西南大学学报 (自然科学版)  2018, Vol. 40 Issue (10): 83-88.  DOI: 10.13718/j.cnki.xdzk.2018.10.014
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  • 非线性复微分方程的解与Hω空间    [PDF全文]
    孙煜1, 龙见仁1, 覃智高1, 胡光明2     
    1. 贵州师范大学 数学科学学院, 贵阳 550001;
    2. 北京航空航天大学 数学与系统科学学院, 北京 100191
    摘要:利用直接的积分估计,研究非线性复微分方程 $ {\left( {{f^{\left( k \right)}}} \right)^{{n_k}}} + {A_{k - 1}}\left( z \right){\left( {{f^{\left( {k - 1} \right)}}} \right)^{{n_{k - 1}}}} + \cdots + {A_1}\left( z \right){\left( {f'} \right)^{{n_1}}} + {A_0}\left( z \right)f = {A_k}\left( z \right) $ 解的函数空间属性,刻画了方程的解析解,以及它们的导数属于Hω空间时系数需要满足的条件.改善及推广了已有的相关结果.
    关键词非线性复微分方程    Hardy空间    解析解    单位圆    

    近年来,关于微分方程的研究已经引起了广泛的关注(参见文献[1-2]).其中一个分支是关于复线性微分方程

    $ {f^{\left( k \right)}} + {A_{k - 1}}\left( z \right){f^{\left( {k - 1} \right)}} + \cdots + {A_1}\left( z \right)f' + {A_0}\left( z \right)f = {A_k}\left( z \right) $ (1)

    解的性质的研究,主要关注的是方程(1)的解的增长性,其中Aj$ \mathscr{H} $(D)(j=0,1,…,k),$\mathscr{ H} $(D)={ffD上是解析的},D={z$ \mathbb{C} $:|z|<1}.通过Nevanlinna理论,文献[3-4]得到了一些关于解的快速增长的结果.文献[5-7]得到了一些关于解的慢速增长结果.文献[8-9]研究了非线性复微分方程

    $ {\left( {{f^{\left( k \right)}}} \right)^{{n_k}}} + {A_{k - 1}}\left( z \right){\left( {{f^{\left( {k - 1} \right)}}} \right)^{{n_{k - 1}}}} + \cdots + {A_1}\left( z \right){\left( {f'} \right)^{{n_1}}} + {A_0}\left( z \right)f = {A_k}\left( z \right) $ (2)

    解的增长性质,其中Aj$ \mathscr{H} $ (D)(j=0,1,…,k),给出了方程(2)的所有解析解属于给定空间(例如Qk空间、Hardy空间等)的一些充分条件.在研究方程(1)的解的慢速增长性中,常用Herold比较定理[10]和一些其它基于Carleson测度的方法[7].本文与以上方法不同,主要基于直接的积分估计.

    文献[11]给出了一些使得方程(1)的所有解和它们的导数属于Hω空间的充分条件,其中

    $ H_\omega ^\infty = \left\{ {f \in {\mathscr{H}}\left( D \right):{{\left\| f \right\|}_{H_\omega ^\infty }}{\rm{ = }}\mathop {\sup }\limits_{z \in D} \left| {f\left( z \right)} \right| \cdot \omega \left( z \right) < \infty } \right\} $

    ω是一个权重函数,满足ωD→(0,∞)是有界可测的.如果对于所有的zD,有ω(z)=ω(|z|),则称ω是径向的.若对于所有的p∈(0,∞),有ω(z)=(1-|z|)p,则Hω=Hp.令

    $ \begin{array}{*{20}{c}} {\omega \left( z \right) = \omega _a^{{h_1}}\left( z \right)\omega _b^{{h_2}}\left( z \right)}&{{h_1},{h_2} \in \mathbb{N}} \end{array} $

    其中$ \mathbb{N} $为自然数集,$ {{\omega }_{a}}(z)={{\left( \text{log}\left( \frac{\text{e}}{1-\left| z \right|} \right) \right)}^{-1}}, {{\omega }_{b}}(z)=1-\left| z \right| $.

