, C, D)\}, $其中

$ \widetilde{1/p_{2}} = \left\{ \begin{array}{ll} 1/p_{2}, \ &x\in J, \\ 0, & x\in J'\setminus J; \end{array} \right. $

$ \widetilde{p_{1}}, \widetilde{q}, \widetilde{w} $可类似定义.

引入一个Banach空间

$ X = L^{1}(a', b')\times L^{1}(a', b')\times L^{1}(a', b')\times L^{1}(a', b') \times{\mathbb R}^{5}\times M_{4\times4}({\mathbb R})\times M_{4\times4}({\mathbb R})\times M_{4\times4}({\mathbb R}) $

及范数

$ \begin{eqnarray*} ||\omega|| = ||\widetilde{\omega}|| & = &\int^{c}_{a} (|\widetilde{1/p_{2}}|+|\widetilde{p_{1}}|+ |\widetilde{q}|+ |\widetilde{w}|){\rm d}x+\int^{b}_{c} (|\widetilde{1/p_{2}}|+|\widetilde{p_{1}}|+ |\widetilde{q}|+ |\widetilde{w}|){\rm d}x\\ &&+|a|+|b|+|\gamma|+|c-|+|c+|+||A||+||C||+||D||, \end{eqnarray*} $

其中$ ||\cdot|| $是任意的矩阵范数. 易知, $ \Omega $中的每一点是$ \Omega $中关于$ X $中范数的聚点. 由于$ 1/p_{2}, p_{1}, q, w $仅仅都属于$ L^{1}_{loc}(J'), $故$ \Omega $不是$ X $的子集, 而$ \widetilde{1/p_{2}}, \widetilde{p_{1}}, \widetilde{q}, \widetilde{w} $属于$ L^{1}(J'), $故$ \widetilde{\Omega} $是$ X $的子集. 由$ \Omega $中范数的定义可知, $ \Omega $与$ \widetilde{\Omega} $在$ X $中有相同的范数, 都是$ X $的子集, 且$ \Omega $中的收敛性由此范数决定. 从而我们得到四阶边值问题(2.1)–(2.9) 特征值与特征函数连续依赖于问题.

定理4.1   令$ \omega_{0} = (1/p_{2_{0}}, p_{1_{0}}, q_{0}, w_{0}, a_{0}, b_{0}, \gamma_{0}, c_{0}-, c_{0}+, A_{0}, C_{0}, D_{0})\in\Omega, $设$ \mu = \lambda(\omega_{0}) $为由$ \omega_{0} $确定的算子$ T $的特征值, 则$ \lambda $在$ \omega_{0} $处是连续的. 即对$ \forall \varepsilon>0, \exists \delta>0, $使得对任意的$ \omega\in\Omega, $当$ ||\omega-\omega_{0}||<\delta $时, 有$ |\lambda(\omega)-\lambda(\omega_{0})|<\varepsilon. $

证  令$ \Delta(\lambda) = \Delta(\omega, \lambda) = {\rm det}(A_{\lambda}+B\Phi(b, \lambda)). $由引理4.1可知, 问题的特征值正是判别函数的零点. 对$ \omega\in\Omega, \lambda(\omega) $是算子$ T $的特征值当且仅当$ \Delta(\omega, \lambda(\omega)) = 0. $对于$ \forall \omega\in\Omega, \Delta(\omega, \lambda) $是$ \lambda $的整函数且在$ \omega $处连续(见文献[33, Theorem 2.7, 2.8]). 又由算子$ T $的自伴性可知, $ \mu $是孤立特征值, 从而$ \Delta(\omega_{0}, \mu) = 0 $且$ \Delta(\omega_{0}, \lambda) $关于$ \lambda $不是常数, 因此, $ \exists \rho>0, $对$ \lambda\in S_{\rho}: = \{\lambda\in {\mathbb C}: |\lambda-\mu| = \rho\}, \Delta(\omega_{0}, \lambda)\neq0. $由方程解对初值和参数的连续性定理(见文献[34, p248, (9.17.4)]), 即可得定理.

引理4.2   假设(2.7)–(2.9) 成立, $ t_{0}\in [a, c)\cup(c, b] \cup\{c+, c-\}, d, k, m, n\in{\mathbb C}. $初值问题

$ \left\{ \begin{array}{ll} (p_{2}(x)y'')''-(p_{1}(x)y')'+q(x)y = \lambda w(x)y, \\ y(t_{0}) = d, y'(t_{0}) = k, (p_{2}y'')(t_{0}) = m, ((p_{2}y'')'-p_{1}y')(t_{0}) = n \end{array} \right. $

的唯一解$ y = y(\cdot, t_{0}, d, k, m, n, C, D, 1/p_{2}, p_{1}, q, w) $是关于任一变量的连续函数. 即对$ \forall \varepsilon>0, $$ \exists \delta>0, $若

$ \begin{eqnarray*} &&|t-t_{0}|+|d-d_{0}|+|k-k_{0}|+|m-m_{0}|+|n-n_{0}|+||C-C_{0}||+||D-D_{0}|| \\ &&+\int_{a}^{c}(|1/p_{2}-1/p_{2_{0}}|+|p_{1}-p_{1_{0}}|+|q-q_{0}|+|w-w_{0}|){\rm d}x\\ &&+\int_{c}^{b}(|1/p_{2}-1/p_{2_{0}}|+|p_{1}-p_{1_{0}}|+|q-q_{0}|+|w-w_{0}|){\rm d}x<\delta, \end{eqnarray*} $

则对$ \forall x\in J $有

$ \begin{eqnarray*} &&|y(x, t, d, k, m, n, C, D, 1/p_{2}, p_{1}, q, w)-y(x, t_{0}, d_{0}, k_{0}, m_{0}, n_{0}, C_{0}, D_{0}, 1/p_{2_{0}}, p_{1_{0}}, q_{0}, w_{0})|< \varepsilon, \\ &&|y'(x, t, d, k, m, n, C, D, 1/p_{2}, p_{1}, q, w)-y'(x, t_{0}, d_{0}, k_{0}, m_{0}, n_{0}, C_{0}, D_{0}, 1/p_{2_{0}}, p_{1_{0}}, q_{0}, w_{0})|< \varepsilon, \\ &&|(p_{2}y'')(x, t, d, k, m, n, C, D, 1/p_{2}, p_{1}, q, w)\\ && -(p_{2}y'')(x, t_{0}, d_{0}, k_{0}, m_{0}, n_{0}, C_{0}, D_{0}, 1/p_{2_{0}}, p_{1_{0}}, q_{0}, w_{0})|< \varepsilon, \\ &&|((p_{2}y'')'-p_{1}y')(x, t, d, k, m, n, C, D, 1/p_{2}, p_{1}, q, w)\\ && -((p_{2}y'')'-p_{1}y')(x, t_{0}, d_{0}, k_{0}, m_{0}, n_{0}, C_{0}, D_{0}, 1/p_{2_{0}}, p_{1_{0}}, q_{0}, w_{0})|< \varepsilon. \end{eqnarray*} $

证  可参考文献[15, Lemma 3.2], 虽然本文讨论的边界条件含有谱参数, 但证明过程类似.

引理4.3   假设$ \omega_{0} = (1/p_{2_{0}}, p_{1_{0}}, q_{0}, w_{0}, a_{0}, b_{0}, \gamma_{0}, c_{0}-, c_{0}+, A_{0}, C_{0}, D_{0})\in\Omega, \lambda = \lambda(\omega) $是算子$ T $的一个特征值. 若$ \lambda(\omega_{0}) $是单重特征值, 则在$ \Omega $中存在$ \omega_{0} $的邻域$ M, $满足对$ \forall \omega\in M, \lambda(\omega) $是单重特征值.

证  若$ \lambda(\omega_{0}) $是单重特征值, 则$ \Delta'(\lambda(\omega_{0}))\neq0 $. 因为$ \Delta(\lambda) $是$ \lambda $的整函数, 由定理4.1可知结论成立.

定义4.1   设$ u $满足边值问题(2.1)–(2.6), $ u_{1} = -M'(u), u_{2} = N'(u), $且有

(4.2) $ \begin{equation} \eta\int_{a}^{c}u\overline{u}w{\rm d}x+\xi\int_{c}^{b}u\overline{u}w{\rm d}x+\eta\rho_{1}u_{1}\overline{u}_{1}+\eta\rho_{2}u_{2}\overline{u}_{2} = 1 \end{equation} $

成立, 则称$ (u, u_{1}, u_{2})^{T} $为正规化特征向量.