    在本文中,总假设径向权重ωD→(0,∞)满足下面两个条件:

    (f1)存在M=M(ω)∈(0,∞),使得

    $ \mathop {\lim \sup}\limits_{s \to {1^ - }} \omega \left( s \right)\int_0^s {\frac{{{\rm{d}}t}}{{\omega \left( t \right)\left( {1 - t} \right)}}} < M < \infty $ (3)

    (f2)存在常数ε∈(0,∞),m=m(ωε)∈(0,∞),使得

    $ \mathop {\lim \sup}\limits_{s \to {1^ - }} \frac{{\omega \left( s \right)}}{{\omega \left( {\frac{{1 + \varepsilon s}}{{1 + \varepsilon }}} \right)}} < m $ (4)

    由(3)式知,存在Mj=Mj(ωj)∈(0,M]和M0=M0(ω)∈(0,∞),使得

    $ \begin{array}{*{20}{c}} {\mathop {\lim \sup}\limits_{s \to {1^ - }} \omega \left( s \right){{\left( {1 - s} \right)}^{j - 1}}\int_0^s {\frac{{{\rm{d}}t}}{{\omega \left( t \right){{\left( {1 - t} \right)}^j}}} < {M_j}} }&{j = 1,2, \cdots ,k} \end{array} $ (5)

    $ \begin{array}{*{20}{c}} {\omega \left( r \right)\int_0^r {\frac{{{\rm{d}}t}}{{\omega \left( t \right)\left( {1 - t} \right)}} < {m_0}} }&{r \in \left( {0,1} \right)} \end{array} $ (6)

    为方便描述,特作以下记号:

    $ {\omega _p}\left( z \right) = \omega \left( z \right){\left( {1 - \left| z \right|} \right)^p} $
    $ {{\dot \omega }_{h\left( {1,2} \right),n\left( {k,j} \right)}}\left( z \right) = {\left( {{\omega _a}\left( z \right)} \right)^{{h_1}\left( {{n_k} - {n_j}} \right)}}{\left( {{\omega _b}\left( z \right)} \right)^{\left( {{h_2} + k} \right){n_k} - \left( {{h_2} + j} \right){n_j}}} $

    j=0时,

    $ {{\dot \omega }_{h\left( {1,2} \right),n\left( {k,0} \right)}}\left( z \right) = {\left( {{\omega _a}\left( z \right)} \right)^{{h_1}\left( {{n_k} - 1} \right)}}{\left( {{\omega _b}\left( z \right)} \right)^{\left( {{h_2} + k} \right){n_k} - {h_2}}} $
    $ {{\tilde \omega }_{h\left( {1,2} \right),n\left( {k,0} \right)}}\left( z \right) = {\left( {{\omega _a}\left( z \right)} \right)^{{h_1}\left( {{n_k} - 1} \right)}}{\left( {{\omega _b}\left( z \right)} \right)^{\left( {{h_2} + k - 1} \right){n_k} - {h_2}}} $

    记号中的ωωaωbωp$ \dot{\omega } $h(1,2),n(kj)$ \widetilde{\omega } $h(1,2),n(kj)均为径向权重,其中n0=1,nj≥1(j=1,2,…,k),njnk(j=1,2,…,k-1).值得注意的是,若方程(2)是线性的,即nk=nj=1(j=0,…,k),则

    $ \begin{array}{*{20}{c}} {{{\dot \omega }_{h\left( {1,2} \right),n\left( {k,j} \right)}}\left( z \right) = {{\left( {1 - \left| z \right|} \right)}^{k - j}}}&{{{\tilde \omega }_{h\left( {1,2} \right),n\left( {k,j} \right)}}\left( z \right) = {{\left( {1 - \left| z \right|} \right)}^{k - j - 1}}} \end{array} $

    类似Hω的定义,我们定义如下函数空间:

    $ H_{{{\dot \omega }_{h\left( {1,2} \right),n\left( {k,j} \right)}}}^\infty = \left\{ {f \in {\mathscr{H}}\left( D \right):{{\left\| f \right\|}_{H_{{{\dot \omega }_{h\left( {1,2} \right),n\left( {k,j} \right)}}}^\infty }} = \mathop {\sup }\limits_{z \in D} \left| {f\left( z \right)} \right|{{\dot \omega }_{h\left( {1,2} \right),n\left( {k,j} \right)}}\left( z \right) < \infty } \right\} $
    $ H_{{{\tilde \omega }_{h\left( {1,2} \right),n\left( {k,j} \right)}}}^\infty = \left\{ {f \in {\mathscr{H}}\left( D \right):{{\left\| f \right\|}_{H_{{{\tilde \omega }_{h\left( {1,2} \right),n\left( {k,j} \right)}}}^\infty }} = \mathop {\sup }\limits_{z \in D} \left| {f\left( z \right)} \right|{{\tilde \omega }_{h\left( {1,2} \right),n\left( {k,j} \right)}}\left( z \right) < \infty } \right\} $
    1 引理和主要结果

    引理 1[12]  设n=1,2,…,Nan≥0,则:

    $ \begin{array}{*{20}{c}} {{{\left( {\sum\limits_{n = 1}^n {{a_n}} } \right)}^p} \le \left( {\sum\limits_{n = 1}^n {{a_n}^p} } \right)}&{0 < p \le 1} \end{array} $
    $ \begin{array}{*{20}{c}} {{{\left( {\sum\limits_{n = 1}^n {{a_n}} } \right)}^p} \le {N^{p - 1}}\left( {\sum\limits_{n = 1}^n {{a_n}^p} } \right)}&{1 \le p < \infty } \end{array} $

    引理 2[11]  设ω:Δ→(0,∞)是一个径向权重且满足(3)式,则对于f$ \mathscr{H} $(D)有

    $ \begin{array}{*{20}{c}} {\left| {f\left( z \right)} \right|\omega \left( z \right) \leqslant {Q_k}\mathop {\sup }\limits_{\left| \xi \right| \leqslant \left| z \right|} \left( {\left| {{f^{\left( k \right)}}\left( \xi \right)} \right|\omega \left( \xi \right){{\left( {1 - \left| \xi \right|} \right)}^k}} \right) + C}&{z \in D,k \in \mathbb{N}} \end{array} $

    其中C∈[0,∞)为不依赖于z的常数,$ {Q_k} = \mathop {\mathop \prod \limits^k }\limits_{j = 1} {\mkern 1mu} {M_j} $Mj为(5)式所定义.

    引理 3[11]  设ωD→(0,∞)是一个径向权重,且存在常数ε∈(0,∞),m=m(ωε)∈(0,∞)满足(4)式,则对于f$ \mathscr{H} $(D)有

    $ \begin{array}{*{20}{c}} {\left| {{f^{\left( k \right)}}\left( z \right)} \right|\omega \left( z \right){{\left( {1 - \left| z \right|} \right)}^{\left( k \right)}} \leqslant k!{{\left( {1 + \varepsilon } \right)}^k}m\mathop {\sup }\limits_{\left| \xi \right| = \rho } \left| {f\left( \xi \right)} \right|\omega \left( \rho \right) + C}&{z \in D,k \in \mathbb{N}} \end{array} $

    其中$ \rho =\rho (\varepsilon , \left| z \right|)=\frac{1+\varepsilon \left| z \right|}{1+\varepsilon } $C≥0为不依赖于z的常数.

    定义如下扩张函数:设f$ \mathscr{H} $(D),令fr(z)=f(rz),其中zDr∈[0,1).

    引理 4[11]  设ωD→(0,∞)是一个径向权重,且存在常数ε∈(0,∞),m=m(ωε)∈(0,∞)满足(4)式.如果$\underset{r\in \left[ 0, 1 \right)}{\mathop{\text{sup}}}\, {{\left\| {{f}_{r}} \right\|}_{H_{\omega }^{\infty }}}<\infty $,则fHω$ {{\left\| {{f}_{r}} \right\|}_{H_{\omega }^{\infty }}}=\underset{r\in \left[ 0, \text{ }1 \right)}{\mathop{\text{sup}}}\, {{\left\| {{f}_{r}} \right\|}_{H_{\omega }^{\infty }}} $.

    引理 5  设径向权重ω(z)=ωah1(z)ωbh2(z),则ω(z)满足(3)式和(4)式.

      设s∈[0,1),h1h2$ \mathbb{N} $,则由:

    $ \begin{array}{*{20}{c}} {1 - \log \left( {1 - s} \right) > - \log \left( {1 - s} \right)}&{{{\left( {\log \left( {\frac{{\rm{e}}}{{1 - s}}} \right)} \right)}^{ - 1}} < {{\left( { - \log \left( {1 - s} \right)} \right)}^{ - 1}}} \end{array} $

    $ {\left( {\log \left( {\frac{{\rm{e}}}{{1 - s}}} \right)} \right)^{ - {h_1}}} < {\left( { - \log \left( {1 - s} \right)} \right)^{ - {h_1}}} $ (7)