定理4.2   假设记号同定理4.1. 设特征值$ \lambda(\omega) (\omega\in \Omega) $是$ \Omega $内$ \omega_{0} $的某个邻域$ M $内所有$ \omega $的$ l\ (l = 1, 2, 3, 4) $重特征值, 令$ (u_{k}(x, \omega_{0}), u_{k1}(\omega_{0}), u_{k2}(\omega_{0}))^{T}\in H, $$ k = 1, 2, \cdots, l $是算子$ T $对应于$ l $重特征值$ \lambda(\omega_{0}) (\omega_{0}\in \Omega) $的线性无关的正规化特征向量. 则存在$ l $个对应于特征值$ \lambda(\omega) $的线性无关的正规化特征向量$ (u_{k}(x, \omega), u_{k1}(\omega), u_{k2}(\omega))^{T}\in H, k = 1, 2, \cdots, l, $使得在$ \Omega $中当$ \omega\rightarrow \omega_{0} $时有

(4.3) $ \begin{eqnarray} &&u_{k}(x, \omega)\rightarrow u_{k}(x, \omega_{0}), u_{k}'(x, \omega)\rightarrow u_{k}'(x, \omega_{0}), (p_{2}u_{k}'')(x, \omega)\rightarrow (p_{2}u_{k}'')(x, \omega_{0}), {}\\ &&((p_{2}u_{k}'')'-p_{1}u_{k}')(x, \omega)\rightarrow ((p_{2}u_{k}'')'-p_{1}u_{k}')(x, \omega_{0}), \\ &&u_{k1}(\omega)\rightarrow u_{k1}(\omega_{0}), u_{k2}(\omega)\rightarrow u_{k2}(\omega_{0}){} \end{eqnarray} $

在区间$ J $上一致成立.

证  设$ \lambda(\omega_{0}) $是算子$ T $的单重特征值, $ (y(x, \omega_{0}), y_{1}(\omega_{0}), y_{2}(\omega_{0}))^{T}\in H $是其对应的特征向量, 且满足

$ \| y(x, \omega_{0})\|^{2} = \eta\int_{a}^{c}| y(x, \omega_{0})|^{2}w{\rm d}x+\xi\int_{c}^{b}| y(x, \omega_{0})|^{2}w{\rm d}x = 1. $

由引理4.3可知, 存在一个$ \omega_{0} $的邻域$ M $使得对任意的$ \omega\in M, \lambda(\omega) $是单重的. 根据定理4.1知, 当$ \omega\rightarrow \omega_{0} $时$ \lambda(\omega)\rightarrow \lambda(\omega_{0}) $成立. 当$ \omega\rightarrow \omega_{0} $时, 边界条件矩阵满足$ (A_{\lambda}, B)_{4\times8}(\omega)\rightarrow (A_{\lambda}, B)_{4\times8}(\omega_{0}). $由文献[4, Theorem 3.2] 可知, 当$ \omega\rightarrow \omega_{0} $时, 存在特征值$ \lambda(\omega) $对应的特征向量$ (y(x, \omega), y_{1}(\omega), y_{2}(\omega))^{T}\in H, $使其第一个分量$ y(x, \omega) $在区间$ J $上满足

(4.4) $ \begin{eqnarray} &&\| y(x, \omega)\|^{2} = \eta\int_{a}^{c}| y(x, \omega)|^{2}w{\rm d}x+\xi\int_{c}^{b}| y(x, \omega)|^{2}w{\rm d}x = 1, {}\\ &&y(x, \omega)\rightarrow y(x, \omega_{0}), y'(x, \omega)\rightarrow y'(x, \omega_{0}), (p_{2}y'')(x, \omega)\rightarrow (p_{2}y'')(x, \omega_{0}), \\ &&((p_{2}y'')'-p_{1}y')(x, \omega)\rightarrow ((p_{2}y'')'-p_{1}y')(x, \omega_{0}).{} \end{eqnarray} $

所以当$ \omega\rightarrow \omega_{0} $时

(4.5) $ \begin{equation} y_{1}(\omega)\rightarrow y_{1}(\omega_{0}), y_{2}(\omega)\rightarrow y_{2}(\omega_{0}). \end{equation} $

下面令$ \lambda(\omega_{0}) $对应的正规化特征向量$ (u(x, \omega_{0}), u_{1}(\omega_{0}), u_{2}(\omega_{0}))^{T} $及第一个分量的导数$ u'(x, \omega_{0}) $和拟导数$ (p_{2}u'')(x, \omega_{0}), ((p_{2}u'')'-p_{1}u')(x, \omega_{0}) $分别为

(4.6) $ \begin{eqnarray} &&(u(x, \omega_{0}), u_{1}(\omega_{0}), u_{2}(\omega_{0}))^{T} = \frac{(y(x, \omega_{0}), y_{1}(\omega_{0}), y_{2}(\omega_{0}))^{T}}{\|(y(x, \omega_{0}), y_{1}(\omega_{0}), y_{2}(\omega_{0}))^{T}\|}, \end{eqnarray} $

(4.7) $ \begin{eqnarray} &&u'(x, \omega_{0}) = \frac{y'(x, \omega_{0})}{\|(y(x, \omega_{0}), y_{1}(\omega_{0}), y_{2}(\omega_{0}))^{T}\|}, \end{eqnarray} $

(4.8) $ \begin{eqnarray} &&(p_{2}u'')(x, \omega_{0}) = \frac{(p_{2}y'')(x, \omega_{0})}{\|(y(x, \omega_{0}), y_{1}(\omega_{0}), y_{2}(\omega_{0}))^{T}\|}, \end{eqnarray} $

(4.9) $ \begin{eqnarray} &&((p_{2}u'')'-p_{1}u')(x, \omega_{0}) = \frac{((p_{2}y'')'-p_{1}y')(x, \omega_{0})}{\|(y(x, \omega_{0}), y_{1}(\omega_{0}), y_{2}(\omega_{0}))^{T}\|}. \end{eqnarray} $

对于$ \lambda(\omega) $对应的正规化特征向量$ (u(x, \omega), u_{1}(\omega), u_{2}(\omega))^{T} $及其第一个分量的导数$ u'(x, \omega) $和拟导数$ (p_{2}u'')(x, \omega), ((p_{2}u'')'-p_{1}u')(x, \omega) $形式与(4.6)–(4.9) 式类似. 再由(4.4)–(4.5) 式知结论成立.

设特征值$ \lambda(\omega) $关于$ \omega_{0}\in\Omega $的某个邻域$ M\subset\Omega $内的所有$ \omega $的重数为$ l (l = 2, 3, 4), $由定理4.1和文献[8, p181, Theorem 3.5] 可知, 当$ \omega\rightarrow \omega_{0} $时, 存在$ l $个线性无关的特征向量$ (y_{k}(x, \omega), y_{k1}(\omega), y_{k2}(\omega))^{T}\in H, k = 1, 2, \cdots, l, $使其第一个分量$ y_{k}(x, \omega) $在区间$ J $上满足

(4.10) $ \begin{eqnarray} &&\| y_{k}(x, \omega)\|^{2} = \eta\int_{a}^{c}| y_{k}(x, \omega)|^{2}w{\rm d}x+\xi\int_{c}^{b}| y_{k}(x, \omega)|^{2}w{\rm d}x = 1, {}\\ &&y_{k}(x, \omega)\rightarrow y_{k}(x, \omega_{0}), y_{k}'(x, \omega)\rightarrow y_{k}'(x, \omega_{0}), (p_{2}y_{k}'')(x, \omega)\rightarrow (p_{2}y_{k}'')(x, \omega_{0}), \\ &&((p_{2}y_{k}'')'-p_{1}y_{k}')(x, \omega)\rightarrow ((p_{2}y_{k}'')'-p_{1}y_{k}')(x, \omega_{0}).{} \end{eqnarray} $

通过类似上面的讨论得定理结论.

在特征值与特征函数关于问题连续依赖性的基础上, 进一步考虑特征值关于问题所有参数的可微性, 即给出特征值关于各个参数的微分表达式. 为此, 我们将用到Fr$ \acute{\rm{e}} $chet导数, 定义如下:

定义5.1^[4]  设 $ {\cal T} $是从Banach空间$ {\mathbb X} $到Banach空间$ {\mathbb Y} $上的映射, 若存在有界线性算子$ d{\cal T}_{x}:{\mathbb X}\rightarrow {\mathbb Y} $, 对$ h\in {\mathbb X}, $当$ h\rightarrow 0 $时, 有

$ |{\cal T}(x+h)-{\cal T}(x)-d{\cal T}(x)| = o(h), \, \, h\rightarrow 0, $

则称映射$ {\cal T} $在点$ x\in {\mathbb X} $处是Fr$ \acute{\rm{e}} $chet可微的.

引理5.1   设函数$ u $和$ v $分别为方程(2.1) 对应于特征值$ \lambda = \mu $和$ \lambda = \nu $的特征函数, 则

(5.1) $ \begin{eqnarray} & &(\nu-\mu)[\eta\int_{a}^{c}u\overline{v}w{\rm d}x+\xi\int_{c}^{b}u\overline{v}w{\rm d}x+\eta\rho_{1}u_{1}\overline{v}_{1}+\eta\rho_{2}u_{2}\overline{v}_{2}]{}\\ & = &\eta[u, v]_{a}^{c}+\xi[u, v]_{c}^{b}+\eta\rho_{1}[-M'(u)M(\overline{v})+M(u)M'(\overline{v})]+\eta\rho_{2}[-N'(u)N(\overline{v})+N(u)N'(\overline{v})]{}\\ & = &\eta[u, v](c-)-\eta[u, v](a)+\xi[u, v](b)-\xi[u, v](c+){}\\ & &+\eta\rho_{1}[-M'(u)M(\overline{v})+M(u)M'(\overline{v})]+\eta\rho_{2}[-N'(u)N(\overline{v})+N(u)N'(\overline{v})]. \end{eqnarray} $

证  由分部积分的方法可得.

引理5.2^[5]   假设函数$ f\in L_{loc}(a', b'), $则

(5.2) $ \begin{eqnarray} \lim\limits_{h\rightarrow0}\frac{1}{h}\int_{x}^{x+h}f = f(x) {\quad} a.e.\ (a', b'). \end{eqnarray} $

证  见文献[5, Lemma 3.2].

定理5.1   令$ \omega = (1/p_{2}, p_{1}, q, w, a, b, \gamma, c-, c+, A, C, D)\in\Omega, $$ \lambda = \lambda(\omega) $是算子$ T $的特征值, $ (u, u_{1}, u_{2})^{T} $是相应的正规化特征向量. 若$ \lambda(\omega) $在$ \omega $的某邻域$ M\subset\Omega $内的几何重数不变, 则$ \lambda $关于边界条件参数$ \gamma, $特征参数依赖的边值条件矩阵$ A $及转移条件的系数矩阵$ C, D $和方程系数函数$ 1/p_{2}, p_{1}, q, w $都是可微的且导数公式如下:

1. 固定$ \omega $中除$ \gamma $之外的所有变量, 令$ \lambda = \lambda(\gamma) $为特征值, 则$ \lambda $是可微的且有

(5.3) $ \begin{eqnarray} \lambda'(\gamma) = \xi\csc^{2}\gamma(|u'(b)|^{2}-|u(b)|^{2}). \end{eqnarray} $

2. 固定$ \omega $中除$ A $之外的所有变量, 令$ \lambda = \lambda(A) $为特征值, 则$ \lambda $是Fr$ \acute{\rm{e}} $chet可微的且有

(5.4) $ \begin{equation} d \lambda_{A}(L) = -\eta(u, -u', p_{2}u'', -((p_{2}u'')'-p_{1}u'))(a)[E-A(A+L)^{-1}] \left(\begin{array}{ccccc} (p_{2}\overline{u}'')'-p_{1}\overline{u}'\\ p_{2}\overline{u}''\\ \overline{u}'\\ \overline{u}\end{array}\right)(a), \end{equation} $

其中$ L $满足$ {\rm det}(A+L) = {\rm det}A = -\theta_{1}\theta_{2}. $

3. 固定$ \omega $中除$ C $之外的所有变量, 在$ C $的邻域内, 对所有满足det$ (C+H) = \eta^{2}, $$ \xi (C+H)J_{0}(C+H)^{T} = \eta DJ_{0}D^{T} $的$ H, \lambda $是Fr$ \acute{\rm{e}} $chet可微的且有

(5.5) $ \begin{eqnarray} {\rm d}\lambda_{C}(H) = \xi U^{\ast}(c+)J_{0}^{-1}D^{-1}HC^{-1}DU(c+). \end{eqnarray} $

4. 固定$ \omega $中除$ D $之外的所有变量, 在$ D $的邻域内, 对所有满足det$ (D+H) = \xi^{2}, $$ \xi CJ_{0}C^{T} = \eta(D+H)J_{0}(D+H)^{T} $的$ H, \lambda $是Fr$ \acute{\rm{e}} $chet可微的且有

(5.6) $ \begin{eqnarray} {\rm d}\lambda_{D}(H) = -\eta U^{\ast}(c-)J_{0}^{-1}C^{-1}HD^{-1}CU(c-). \end{eqnarray} $

5. 固定$ \omega $中除$ 1/p_{2} $之外的所有变量, 令$ \lambda = \lambda(1/p_{2}) $为特征值, 则$ \lambda $是Fr$ \acute{\rm{e}} $chet可微的且有

(5.7) $ \begin{eqnarray} {\rm d}\lambda_{(1/p_{2})}(h) = -\eta\int_{a}^{c}|p_{2}u''|^{2}h{\rm d}x-\xi\int_{c}^{b}|p_{2}u''|^{2}h{\rm d}x, \, \, \, h\in L^{1}(a, b). \end{eqnarray} $

6. 固定$ \omega $中除$ p_{1} $之外的所有变量, 令$ \lambda = \lambda(p_{1}) $为特征值, 则$ \lambda $是Fr$ \acute{\rm{e}} $chet可微的且有

(5.8) $ \begin{eqnarray} {\rm d}\lambda_{p_{1}}(h) = \eta\int_{a}^{c}|u'|^{2}h{\rm d}x+\xi\int_{c}^{b}|u'|^{2}h{\rm d}x, \, \, \, h\in L^{1}(a, b). \end{eqnarray} $

7. 固定$ \omega $中除$ q $之外的所有变量, 令$ \lambda = \lambda(q) $为特征值, 则$ \lambda $是Fr$ \acute{\rm{e}} $chet可微的且有

(5.9) $ \begin{eqnarray} {\rm d}\lambda_{q}(h) = \eta\int_{a}^{c}|u|^{2}h{\rm d}x+\xi\int_{c}^{b}|u|^{2}h{\rm d}x, \, \, \, h\in L^{1}(a, b). \end{eqnarray} $

8. 固定$ \omega $中除$ w $之外的所有变量, 令$ \lambda = \lambda(w) $为特征值, 则$ \lambda $是Fr$ \acute{\rm{e}} $chet可微的且有

(5.10) $ \begin{eqnarray} {\rm d}\lambda_{w}(h) = -\eta\lambda\int_{a}^{c}|u|^{2}h{\rm d}x-\xi\lambda\int_{c}^{b}|u|^{2}h{\rm d}x, \, \, \, h\in L^{1}(a, b). \end{eqnarray} $

证  1. 取$ \varepsilon $充分小, 令特征值$ \mu = \lambda(\gamma), \nu = \lambda(\gamma+\varepsilon) $对应的正规化特征向量分别为$ (u, u_{1}, u_{2})^{T}, (v, v_{1}, v_{2})^{T}, $其中$ u = u(x, \gamma), v = u(x, \gamma+\varepsilon), $由(2.1) 式我们有

(5.11) $ \begin{equation} (p_{2}u'')''-(p_{1}u')'+qu = \lambda(\gamma)wu, \end{equation} $

(5.12) $ \begin{equation} (p_{2}\overline{v}'')''-(p_{1}\overline{v}')'+q\overline{v} = \lambda(\gamma+\varepsilon)w\overline{v}, \end{equation} $

根据(5.11) 式和(5.12) 式有

(5.13) $ \begin{eqnarray} [\lambda(\gamma+\varepsilon)-\lambda(\gamma)]u\overline{v}w = [(p_{2}\overline{v}'')''- (p_{1}\overline{v}')']u-[(p_{2}u'')''-(p_{1}u')']\overline{v}, \end{eqnarray} $

将(5.13) 式分别从$ a $到$ c $和$ c $到$ b $积分且使用分部积分法, 我们得到

(5.14) $ \begin{eqnarray} &&[\lambda(\gamma+\varepsilon)-\lambda(\gamma)][\eta\int_{a}^{c}u\overline{v}w{\rm d}x+\xi\int_{c}^{b}u\overline{v}w{\rm d}x]{}\\ & = &\eta\bigg[\int_{a}^{c}[(p_{2}\overline{v}'')''-(p_{1}\overline{v}')']u{\rm d}x-\int_{a}^{c}[(p_{2}u'')''-(p_{1}u')']\overline{v}{\rm d}x\bigg]{}\\ &&+\xi\bigg[\int_{c}^{b}[(p_{2}\overline{v}'')''-(p_{1}\overline{v}')']u{\rm d}x-\int_{c}^{b}[(p_{2}u'')''-(p_{1}u')']\overline{v}{\rm d}x\bigg]{}\\ & = &\eta[u, v](c-)-\eta[u, v](a)+\xi[u, v](b)-\xi[u, v](c+). \end{eqnarray} $