    做辅助函数F(s)=(log(1-s))h1(1-s)-h2,则

    $ F'\left( s \right) = - {h_1}{\left( {\log \left( {1 - s} \right)} \right)^{{h_1} - 1}}{\left( {1 - s} \right)^{{h_2} - 1}} - {h_2}{\left( {\log \left( {1 - s} \right)} \right)^{{h_1}}}{\left( {1 - s} \right)^{{h_2} - 1}} $

    两边求积分,得

    $ - {\left( {\log \left( {1 - r} \right)} \right)^{{h_1}}}{\left( {1 - r} \right)^{ - {h_2}}} = {h_1}\int_0^r {{{\left( {\log \left( {1 - s} \right)} \right)}^{{h_1} - 1}}{{\left( {1 - s} \right)}^{{h_2} - 1}}{\rm{d}}s} + {h_2}\int_0^r \\ {{{\left( {\log \left( {1 - s} \right)} \right)}^{{h_1}}}{{\left( {1 - s} \right)}^{{h_2} - 1}}{\rm{d}}s} $

    两边同乘-(log(1-r))-h1(1-r)h2,得

    $ \begin{array}{*{20}{c}} {{h_1}{{\left( {\log \left( {1 - r} \right)} \right)}^{ - {h_1}}}{{\left( {1 - r} \right)}^{{h_2}}}\int_0^r {{{\left( {\log \left( {1 - s} \right)} \right)}^{{h_1} - 1}}{{\left( {1 - s} \right)}^{{h_2} - 1}}{\rm{d}}s} + }\\ {{h_2}{{\left( {\log \left( {1 - r} \right)} \right)}^{ - {h_1}}}{{\left( {1 - r} \right)}^{{h_2}}}\int_0^r {{{\left( {\log \left( {1 - s} \right)} \right)}^{{h_1}}}{{\left( {1 - s} \right)}^{{h_2} - 1}}} {\rm{d}}s = - 1} \end{array} $

    重复以上过程h1次,得

    $ \mathop {\lim \sup}\limits_{r \to {1^ - }} {\left( {\log \left( {1 - r} \right)} \right)^{ - {h_1}}}{\left( {1 - r} \right)^{{h_2}}}\int_0^r {{{\left( {\log \left( {1 - s} \right)} \right)}^{{h_1}}}} {\left( {1 - s} \right)^{{h_2} - 1}}{\rm{d}}s < \infty $

    由(7)式得

    $ \mathop {\lim \sup}\limits_{r \to {1^ - }} \omega \left( r \right)\int_0^r {\frac{{{\rm{d}}s}}{{\omega \left( s \right)\left( {1 - s} \right)}} < \infty } $
    $ \begin{array}{l} \mathop {\lim }\limits_{r \to {1^ - }} \frac{{\omega \left( r \right)}}{{\omega \left( {\frac{{1 + \varepsilon r}}{{1 + \varepsilon }}} \right)}} = \mathop {\lim }\limits_{r \to {1^ - }} \frac{{{{\left( {{\omega _a}\left( r \right)} \right)}^{{h_1}}}{{\left( {{\omega _b}\left( r \right)} \right)}^{{h_2}}}}}{{{{\left( {{\omega _a}\left( {\frac{{1 + \varepsilon r}}{{1 + \varepsilon }}} \right)} \right)}^{{h_1}}}{{\left( {{\omega _b}\left( {\frac{{1 + \varepsilon r}}{{1 + \varepsilon }}} \right)} \right)}^{{h_2}}}}} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathop {\lim }\limits_{r \to {1^ - }} \frac{{{{\left( {\log \frac{{\rm{e}}}{{1 - r}}} \right)}^{ - {h_1}}}}}{{{{\left( {\log \frac{{\rm{e}}}{{1 - \frac{{1 + \varepsilon r}}{{1 + \varepsilon }}}}} \right)}^{ - {h_1}}}}}{\left( {\frac{{1 + \varepsilon }}{\varepsilon }} \right)^{{h_2}}} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathop {\lim }\limits_{r \to {1^ - }} {\left( {\frac{{1 - \log \left( {1 - \frac{{1 + \varepsilon r}}{{1 + \varepsilon }}} \right)}}{{1 - \log \left( {1 - r} \right)}}} \right)^{{h_1}}}{\left( {\frac{{1 + \varepsilon }}{\varepsilon }} \right)^{{h_2}}} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathop {\lim }\limits_{r \to {1^ - }} {\left( {1 - \frac{{\log \varepsilon + \log \left( {1 + \varepsilon } \right)}}{{1 - \log \left( {1 - r} \right)}}} \right)^{{h_1}}}{\left( {\frac{{1 + \varepsilon }}{\varepsilon }} \right)^{{h_2}}} < \infty \end{array} $

    于是径向权重ω满足(3)式和(4)式.