由边界条件(2.2) 我们得到

$ \lambda(\gamma+\varepsilon)(-\alpha_{1}\overline{v}(a)+\alpha_{2}((p_{2}\overline{v}'')'-p_{1}\overline{v}')(a)) = \beta_{1}\overline{v}(a)-\beta_{2}((p_{2}\overline{v}'')'-p_{1}\overline{v}')(a), $

$ \lambda(\gamma)(-\alpha_{1}u(a)+\alpha_{2}((p_{2}u'')'-p_{1}u')(a)) = \beta_{1}u(a)-\beta_{2}((p_{2}u'')'-p_{1}u')(a), $

因此

(5.15) $ \begin{eqnarray} & &[\lambda(\gamma+\varepsilon)-\lambda(\gamma)]\eta\rho_{1}u_{1}\overline{v}_{1}{}\\ & = &\eta\rho_{1}u_{1}(\beta_{1}\overline{v}(a)-\beta_{2}((p_{2}\overline{v}'')'-p_{1}\overline{v}')(a)) -\eta\rho_{1}(\beta_{1}u(a)-\beta_{2}((p_{2}u'')'-p_{1}u')(a))\overline{v}_{1}{}\\ & = &\eta[u(a)((p_{2}\overline{v}'')'-p_{1}\overline{v}')(a)-((p_{2}u'')'-p_{1}u')(a)\overline{v}(a)]. \end{eqnarray} $

再由边界条件(2.3), 我们得到

$ \lambda(\gamma+\varepsilon)(\alpha_{3}\overline{v}'(a)-\alpha_{4}(p_{2}\overline{v}'')(a)) = -\beta_{3}\overline{v}'(a)+\beta_{4}(p_{2}\overline{v}'')(a), $

$ \lambda(\gamma)(\alpha_{3}u'(a)-\alpha_{4}(p_{2}u'')(a)) = -\beta_{3}u'(a)+\beta_{4}(p_{2}u'')(a), $

因此

(5.16) $ \begin{eqnarray} &&[\lambda(\gamma+\varepsilon)-\lambda(\gamma)]\eta\rho_{2}u_{2}\overline{v}_{2}{}\\ & = &\eta\rho_{2}u_{2}(-\beta_{3}\overline{v}'(a)+\beta_{4}(p_{2}\overline{v}'')(a)) -\eta\rho_{2}(-\beta_{3}u'(a)+\beta_{4}(p_{2}u'')(a))\overline{v}_{2}{}\\ & = &-\eta[u'(a)(p_{2}\overline{v}'')(a)-(p_{2}u'')(a)\overline{v}'(a)]. \end{eqnarray} $

我们发现(5.15) 式与(5.16) 式的和正是$ \eta[u, v](a), $综合(5.14)–(5.16) 式及边界条件(2.4)–(2.5) 和(3.5) 式, 我们得到

$ \begin{eqnarray*} &&[\lambda(\gamma+\varepsilon)-\lambda(\gamma)][\eta\int_{a}^{c}u\overline{v}w{\rm d}x+\xi\int_{c}^{b}u\overline{v}w{\rm d}x+\eta\rho_{1}u_{1}\overline{v}_{1}+\eta\rho_{2}u_{2}\overline{v}_{2}]{\nonumber}\\ & = &\xi[u(b)((p_{2}\overline{v}'')'-p_{1}\overline{v}')(b)-u'(b)(p_{2}\overline{v}'')(b)+(p_{2}u'')(b)\overline{v}'(b)-((p_{2}u'')'-p_{1}u')(b)\overline{v}(b)]{\nonumber}\\ & = &\xi[u(b)\cot(\gamma+\varepsilon)\overline{v}(b)-u'(b)\cot(\gamma+\varepsilon)\overline{v}'(b)+\cot \gamma u'(b)\overline{v}'(b)-\cot \gamma u(b)\overline{v}(b)]{\nonumber}\\ & = &\xi[(\cot(\gamma+\varepsilon)-\cot \gamma )(u(b)\overline{v}(b)-u'(b)\overline{v}'(b))]. \end{eqnarray*} $

上述等式两端同时除以$ \varepsilon, $当$ \varepsilon\rightarrow 0 $时, 取其极限, 由定义4.1和定理4.2可得(5.3) 式成立.

2. 令特征值$ \mu = \lambda(A), \nu = \lambda(A+L) $对应的正规化特征向量分别为$ (u, u_{1}, u_{2})^{T}, $$ (v, v_{1}, v_{2})^{T}, $其中$ u = u(x, A), v = u(x, A+L), $直接计算得

(5.17) $ \begin{eqnarray} &&[\lambda(A+L)-\lambda(A)][\eta\int_{a}^{c}u\overline{v}w{\rm d}x+\xi\int_{c}^{b}u\overline{v}w{\rm d}x] = -\eta[u, v](a){}\\ & = &-\eta(u, -u', p_{2}u'', -((p_{2}u'')'-p_{1}u'))(a)E \left(\begin{array}{ccccc} (p_{2}\overline{v}'')'-p_{1}\overline{v}'\\ p_{2}\overline{v}''\\ \overline{v}'\\ \overline{v}\end{array}\right)(a). \end{eqnarray} $

令$ A+L = \left(\begin{array}{ccccc} \widetilde{\alpha_{1}}&0&0&\ \widetilde{\beta_{1}}\\ 0&\ \widetilde{\alpha_{3}}\ &\widetilde{\beta_{3}}&\ 0\\ 0&\widetilde{\alpha_{4}}&\widetilde{\beta_{4}}&\ 0\\ \widetilde{\alpha_{2}}&0&0&\ \widetilde{\beta_{2}}\end{array}\right) $, 然后根据边界条件(2.2) 得

$ \begin{aligned}&\lambda(A+L)\left(-\widetilde{\alpha_{1}} \bar{v}(a)+\widetilde{\alpha_{2}}\left(\left(p_{2} \bar{v}^{\prime \prime}\right)^{\prime}-p_{1} \bar{v}^{\prime}\right)(a)\right)=\widetilde{\beta_{1}} \bar{v}(a)-\widetilde{\beta_{2}}\left(\left(p_{2} \bar{v}^{\prime \prime}\right)^{\prime}-p_{1} \bar{v}^{\prime}\right)(a), \\&\lambda(A+L) \rho_{1} u_{1} \bar{v}_{1}=\rho_{1} u_{1}\left(\widetilde{\beta_{1}} \bar{v}(a)-\widetilde{\beta_{2}}\left(\left(p_{2} \bar{v}^{\prime \prime}\right)^{\prime}-p_{1} \bar{v}^{\prime}\right)(a)\right), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\rm{i})\\&\lambda(A)\left(-\alpha_{1} u(a)+\alpha_{2}\left(\left(p_{2} u^{\prime \prime}\right)^{\prime}-p_{1} u^{\prime}\right)(a)\right)=\beta_{1} u(a)-\beta_{2}\left(\left(p_{2} u^{\prime \prime}\right)^{\prime}-p_{1} u^{\prime}\right)(a) \\&\lambda(A) \rho_{1} u_{1} \bar{v}_{1}=\rho_{1}\left(\beta_{1} u(a)-\beta_{2}\left(\left(p_{2} u^{\prime \prime}\right)^{\prime}-p_{1} u^{\prime}\right)(a)\right) \bar{v}_{1}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\rm{ii})\end{aligned}$

将(i)–(ii) 式的右端整理成如下矩阵形式

(5.18) $ \begin{eqnarray} &&[\lambda(A+L)-\lambda(A)]\eta\rho_{1}u_{1}\overline{v}_{1}{}\\ & = &\eta\rho_{1}u_{1}(\widetilde{\beta_{1}}\overline{v}(a)-\widetilde{\beta_{2}}((p_{2}\overline{v}'')'-p_{1}\overline{v}')(a)) -\eta\rho_{1}(\beta_{1}u(a)-\beta_{2}((p_{2}u'')'-p_{1}u')(a))\overline{v}_{1}{}\\ & = &\eta\rho_{1}(u, -u', p_{2}u'', -((p_{2}u'')'-p_{1}u'))(a) \left(\begin{array}{ccccc} -\alpha_{1}\\0\\0\\ -\alpha_{2}\end{array}\right) (-\widetilde{\beta_{2}}, 0, 0, \widetilde{\beta_{1}}) \left(\begin{array}{ccccc}(p_{2}\overline{v}'')'-p_{1}\overline{v}'\\p_{2}\overline{v}''{\nonumber}\\ \overline{v}'\\ \overline{v}\end{array}\right)(a){}\\ &&+\eta\rho_{1}(u, -u', p_{2}u'', -((p_{2}u'')'-p_{1}u'))(a) \left(\begin{array}{ccccc}-\beta_{1}\\0\\0\\ -\beta_{2} \end{array}\right)(\widetilde{\alpha_{2}}, 0, 0, -\widetilde{\alpha_{1}}) \left(\begin{array}{ccccc} (p_{2}\overline{v}'')'-p_{1}\overline{v}'\\p_{2}\overline{v}''\\ \overline{v}'{\nonumber}\\ \overline{v}\end{array}\right)(a){}\\ & = &\eta\rho_{1}(u, -u', p_{2}u'', -((p_{2}u'')'-p_{1}u'))(a){}\\ &&\cdot\left(\begin{array}{ccccc} \alpha_{1}\widetilde{\beta_{2}}-\beta_{1}\widetilde{\alpha_{2}}&\ 0\ &0&\ -\alpha_{1}\widetilde{\beta_{1}}+\beta_{1}\widetilde{\alpha_{1}}\\ 0&0&0&\ 0\\ 0&0&0&\ 0\\ \alpha_{2}\widetilde{\beta_{2}}-\beta_{2}\widetilde{\alpha_{2}}&0&0& \ -\alpha_{2}\widetilde{\beta_{1}}+\beta_{2}\widetilde{\alpha_{1}}\end{array}\right) \left(\begin{array}{ccccc} (p_{2}\overline{v}'')'-p_{1}\overline{v}'\\p_{2}\overline{v}''\\ \overline{v}'\\ \overline{v}\end{array}\right)(a). \end{eqnarray} $

同理根据边界条件(2.3), 同(i)–(ii) 式方法类似也整理得如下矩阵形式

(5.19) $ \begin{eqnarray} &&[\lambda(A+L)-\lambda(A)]\eta\rho_{2}u_{2}\overline{v}_{2}{}\\ & = &\eta\rho_{2}u_{2}(-\widetilde{\beta_{3}}\overline{v}'(a)+\widetilde{\beta_{4}}(p_{2}\overline{v}'')(a)) -\eta\rho_{2}(-\beta_{3}u'(a)+\beta_{4}(p_{2}u'')(a))\overline{v}_{2}{}\\ & = &\eta\rho_{2}(u, -u', p_{2}u'', -((p_{2}u'')'-p_{1}u'))(a)\left(\begin{array}{ccccc}0\\ -\alpha_{3}\\ -\alpha_{4}\\0\end{array}\right)(0, \widetilde{\beta_{4}}, -\widetilde{\beta_{3}}, 0)\left(\begin{array}{ccccc}(p_{2}\overline{v}'')'-p_{1}\overline{v}'\\p_{2}\overline{v}''\\ \overline{v}'\\ \overline{v}\end{array}\right)(a){}\\ &&+\eta\rho_{2}(u, -u', p_{2}u'', -((p_{2}u'')'-p_{1}u'))(a)\left(\begin{array}{ccccc}0\\ -\beta_{3}\\ -\beta_{4}\\0\end{array}\right)(0, -\widetilde{\alpha_{4}}, \widetilde{\alpha_{3}}, 0)\left(\begin{array}{ccccc}(p_{2}\overline{v}'')'-p_{1}\overline{v}'\\p_{2}\overline{v}''\\ \overline{v}'\\ \overline{v}\end{array}\right)(a){}\\ & = &\eta\rho_{2}(u, -u', p_{2}u'', -((p_{2}u'')'-p_{1}u'))(a){}\\ &&\cdot\left(\begin{array}{ccccc}0&0&0&0\\ 0&\ -\alpha_{3}\widetilde{\beta_{4}}+\beta_{3}\widetilde{\alpha_{4}}\ & \alpha_{3}\widetilde{\beta_{3}}-\beta_{3}\widetilde{\alpha_{3}}&\ 0\\ 0&-\alpha_{4}\widetilde{\beta_{4}} +\beta_{4}\widetilde{\alpha_{4}}& \alpha_{4}\widetilde{\beta_{3}}-\beta_{4}\widetilde{\alpha_{3}}&\ 0\\ 0&0&0&\ 0\end{array}\right) \left(\begin{array}{ccccc}(p_{2}\overline{v}'')'-p_{1}\overline{v}'\\p_{2}\overline{v}''\\ \overline{v}'\\ \overline{v}\end{array}\right)(a). \end{eqnarray} $

结合(5.17)–(5.19) 式, 我们得到

$ \begin{eqnarray*} &&[\lambda(A+L)-\lambda(A)][\eta\int_{a}^{c}u\overline{v}w{\rm d}x+\xi\int_{c}^{b}u\overline{v}w{\rm d}x +\eta\rho_{1}u_{1}\overline{v}_{1}+\eta\rho_{2}u_{2}\overline{v}_{2}]\\ & = &-\eta(u, -u', p_{2}u'', -((p_{2}u'')'-p_{1}u'))(a)\bigg[E-\rho_{1}\left(\begin{array}{ccccc} \alpha_{1}\widetilde{\beta_{2}}-\beta_{1}\widetilde{\alpha_{2}}&\ 0\ &0&\ -\alpha_{1}\widetilde{\beta_{1}} +\beta_{1}\widetilde{\alpha_{1}}\\0&0&0&\ 0\\0&0&0&\ 0\\ \alpha_{2}\widetilde{\beta_{2}}-\beta_{2}\widetilde{\alpha_{2}}&0&0&\ -\alpha_{2}\widetilde{\beta_{1}} +\beta_{2}\widetilde{\alpha_{1}}\end{array}\right)\\ &&-\rho_{2}\left(\begin{array}{ccccc}0&0&0&\ 0\\ 0&\ -\alpha_{3}\widetilde{\beta_{4}}+\beta_{3}\widetilde{\alpha_{4}}\ & \alpha_{3}\widetilde{\beta_{3}}-\beta_{3}\widetilde{\alpha_{3}}&\ 0\\ 0&-\alpha_{4}\widetilde{\beta_{4}} +\beta_{4}\widetilde{\alpha_{4}}& \alpha_{4}\widetilde{\beta_{3}}-\beta_{4}\widetilde{\alpha_{3}}&\ 0\\ 0&0&0&\ 0\end{array}\right)\bigg] \left(\begin{array}{ccccc}(p_{2}\overline{v}'')'-p_{1}\overline{v}'\\p_{2}\overline{v}''\\ \overline{v}'\\ \overline{v}\end{array}\right)(a)\\ & = &-\eta(u, -u', p_{2}u'', -((p_{2}u'')'-p_{1}u'))(a)[E-A(A+L)^{-1}]\left(\begin{array}{ccccc}(p_{2}\overline{v}'')'-p_{1}\overline{v}'\\p_{2}\overline{v}''\\ \overline{v}'\\ \overline{v}\end{array}\right)(a). \end{eqnarray*} $

令$ L\rightarrow0 $, 由定义4.1和定理4.2可得到(5.4) 式成立.

3. 令特征值$ \mu = \lambda(C), $$ \nu = \lambda(C+H) $对应的正规化特征向量分别为$ (u, u_{1}, u_{2})^{T}, $$ (v, v_{1}, v_{2})^{T}, $其中$ u = u(x, C), v = u(x, C+H), $我们有

(5.20) $ \begin{eqnarray} &&[\lambda(C+H)-\lambda(C)][\eta\int_{a}^{c}u\overline{v}w{\rm d}x+\xi\int_{c}^{b}u\overline{v}w{\rm d}x+\eta\rho_{1}u_{1}\overline{v}_{1}+\eta\rho_{2}u_{2}\overline{v}_{2}]{}\\ & = &\eta[u, v](c-)-\xi[u, v](c+){}\\ & = &\eta((p_{2}\overline{v}'')'-p_{1}\overline{v}', -p_{2}\overline{v}'', \overline{v}', -\overline{v})(c-)U(c-){}\\ &&-\xi((p_{2}\overline{v}'')'-p_{1}\overline{v}', -p_{2}\overline{v}'', \overline{v}', -\overline{v})(c+)U(c+), \end{eqnarray} $

由$ \xi(C+H)J_{0}(C+H)^{T} = \eta DJ_{0}D^{T} $可得$ \frac{\xi}{\eta}D^{-1}(C+H)J_{0}(C+H)^{T} = J_{0}D^{T}, $因此