    本文的主要目的是研究方程(2)的解析解,以及它们的导数属于空间Hω时系数需要满足的条件,主要证明了下面的结果:

    定理 1  设径向权重ω在单位圆区域D上满足(3)式和(4)式.如果$ {{A}_{j}}\in H_{{{{\dot{\omega }}}_{h\left( 1, 2 \right), n\left( k, j \right)}}}^{\infty }\left( j=0, 1, \cdots , k \right)$,且

    $ E = {Q_k}\left( {\left\| {{A_0}} \right\|_{H_{{{\dot \omega }_{h\left( {1,2} \right),n\left( {k,0} \right)}}}^\infty }^{\frac{1}{{{n_k}}}} + \sum\limits_{j = 1}^{k - 1} {j!} {{\left( {1 + \varepsilon } \right)}^j}m\left\| {{A_j}} \right\|{{_{H_{{{\dot \omega }_{h\left( {1,2} \right),n\left( {k,j} \right)}}}^\infty }^{\frac{1}{{{n_k}}}}}^{\frac{1}{{{n_k}}}}}} \right) < 1 $

    其中$ {Q_k} = \mathop {\mathop \prod \limits^k }\limits_{j = 1} {\mkern 1mu} {M_j} $Mj为(5)式所定义,mε为(4)式所定义,则方程(2)的所有解析解属于Hω.

    定理 2  设径向权重ω在单位圆区域D上满足(3)式和(4)式.如果${{A}_{j}}\in H_{{{\widetilde{\omega }}_{n, k, j}}}^{\infty }\left( j=0, 1, \cdots , k \right) $,且

    $ \begin{array}{l} F = {Q_{k - 1}}\left( {\mathop {\sup }\limits_{z \in D} \omega \left( z \right)\left\| {{A_0}} \right\|_{H_{{{\tilde \omega }_{h\left( {1,2} \right),n\left( {k,0} \right)}}}^\infty }^{\frac{1}{{{n_k}}}} + \int_0^{\left| z \right|} {\frac{{{\rm{d}}r}}{{\omega \left( r \right)}}} \left\| {{A_1}} \right\|_{H_{\tilde \omega h\left( {1,2} \right),n\left( {k,1} \right)}^\infty }^{\frac{1}{{{n_k}}}} + } \right.\\ \left. {\;\;\;\;\;\;\sum\limits_{j = 1}^{k - 1} {\left( {j - 1} \right)!} {{\left( {1 + \varepsilon } \right)}^{j - 1}}m\left\| {{A_{j + 1}}} \right\|_{H_{{{\tilde \omega }_{h\left( {1,2} \right),n\left( {k,j} \right)}}}^\infty }^{\frac{1}{{{n_k}}}}} \right) < 1 \end{array} $

    其中$ {Q_k} = \mathop {\mathop \prod \limits^k }\limits_{j = 1} {\mkern 1mu} {M_j}$Mj为(5)式所定义,mε为(4)式所定义,则方程(2)的每个解析解的导数属于Hω.

    2 主要结果的证明

    定理1的证明  设f是方程(2)的解析解,则

    $ \begin{array}{*{20}{c}} {{{\left( {{f_r}^{\left( k \right)}\left( z \right)} \right)}^{{n_k}}} + \sum\limits_{j = 0}^{k - 1} {{B_j}\left( z \right){{\left( {f_r^{\left( j \right)}\left( z \right)} \right)}^{{n_j}}} = 0} }&{z \in D} \end{array} $ (8)