(5.21) $ \begin{eqnarray} &&((p_{2}\overline{v}'')'-p_{1}\overline{v}', -p_{2}\overline{v}'', \overline{v}', -\overline{v})(c-){}\\ & = &(\overline{v}, \overline{v}', p_{2}\overline{v}'', (p_{2}\overline{v}'')'-p_{1}\overline{v}')(c-)J_{0}^{-1}{}\\ & = &-(\overline{v}, \overline{v}', p_{2}\overline{v}'', (p_{2}\overline{v}'')'-p_{1}\overline{v}')(c+)D^{\ast}[(C+H)^{\ast}]^{-1}J_{0}^{-1}{}\\ & = &-((p_{2}\overline{v}'')'-p_{1}\overline{v}', -p_{2}\overline{v}'', \overline{v}', -\overline{v})(c+)J_{0}D^{T}[(C+H)^{T}]^{-1}J_{0}^{-1}{}\\ & = &-\frac{\xi}{\eta}V^{\ast}(c+)J_{0}^{-1}D^{-1}(C+H), \end{eqnarray} $

将(5.21) 式代入(5.20) 式可得

$ \begin{eqnarray*} &&[\lambda(C+H)-\lambda(C)][\eta\int_{a}^{c}u\overline{v}w{\rm d}x+\xi\int_{c}^{b}u\overline{v}w{\rm d}x+\eta\rho_{1}u_{1}\overline{v}_{1}+\eta\rho_{2}u_{2}\overline{v}_{2}]\\ & = &-\xi V^{\ast}(c+)J_{0}^{-1}D^{-1}(C+H)U(c-)+\xi V^{\ast}(c+)J_{0}^{-1}D^{-1}CU(c-)\\ & = &-\xi V^{\ast}(c+)J_{0}^{-1}D^{-1}HU(c-)\\ & = &\xi V^{\ast}(c+)J_{0}^{-1}D^{-1}HC^{-1}DU(c+). \end{eqnarray*} $

令$ H\rightarrow0 $, 由Fr$ \acute{\rm{e}} $chet导数的定义得(5.5) 式成立. 对(5.6) 式的证明类似, 因此省略.

4. 取$ h $充分小, 令特征值$ \mu = \lambda(\frac{1}{p_{2}}), $$ \nu = \lambda(\frac{1}{p_{2h}}) $对应的正规化特征向量分别为$ (u, u_{1}, u_{2})^{T}, $$ (v, v_{1}, v_{2})^{T}, $其中$ u = u(x, \frac{1}{p_{2}}), $$ v = u(x, \frac{1}{p_{2h}}), $$ \frac{1}{p_{2h}} = \frac{1}{p_{2}}+h, $$ h\in L^{1}(a, b), $则$ p_{2}-p_{2h} = p_{2}p_{2h} h. $

由(2.1) 式和分部积分法得

$ \begin{eqnarray*} &&[\lambda(\frac{1}{p_{2h}})-\lambda(\frac{1}{p_{2}})][\eta\int_{a}^{c}u\overline{v}w{\rm d}x+\xi\int_{c}^{b}u\overline{v}w{\rm d}x]\\ & = &\eta\int_{a}^{c}\lambda(\frac{1}{p_{2h}})u\overline{v}w{\rm d}x-\eta\int_{a}^{c}\lambda(\frac{1}{p_{2}})u\overline{v}w{\rm d}x+\xi\int_{c}^{b}\lambda(\frac{1}{p_{2h}})u\overline{v}w{\rm d}x-\xi\int_{c}^{b}\lambda(\frac{1}{p_{2}})u\overline{v}w{\rm d}x\\ & = &\eta[u, v](c-)-\eta[u, v](a)+\eta\int_{a}^{c}(p_{2h}-p_{2})u''\overline{v}''{\rm d}x\\ &&+\xi[u, v](b)-\xi[u, v](c+)+\xi\int_{c}^{b}(p_{2h}-p_{2})u''\overline{v}''{\rm d}x, \end{eqnarray*} $

根据边界条件(2.2) 和(2.3) 有

$ [\lambda(\frac{1}{p_{2h}})-\lambda(\frac{1}{p_{2}})][\eta\rho_{1}u_{1}\overline{v}_{1}+\eta\rho_{2}u_{2}\overline{v}_{2}] = \eta[u, v](a), $

再结合边界条件(2.4), (2.5) 和(3.5) 式可得

$ \begin{eqnarray*} &&[\lambda(\frac{1}{p_{2h}})-\lambda(\frac{1}{p_{2}})][\eta\int_{a}^{c}u\overline{v}w{\rm d}x+\xi\int_{c}^{b}u\overline{v}w{\rm d}x+\eta\rho_{1}u_{1}\overline{v}_{1}+\eta\rho_{2}u_{2}\overline{v}_{2}]\\ & = &\eta\int_{a}^{c}(p_{2h}-p_{2})u''\overline{v}''{\rm d}x+\xi\int_{c}^{b}(p_{2h}-p_{2})u''\overline{v}''{\rm d}x\\ & = &-\eta\int_{a}^{c}(p_{2}u'')(p_{2h}\overline{v}'')h{\rm d}x-\xi\int_{c}^{b}(p_{2}u'')(p_{2h}\overline{v}'')h{\rm d}x. \end{eqnarray*} $

令$ h\rightarrow0 $, 由定义4.1和定理4.2知, (5.7) 式成立.

5. 取$ h $充分小, 令特征值$ \mu = \lambda(p_{1}), $$ \nu = \lambda(p_{1}+h) $对应的正规化特征向量分别为$ (u, u_{1}, u_{2})^{T}, $$ (v, v_{1}, v_{2})^{T}, $其中$ u = u(x, p_{1}), v = u(x, p_{1}+h), $有

$ \begin{eqnarray*} &&[\lambda(p_{1}+h)-\lambda(p_{1})][\eta\int_{a}^{c}u\overline{v}w{\rm d}x+\xi\int_{c}^{b}u\overline{v}w{\rm d}x]\\ & = &\eta\int_{a}^{c}\lambda(p_{1}+h)u\overline{v}w{\rm d}x-\eta\int_{a}^{c}\lambda(p_{1})u\overline{v}w{\rm d}x+\xi\int_{c}^{b}\lambda(p_{1}+h)u\overline{v}w{\rm d}x-\xi\int_{c}^{b}\lambda(p_{1})u\overline{v}w{\rm d}x\\ & = &\eta[u, v](c-)-\eta[u, v](a)+\xi[u, v](b)-\xi[u, v](c+)\\ &&-\eta\int_{a}^{c}((p_{1}+h)\overline{v}')'u{\rm d}x+\eta\int_{a}^{c}(p_{1}\overline{v}')'u{\rm d}x-\xi\int_{c}^{b}((p_{1}+h)\overline{v}')'u{\rm d}x+\xi\int_{c}^{b}(p_{1}\overline{v}')'u{\rm d}x, \end{eqnarray*} $

根据边界条件(2.2) 和(2.3) 有

$ [\lambda(p_{1}+h)-\lambda(p_{1})][\eta\rho_{1}u_{1}\overline{v}_{1}+\eta\rho_{2}u_{2}\overline{v}_{2}] = \eta[u, v](a), $

再由边界条件(2.4), (2.5) 及(3.5) 式, 并用分部积分法可得

$ \begin{eqnarray*} &&[\lambda(p_{1}+h)-\lambda(p_{1})][\eta\int_{a}^{c}u\overline{v}w{\rm d}x+\xi\int_{c}^{b}u\overline{v}w{\rm d}x+\eta\rho_{1}u_{1}\overline{v}_{1}+\eta\rho_{2}u_{2}\overline{v}_{2}]\\ & = &-\eta\int_{a}^{c}((p_{1}+h)\overline{v}')'u{\rm d}x+\eta\int_{a}^{c}(p_{1}\overline{v}')'u{\rm d}x-\xi\int_{c}^{b}((p_{1}+h)\overline{v}')'u{\rm d}x+\xi\int_{c}^{b}(p_{1}\overline{v}')'u{\rm d}x\\ & = &\eta[-u((p_{1}+h)\overline{v}')+u(p_{1}\overline{v}')]_{a}^{c}+\eta\int_{a}^{c}u'h\overline{v}'{\rm d}x\\ &&+\xi[-u((p_{1}+h)\overline{v}')+u(p_{1}\overline{v}')]_{c}^{b}+\xi\int_{c}^{b}u'h\overline{v}'{\rm d}x. \end{eqnarray*} $

令$ h\rightarrow0 $, 由定义4.1和定理4.2知, (5.8) 式成立. (5.9)和(5.10) 式证明与此类似, 省略.

接下来我们考虑$ \lambda $关于$ c-, c+ $的微分表达式, 令$ c_{1} = c-, c_{2} = c+, $然后有

定理5.2   令(2.7)–(2.9) 式成立, $ \omega = (1/p_{2}, p_{1}, q, w, a, b, \gamma, c-, c+, A, C, D)\in\Omega, $$ \lambda = \lambda(\omega) $是算子$ T $的特征值, $ (u, u_{1}, u_{2})^{T} $是相应的正规化特征向量. 若$ \lambda(\omega) $在$ \omega $的某个邻域$ M\subset\Omega $内的几何重数不变, 则$ \lambda $关于内部不连续点$ c $左右两侧是可微的且有如下表达式:

1. 固定$ \omega $中除$ c_{1} $之外的所有变量, 令$ \lambda = \lambda(c_{1}) $为特征值, 则$ \lambda $是可微的且有

(5.22) $ \begin{eqnarray} \lambda'(c_{1}) = \eta(U')^{\ast}(c_{1}, c_{1})J_{0}U(c_{1}, c_{1}){\quad} a.e.\ c_{1}\in[a, c). \end{eqnarray} $

特别地, 如果$ p_{2}^{-1}, p_{1}, q, w $在$ c_{1} $处连续且$ p_{2}(c_{1})\neq0, $则(5.22) 式在$ c_{1} $处也成立. 其中, $ U' $表示向量$ U $的导数.

2. 固定$ \omega $中除$ c_{2} $之外的所有变量, 令$ \lambda = \lambda(c_{2}) $为特征值, 则$ \lambda $是可微的且有

(5.23) $ \begin{eqnarray} \lambda'(c_{2}) = -\xi(U')^{\ast}(c_{2}, c_{2})J_{0}U(c_{2}, c_{2}) {\quad} a.e.\ c_{2}\in(c, b]. \end{eqnarray} $

特别地, 如果$ p_{2}^{-1}, p_{1}, q, w $在$ c_{2} $处连续且$ p_{2}(c_{2})\neq0, $则(5.23) 式在$ c_{2} $处也成立.

证  因为(5.22)和(5.23) 式证明类似, 我们只证明(5.22) 式. 让$ |h| $充分小($ h<0 $), 令特征值$ \mu = \lambda(c_{1}), \nu = \lambda(c_{1}+h) $对应的正规化特征向量分别为$ (u, u_{1}, u_{2})^{T}, (v, v_{1}, v_{2})^{T}, $其中$ u = u(x, c_{1}), v = u(x, c_{1}+h), $我们有

(5.24) $ \begin{eqnarray} &&[\lambda(c_{1}+h)-\lambda(c_{1})][\eta\int_{a}^{c}u\overline{v}w{\rm d}x+\xi\int_{c}^{b}u\overline{v}w{\rm d}x+\eta\rho_{1}u_{1}\overline{v}_{1}+\eta\rho_{2}u_{2}\overline{v}_{2}]{}\\ & = &\eta[u, v](c-)-\xi[u, v](c+){}\\ & = &\eta\big\{u(c_{1}, c_{1})[((p_{2}\overline{u}'')'-p_{1}\overline{u}')(c_{1}, c_{1}+h)-((p_{2}\overline{u}'')'-p_{1}\overline{u}')(c_{1}+h, c_{1}+h)]{}\\ &&-u'(c_{1}, c_{1})[(p_{2}\overline{u}'')(c_{1}, c_{1}+h)-(p_{2}\overline{u}'')(c_{1}+h, c_{1}+h)]{}\\ &&+(p_{2}u'')(c_{1}, c_{1})[\overline{u}'(c_{1}, c_{1}+h)-\overline{u}'(c_{1}+h, c_{1}+h)]{}\\ &&-((p_{2}u'')'-p_{1}u')(c_{1}, c_{1})[\overline{u}(c_{1}, c_{1}+h)-\overline{u}(c_{1}+h, c_{1}+h)]\big\}, \end{eqnarray} $

(5.25) $ \begin{eqnarray} &&((p_{2}\overline{u}'')'-p_{1}\overline{u}')(c_{1}, c_{1}+h)-((p_{2}\overline{u}'')'-p_{1}\overline{u}')(c_{1}+h, c_{1}+h){}\\ & = &-\int_{c_{1}}^{c_{1}+h}[(p_{2}\overline{u}'')''-(p_{1}\overline{u}')'](s, c_{1}+h){\rm d}s{}\\ & = &-\int_{c_{1}}^{c_{1}+h}[\lambda(c_{1}+h)w(s)\overline{u}(s, c_{1}+h)-q(s)\overline{u}(s, c_{1}+h)]{\rm d}s{}\\ & = &-\lambda(c_{1}+h)\int_{c_{1}}^{c_{1}+h}w(s)\overline{u}(s, c_{1}){\rm d}s+\lambda(c_{1}+h)\int_{c_{1}}^{c_{1}+h}w(s)[\overline{u}(s, c_{1})-\overline{u}(s, c_{1}+h)]{\rm d}s{}\\ &&+\int_{c_{1}}^{c_{1}+h}q(s)\overline{u}(s, c_{1}){\rm d}s-\int_{c_{1}}^{c_{1}+h}q(s)[\overline{u}(s, c_{1}) -\overline{u}(s, c_{1}+h)]{\rm d}s \end{eqnarray} $

和

(5.26) $ \begin{eqnarray} &&(p_{2}\overline{u}'')(c_{1}, c_{1}+h)-(p_{2}\overline{u}'')(c_{1}+h, c_{1}+h){}\\ & = &-\int_{c_{1}}^{c_{1}+h}(p_{2}\overline{u}'')'(s, c_{1}+h){\rm d}s{}\\ & = &-\int_{c_{1}}^{c_{1}+h}(p_{2}\overline{u}'')'(s, c_{1}){\rm d}s+ \int_{c_{1}}^{c_{1}+h}\{[((p_{2}\overline{u}'')'-p_{1}\overline{u}')(s, c_{1}){}\\ &&-((p_{2}\overline{u}'')'-p_{1}\overline{u}')(s, c_{1}+h)]-p_{1}[\overline{u}'(s, c_{1}+h)-\overline{u}'(s, c_{1})]\}{\rm d}s, \end{eqnarray} $

注意到当$ h\rightarrow 0 $时, (5.25) 和(5.26) 式中$ \overline{u}(s, c_{1})-\overline{u}(s, c_{1}+h)\rightarrow 0, ((p_{2}\overline{u}'')'-p_{1}\overline{u}')(s, c_{1})-((p_{2}\overline{u}'')'-p_{1}\overline{u}')(s, c_{1}+h)\rightarrow 0, \overline{u}'(s, c_{1}+h)-\overline{u}'(s, c_{1})\rightarrow 0, $由引理5.2可知

$ \begin{eqnarray*} &&\lim\limits_{h\rightarrow 0}\frac{((p_{2}\overline{u}'')'-p_{1}\overline{u}')(c_{1}, c_{1}+h)-((p_{2}\overline{u}'')'-p_{1}\overline{u}')(c_{1}+h, c_{1}+h)}{h}\\ & = &-\overline{u}(c_{1}, c_{1})[\lambda(c_{1})w(c_{1})-q(c_{1})], \\ && \lim\limits_{h\rightarrow 0}\frac{(p_{2}\overline{u}'')(c_{1}, c_{1}+h)-(p_{2}\overline{u}'')(c_{1}+h, c_{1}+h)}{h} = -(p_{2}\overline{u}'')'(c_{1}, c_{1}). \end{eqnarray*} $

类似的

$ \lim\limits_{h\rightarrow 0}\frac{\overline{u}'(c_{1}, c_{1}+h)-\overline{u}'(c_{1}+h, c_{1}+h)}{h} = -\frac{1}{p_{2}(c_{1})}(p_{2}\overline{u}'')(c_{1}, c_{1}), $

$ \lim\limits_{h\rightarrow 0}\frac{\overline{u}(c_{1}, c_{1}+h)-\overline{u}(c_{1}+h, c_{1}+h)}{h} = -\overline{u}'(c_{1}, c_{1}). $

对(5.24) 式两边同时除以$ h, $并取极限$ h\rightarrow 0, $我们得到(5.22) 式.

定理5.3  令(2.7)–(2.9) 式成立, $ \gamma = \pi, \lambda = \lambda(\omega) $是算子$ T $的特征值, $ (u, u_{1}, u_{2})^{T} $是相应的正规化特征向量. 若$ \lambda(\omega) $在$ \omega $的某个邻域$ M\subset\Omega $内的几何重数不变. 固定$ \omega $中除$ b $之外的所有变量, 令$ \lambda = \lambda(b) $为特征值, 则$ \lambda $是可微的且有

(5.27) $ \begin{eqnarray} \lambda'(b) = \xi[-\frac{1}{p_{2}(b)}|p_{2}u''|^{2}(b, b)+((p_{2}u'')'-p_{1}u')(b, b)\overline{u}'(b, b)]{\quad} a.e.\ (c, b'). \end{eqnarray} $

特别地, 如果$ p_{2}^{-1}, p_{1} $在$ b $处是连续的, $ p_{2}(b)\neq0, $则(5.27) 式在$ b $处也成立.