    其中Bj(z)=Bj(zr)=rknk-jnjAj(r z),r∈[0,1).由引理5知,ω满足(3)式和(4)式.再由(8)式和引理2,有

    $ \begin{array}{l} \left| {{f_r}\left( z \right)} \right|\omega \left( z \right) \le {Q_k}\mathop {\sup }\limits_{\left| \xi \right| \le \left| z \right|} \left( {\left| {f_r^{\left( k \right)}\left( \xi \right)} \right|{{\left( {{\omega _a}\left( \xi \right)} \right)}^{{h_1}}}{{\left( {1 - \left| \xi \right|} \right)}^{k + {h_2}}}} \right) + {C_{{t_1}}} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{Q_k}\mathop {\sup }\limits_{\left| \xi \right| \le \left| z \right|} {\left( {\left| {\sum\limits_{j = 0}^{k - 1} {{B_j}} \left( \xi \right){{\left( {f_r^{\left( j \right)}\left( \xi \right)} \right)}^{{n_j}}}} \right|} \right)^{\frac{1}{{{n_k}}}}}{\left( {{\omega _a}\left( \xi \right)} \right)^{{h_1}}}{\left( {1 - \left| \xi \right|} \right)^{k + {h_2}}} + {C_{{t_1}}} \end{array} $

    运用引理1,有

    $ \begin{array}{l} \left| {{f_r}\left( z \right)} \right|\omega \left( z \right) \le {Q_k}\mathop {\sup }\limits_{\left| \xi \right| \le \left| z \right|} \sum\limits_{j = 0}^{k - 1} {{{\left| {{B_j}\left( \xi \right)} \right|}^{\frac{1}{{{n_k}}}}}{{\left| {f_r^{\left( j \right)}\left( \xi \right)} \right|}^{\frac{{{n_j}}}{{{n_k}}}}}} {\left( {{\omega _a}\left( \xi \right)} \right)^{{h_1}}}{\left( {1 - \left| \xi \right|} \right)^{k + {h_2}}} + {C_{{t_1}}} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{Q_k}\mathop {\sup }\limits_{\left| \xi \right| \le \left| z \right|} \left[ {{{\left( {\left| {{B_0}\left( \xi \right)} \right|{{\left( {{\omega _a}\left( \xi \right)} \right)}^{{h_1}\left( {{n_k} - 1} \right)}}{{\left( {1 - \left| \xi \right|} \right)}^{{h_2}\left( {{n_k} - 1} \right) + k{n_k}}}} \right)}^{\frac{1}{{{n_k}}}}}\left\| {{f_r}} \right\|_{H_\omega ^\infty }^{\frac{1}{{{n_k}}}} + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\sum\limits_{j = 1}^{k - 1} {{{\left| {{B_j}\left( \xi \right)} \right|}^{\frac{1}{{{n_k}}}}}} {{\left| {f_\rho ^{\left( j \right)}\left( \xi \right)} \right|}^{\frac{{{n_j}}}{{{n_k}}}}}{{\left( {{\omega _a}\left( \xi \right)} \right)}^{{h_1}}}{{\left( {1 - \left| \xi \right|} \right)}^{{h_2} + k}}} \right] + {c_{{t_1}}} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{Q_k}\left\| {{B_0}} \right\|_{H_{{{\mathit{\dot \Psi }}_{n,k,0}}}^\infty }^{\frac{1}{{{n_k}}}} \cdot \left\| {{f_r}} \right\|_{H_\omega ^\infty }^{\frac{1}{{{n_k}}}} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{Q_k}\mathop {\sup }\limits_{\left| \xi \right| \le \left| z \right|} \sum\limits_{j = 1}^{k - 1} {{{\left| {{B_j}\left( \xi \right)} \right|}^{\frac{1}{{{n_k}}}}}} {\left| {f_r^{\left( j \right)}\left( \xi \right)} \right|^{\frac{{{n_j}}}{{{n_k}}}}}{\left( {{\omega _a}\left( \xi \right)} \right)^{{h_1}}}{\left( {1 - \left| \xi \right|} \right)^{k + {h_2}}} + {C_{{t_1}}} \end{array} $