证  让$ h $充分小, 令特征值$ \mu = \lambda(b), \nu = \lambda(b+h) $对应的正规化特征向量分别为$ (u, u_{1}, u_{2})^{T}, (v, v_{1}, v_{2})^{T}, $其中$ u = u(x, b), v = u(x, b+h) $. 由边界条件(2.4)–(2.5) 和(3.5) 式得到

$ \eta[u, v](c-) = \xi[u, v](c+), u(b, b) = 0, u'(b, b) = 0, $

我们有

(5.28) $ \begin{eqnarray} &&[\lambda(b+h)-\lambda(b)][\eta\int_{a}^{c}u\overline{v}w{\rm d}x+\xi\int_{c}^{b}u\overline{v}w{\rm d}x+\eta\rho_{1}u_{1}\overline{v}_{1}+\eta\rho_{2}u_{2}\overline{v}_{2}]{}\\ & = &-\eta[u, v](a)+\xi[u, v](b)+\eta[u, v](a){}\\ & = &\xi[(p_{2}u'')(b, b)\overline{u}'(b, b+h)-((p_{2}u'')'-p_{1}u')(b, b)\overline{u}(b, b+h)], \end{eqnarray} $

其中

$ \begin{eqnarray*} &&\overline{u}'(b, b+h) = \overline{u}'(b, b+h)-\overline{u}'(b+h, b+h)\\ & = &-\int_{b}^{b+h}\frac{1}{p_{2}(s)}(p_{2}\overline{u}'')(s, b+h){\rm d}s\\ & = &-\int_{b}^{b+h}\frac{1}{p_{2}(s)}(p_{2}\overline{u}'')(s, b){\rm d}s+\int_{b}^{b+h}\frac{1}{p_{2}(s)}[(p_{2}\overline{u}'')(s, b)-(p_{2}\overline{u}'')(s, b+h)]{\rm d}s, \end{eqnarray*} $

注意到当$ h\rightarrow0 $时, $ (p_{2}\overline{u}'')(s, b)-(p_{2}\overline{u}'')(s, b+h)\rightarrow0, $由引理5.2可知

$ \lim\limits_{h\rightarrow 0}\frac{\overline{u}'(b, b+h)}{h} = -\frac{1}{p_{2}(b)}(p_{2}\overline{u}'')(b, b). $

类似的, 有

$ \lim\limits_{h\rightarrow 0}\frac{\overline{u}(b, b+h)}{h} = -\overline{u}'(b, b). $

对(5.28) 式两边同时除以$ h, $并且取极限$ h\rightarrow0, $我们得到(5.27) 式.

定理5.4   令(2.7)–(2.9) 式成立, $ \gamma = \frac{\pi}{2}, \lambda = \lambda(\omega) $是算子$ T $的特征值, $ (u, u_{1}, u_{2})^{T} $是相应的正规化特征向量. 若$ \lambda(\omega) $在$ \omega $的某个邻域$ M\subset\Omega $内的几何重数不变. 固定$ \omega $中除$ b $之外的所有变量, 令$ \lambda = \lambda(b) $为特征值, 则$ \lambda $是可微的且有

(5.29) $ \begin{eqnarray} \lambda'(b) = \xi[-|u|^{2}(b, b)[\lambda(b)w(b)-q(b)]+u'(b, b)(p_{2}\overline{u}'')'(b, b)] {\quad} a.e.\ (c, b'). \end{eqnarray} $

特别地, 如果$ p_{2}, q, w $在$ b $处是连续的, 则(5.29) 式在$ b $处也成立.

证  让$ h $充分小, 令特征值$ \mu = \lambda(b), \nu = \lambda(b+h) $对应的正规化特征向量分别为$ (u, u_{1}, u_{2})^{T}, (v, v_{1}, v_{2})^{T}, $其中$ u = u(x, b), v = u(x, b+h) $. 由边界条件(2.4)–(2.5) 和(3.5) 式得到

$ \begin{eqnarray*} \eta[u, v](c-) = \xi[u, v](c+), (p_{2}u'')(b, b) = 0, ((p_{2}u'')'-p_{1}u')(b, b) = 0, \end{eqnarray*} $

我们有

(5.30) $ \begin{eqnarray} &&[\lambda(b+h)-\lambda(b)][\eta\int_{a}^{c}u\overline{v}w{\rm d}x+\xi\int_{c}^{b}u\overline{v}w{\rm d}x+\eta\rho_{1}u_{1}\overline{v}_{1}+\eta\rho_{2}u_{2}\overline{v}_{2}]{}\\ & = &-\eta[u, v](a)+\xi[u, v](b)+\eta[u, v](a){}\\ & = &\xi[u(b, b)((p_{2}\overline{u}'')'-p_{1}\overline{u}')(b, b+h)-u'(b, b)(p_{2}\overline{u}'')(b, b+h)], \end{eqnarray} $

其中

$ \begin{eqnarray*} &&((p_{2}\overline{u}'')'-p_{1}\overline{u}')(b, b+h)\\ & = &((p_{2}\overline{u}'')'-p_{1}\overline{u}')(b, b+h)-((p_{2}\overline{u}'')'-p_{1}\overline{u}')(b+h, b+h)\\ & = &-\int_{b}^{b+h}[(p_{2}\overline{u}'')''-(p_{1}\overline{u}')'](s, b+h){\rm d}s\\ & = &-\int_{b}^{b+h}[\lambda(b+h)w(s)\overline{u}(s, b+h)-q(s)\overline{u}(s, b+h)]{\rm d}s\\ & = &-\lambda(b+h)\int_{b}^{b+h}w(s)\overline{u}(s, b){\rm d}s+\lambda(b+h)\int_{b}^{b+h}w(s)[\overline{u}(s, b)-\overline{u}(s, b+h)]{\rm d}s\\ & &+\int_{b}^{b+h}q(s)\overline{u}(s, b){\rm d}s-\int_{b}^{b+h}q(s)[\overline{u}(s, b)-\overline{u}(s, b+h)]{\rm d}s, \end{eqnarray*} $

注意到当$ h\rightarrow 0 $时, $ \overline{u}(s, b)-\overline{u}(s, b+h)\rightarrow0, $由引理5.2可知

$ \lim\limits_{h\rightarrow 0}\frac{((p_{2}\overline{u}'')'-p_{1}\overline{u}')(b, b+h)}{h} = -\overline{u}(b, b)[\lambda(b)w(b)-q(b)]. $

类似的, 有

$ \lim\limits_{h\rightarrow 0}\frac{(p_{2}\overline{u}'')(b, b+h)}{h} = -(p_{2}\overline{u}'')'(b, b). $

对(5.30) 式两边同时除以$ h, $并且取极限$ h\rightarrow0, $我们得到(5.29) 式.

定理5.5   令(2.7)–(2.9) 式成立, $ 0<\gamma\leq\pi, \lambda = \lambda(\omega) $是算子$ T $的特征值, $ (u, u_{1}, u_{2})^{T} $是相应的正规化特征向量. 若$ \lambda(\omega) $在$ \omega $的某个邻域$ M\subset\Omega $内的几何重数不变, 则$ \lambda $关于边界点$ a, b $是可微的且有如下微分表达式:

1. 固定$ \omega $中除$ a $之外的所有变量, 令特征值$ \lambda = \lambda(a) $, 则$ \lambda $是可微的且有

(5.31) $ \begin{eqnarray} \lambda'(a) = -\eta(U')^{\ast}(a, a)J_{0}U(a, a){\quad} a.e.\ (a', c). \end{eqnarray} $

特别地, 如果$ p_{2}^{-1}, p_{1}, q, w $在$ a $处是连续的, $ p_{2}(a)\neq0, $则(5.31) 式在$ a $处也成立.

2. 固定$ \omega $中除$ b $之外的所有变量, 令特征值$ \lambda = \lambda(b) $, 则$ \lambda $是可微的且有

(5.32) $ \begin{eqnarray} \lambda'(b) = \xi(U')^{\ast}(b, b)J_{0}U(b, b) {\quad} a.e.\ (c, b'). \end{eqnarray} $

特别地, 如果$ p_{2}^{-1}, p_{1}, q, w $在$ b $处是连续的, $ p_{2}(b)\neq0, $则(5.32) 式在$ b $处也成立.

证  显然(5.32) 式是(5.27) 式和(5.29) 式的结合, 且(5.31) 式的证明同(5.32) 式类似, 因此证明过程省略.