    再运用引理3,得

    $ \begin{array}{l} \left| {{f_r}\left( z \right)} \right|\omega \left( z \right) \le {Q_k}\left\| {{B_0}} \right\|_{H_{{{\dot \omega }_{h\left( {1,2} \right),n\left( {k,0} \right)}}}^\infty }^{\frac{1}{{{n_k}}}}\left\| {{f_r}} \right\|_{H_\omega ^\infty }^{\frac{1}{{{n_k}}}} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{Q_k}\mathop {\sup }\limits_{\left| \xi \right| \le \left| z \right|} \sum\limits_{j = 1}^{k - 1} {{{\left( {\left| {{B_j}\left( \xi \right)} \right|{{\left( {{\omega _a}\left( \xi \right)} \right)}^{{h_1}\left( {{n_k} - {n_j}} \right)}}{{\left( {1 - \left| \xi \right|} \right)}^{\left( {{h_2} + k} \right){n_k} - \left( {{h_2} + j} \right){n_j}}}} \right)}^{\frac{1}{{{n_k}}}}} \cdot } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left( {j!{{\left( {1 + \varepsilon } \right)}^j}m\mathop {\sup }\limits_{\left| \xi \right| = \left| \rho \right|} \left| {{f_r}\left( \xi \right)} \right|\omega \left( \xi \right) + {C_j}} \right)^{\frac{{{n_j}}}{{{n_k}}}}} + {C_{{t_1}}} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{Q_k}\left[ {\left\| {{B_0}} \right\|_{H_{{{\dot \omega }_{h\left( {1,2} \right),n\left( {k,0} \right)}}}^\infty }^{\frac{1}{{{n_k}}}}\left\| {{f_r}} \right\|_{H_\omega ^\infty }^{\frac{1}{{{n_k}}}} + } \right.\\ \left. {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathop {\sup }\limits_{\left| \xi \right| \le \left| z \right|} \sum\limits_{j = 1}^{k - 1} {\left\| {{B_j}} \right\|_{H_{{{\dot \omega }_{h\left( {1,2} \right),n\left( {k,j} \right)}}}^\infty }^{\frac{1}{{{n_k}}}}j!{{\left( {1 + \varepsilon } \right)}^j}m} \left\| {{f_r}} \right\|_{H_\omega ^\infty }^{\frac{{{n_j}}}{{{n_k}}}} + {C_j}} \right] + {C_{{t_1}}} \end{array} $ (9)

    其中CjCt1∈(0,∞)为不依赖于z的常数(j=1,…,k-1).若‖frHω≤1,则结论显然成立.因此设‖frHω>1,由(9)式得

    $ \left| {{f_r}\left( z \right)} \right|\omega \left( z \right) \le E{\left\| {{f_r}} \right\|_{H_\omega ^\infty }} + {C_{{t_1}}} $

    $ \mathop {\sup }\limits_{r \in \left[ {0,1} \right)} {\left\| {{f_r}} \right\|_{H_\omega ^\infty }} \le \frac{{{C_{{t_1}}}}}{{1 - E}} < \infty $

    由引理4有fHω.

    定理2的证明  设f是方程(2)的解析解,由

    $ \begin{array}{*{20}{c}} {f\left( z \right) = \int_0^z {f'\left( \xi \right){\rm{d}}\xi {{ + f}}\left( 0 \right)} }&{z \in D} \end{array} $

    $ \begin{array}{l} \left| {f\left( z \right)} \right|\omega \left( z \right) \le \int_0^{\left| z \right|} {\frac{{f'\left( \xi \right)|\omega \left( \xi \right)}}{{\omega \left( \xi \right)}}} \left| {{\rm{d}}\xi } \right|\omega \left( z \right) + \left| {f\left( 0 \right)} \right|\omega \left( z \right) \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathop {\sup }\limits_{\left| \xi \right| \le \left| z \right|} \left| {f'\left( \xi \right)} \right|\omega \left( \xi \right)\int_0^{\left| z \right|} {\frac{{{\rm{d}}r}}{{\omega \left( r \right)}}\omega \left( z \right) + \left| {f\left( 0 \right)} \right|\omega \left( z \right)} \;\;\;\;\;\;\;z \in D \end{array} $ (10)

    运用引理2把fk分别替换成f′和k-1,则有

    $ \left| {{{f'}_r}\left( z \right)} \right|\omega \left( z \right) \le {Q_{k - 1}}\mathop {\sup }\limits_{\left| \xi \right| \le \left| z \right|} \left( {\left| {f_r^{\left( k \right)}\left( \xi \right)} \right|{\omega _{k - 1}}\left( \xi \right)} \right) + {C_{{t_2}}} $ (11)

    由引理5,ω满足(3)式和(4)式,结合引理1、引理3、(8),(10)和(11)式,得

    $ \begin{array}{l} \left| {{{f'}_r}\left( z \right)} \right|\omega \left( z \right) \le {Q_{k - 1}}\mathop {\sup }\limits_{\left| \xi \right| \le \left| z \right|} \sum\limits_{j = 0}^{k - 1} {{{\left| {{B_j}\left( \xi \right)} \right|}^{\frac{1}{{{n_k}}}}}{{\left| {f_r^{\left( j \right)}\left( \xi \right)} \right|}^{\frac{{{n_j}}}{{{n_k}}}}}} \omega \left( \xi \right){\left( {1 - \left| \xi \right|} \right)^{k - 1}} + {C_{{t_2}}} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{Q_{k - 1}}\mathop {\sup }\limits_{\left| \xi \right| \le \left| z \right|} \left[ {\left\| {{B_0}\left( \xi \right)} \right\|_{H_{{{\tilde \omega }_{h\left( {1,2} \right),n\left( {k,0} \right)}}}^\infty }^{\frac{1}{{{n_k}}}}\int_0^{\left| z \right|} {\frac{{{\rm{d}}r}}{{\omega \left( r \right)}}\omega \left( z \right)\left\| {{{f'}_r}} \right\|_{H_\omega ^\infty }^{\frac{1}{{{n_k}}}}} } \right. + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left\| {{B_1}\left( \xi \right)} \right\|_{H_{{{\tilde \omega }_{h\left( {1,2} \right),n\left( {k,1} \right)}}}^\infty }^{\frac{1}{{{n_k}}}}\left\| {{{f'}_r}} \right\|_{H_\omega ^\infty }^{\frac{{n1}}{{{nk}}}} + \\ \left. {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{j = 2}^{k - 1} {\left\| {{B_j}\left( \xi \right)} \right\|_{H_{{{\tilde \omega }_{h\left( {1,2} \right),n\left( {k,0} \right)}}}^\infty }^{\frac{1}{{{n_k}}}}\left( {j - 1} \right)!{{\left( {1 + \varepsilon } \right)}^{j - 1}}m\left\| {{{f'}_r}} \right\|_{H_\omega ^\infty }^{\frac{{{n_j}}}{{{n_k}}}}} + {C_j}} \right] + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{C_{{t_2}}} + {C_{{t_{21}}}} \end{array} $ (12)

    其中$ {{C}_{{{t}_{21}}}}=\left\| {{B}_{0}}\left( \xi \right) \right\|_{H_{{{\widetilde{\omega }}_{h\left( 1, 2 \right), n\left( k, 0 \right)}}}^{\infty }}^{\frac{1}{{{n}_{k}}}}{{\left| f\left( 0 \right) \right|}^{\frac{1}{{{n}_{k}}}}}\omega {{\left( \xi \right)}^{\frac{1}{{{n}_{k}}}}}, {{C}_{j}}, {{C}_{{{t}_{2}}}}\in \left( 0, \infty \right) $为不依赖于z的常数,j=0,…,k-1.若‖fr′‖Hω≤1,则结论显然成立.因此设‖fr′‖Hω>1,由(12)式得

    $ \left| {{{f'}_r}\left( z \right)} \right|\omega \left( z \right) \le F{\left\| {f'} \right\|_{H_\omega ^\infty }} + {C_{{t_2}}} + {C_{{t_{21}}}} $

    $ \mathop {\sup }\limits_{r \in \left[ {0,1} \right)} {\left\| {f'} \right\|_{H_\omega ^\infty }} \le \frac{{{C_{{t_2}}} + {C_{{t_{21}}}}}}{{1 - F}} < \infty $

    由引理4有f′∈Hω.

    参考文献
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    The Solutions of Nonlinear Complex Differential Equations and Hω Space
    SUN Yu1, LONG Jian-ren1, QIN Zhi-gao1, HU Guang-ming2     
    1. School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, China;
    2. School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
    Abstract: Based on the straightforward integral estimate, the properties of function spaces of solutions of the nonlinear differential equation $ {\left( {{f^{\left( k \right)}}} \right)^{{n_k}}} + {A_{k - 1}}\left( z \right){\left( {{f^{\left( {k - 1} \right)}}} \right)^{{n_{k - 1}}}} + \cdots + {A_1}\left( z \right){\left( {f'} \right)^{{n_1}}} + {A_0}\left( z \right)f = {A_k}\left( z \right) $ are studied. The sufficient conditions of the coefficients for the derivatives and analytic solutions of the above equation to be in Hω are given in this paper, which improves and extends previous results from Huusko-Korhonen-Reijonen.
    Key words: nonlinear complex differential equation    Hardy space    analytic solution    unit disc    
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