西南大学学报 (自然科学版)  2017, Vol. 39 Issue (8): 89-96.  DOI: 10.13718/j.cnki.xdzk.2017.08.013
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  • 含两个非线性项的Gronwall-Bellman型非连续函数积分不等式的推广    [PDF全文]
    李自尊, 柳长青     
    百色学院 数学与统计学院,广西 百色 533000
    摘要:研究了含有未知函数的两个非线性项的非连续函数积分不等式,利用分析技巧给出了未知函数的上界估计,并利用此结果估计了脉冲微分方程的上界.
    关键词非连续函数积分不等式    未知函数估计    脉冲微分系统    

    积分不等式是研究微分方程和积分方程的重要工具.通过对积分不等式中未知函数的估计,可以研究某些微分方程解的存在性、有界性、唯一性和稳定性等定性性质[1-17].通过对非连续函数积分不等式中未知函数进行估计,可以研究某些脉冲微分方程和解的一些性质.

    文献[3]研究了积分不等式

    $ u\left( t \right) \le \varphi \left( t \right) + \int_{{t_0}}^t {g\left( s \right){u^m}\left( s \right){\rm{d}}s} + \sum\limits_{{t_0} < {t_i} < t} {{\beta _i}{u^m}\left( {{t_i} - 0} \right)} \;\;\;\;\;\;\;\;\;\;\forall t \ge {t_0} $

    文献[7]研究了下面的非连续函数积分不等式

    $ \begin{array}{l} u\left( t \right) \le a\left( t \right) + q\left( t \right)\left[{\int_{{t_0}}^t {f\left( s \right)u\left( {\tau \left( s \right)} \right){\rm{d}}s} + \int_{{t_0}}^t {f\left( s \right)\left( {\int_{{t_0}}^s {g\left( t \right)u\left( {\tau \left( t \right)} \right){\rm{d}}t} } \right){\rm{d}}s} + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\sum\limits_{{t_0} < {t_i} < t} {{\beta _i}{u^m}\left( {{t_i}-0} \right)} } \right]\;\;\;\;\;\;\;\;\;\;\forall t \ge {t_0} \end{array} $

    其中,a(t)>0,q(t)≥1,f(t)≥0,g(t)≥0,βi≥0.

    文献[16]研究了含有时滞的脉冲积分不等式

    $ \begin{array}{l} u\left( t \right) \le a\left( t \right) + \int_{{t_0}}^t {f\left( {t, s} \right)u\left( {\tau \left( s \right)} \right){\rm{d}}s} + \int_{{t_0}}^t {f\left( {t, s} \right)\left( {\int_{{t_0}}^s {g\left( {s, \theta } \right)u\left( {\tau \left( \theta \right)} \right){\rm{d}}\theta } } \right){\rm{d}}s} + \\ \;\;\;\;\;\;\;\;\;\left. {q\left( t \right)\sum\limits_{{t_0} < {t_i} < t} {{\beta _i}{u^m}\left( {{t_i} -0} \right)} } \right]\;\;\;\;\;\;\;\;\;\;\forall t \ge {t_0} \end{array} $

    文献[12]研究了含有未知函数的复合函数的积分不等式

    $ \begin{array}{*{20}{c}} {u\left( t \right) \le a\left( t \right) + \int_{{t_0}}^t {f\left( {t, s} \right)} \int_{{t_0}}^s {g\left( {s, \tau } \right)w\left( {u\left( \tau \right)} \right){\rm{d}}\tau {\rm{d}}s} + }\\ {q\left( t \right)\sum\limits_{{t_0} < {t_i} < t} {{\beta _i}{u^m}\left( {{t_i} - 0} \right)} \;\;\;\;\;\;\;\;\;\;\forall t \ge {t_0}} \end{array} $

    这里w(u)是定义在[0,∞)上的单调不减连续函数且当u>0时,w(u)>0.本文在上述研究成果的基础上,研究了一类含三项未知函数复合的非连续函数积分不等式

    $ \begin{array}{l} \phi \left( {u\left( t \right)} \right) \le a\left( t \right) + \int_{{t_0}}^t {{f_1}\left( {t, s} \right){w_1}\left( {u\left( \tau \right)} \right){\rm{d}}s} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\int_{{t_0}}^t {f\left( {t, s} \right)} \left( {\int_{{t_0}}^s {g\left( {s, \tau } \right){w_2}\left( {u\left( \tau \right)} \right){\rm{d}}\tau } } \right){\rm{d}}s + \sum\limits_{{t_0} < {t_i} < t} {{\beta _i}\phi \left( {u\left( {{t_i}- 0} \right)} \right)} \end{array} $ (1)

    其中,u(t)定义在是[t0,∞)上的只有第一类不连续点$\left\{ {{t}_{i}}:{{t}_{0}} < {{t}_{1}} < {{t}_{2}}\cdots, \mathop {\lim }\limits_{i \to \infty } {\mkern 1mu} {t_i}=\infty \right\} $的非负逐段连续函数,ϕ(u)是定义在[0,∞)上的正的严格单调递增函数,m>1,βi≥0,mβi是给定的常数.

    1 主要结论

    假设

    (H1) ϕ在[0,∞)是严格增的连续函数,对任意的u>0,ψ(u)>0;

    (H2) wi (i=1,2) 在[0,∞)上是连续不减函数,在(0,∞)上是正的,且$\frac{{{w}_{2}}}{{{w}_{1}}}$是不减的;

    (H3) a(t)是定义在[t0,∞)上的连续函数,a(t0)≠0;

    (H4) fi(ts) (i=1,2) 和f(ts),g(st)是定义在[t0,∞)×[t0,∞)上的非负连续函数;

    (H5) βi≥0是常数.

    定理1  具有第一类不连续点$\left\{ {{t}_{i}}:{{t}_{0}} < {{t}_{1}} < {{t}_{2}}\cdots, \mathop {\lim }\limits_{i \to \infty } {\mkern 1mu} {t_i}=\infty \right\}$的非负逐段连续函数u(t) (tt0≥0) 满足积分不等式(1),则函数u(t)有下面的估计式:

    $ u\left( t \right) \le {\phi ^{ - 1}}\left( {W_2^{ - 1}\left( {{e_3}\left( t \right)} \right)} \right)\;\;\;\;\;\;\;\;\forall t \in \left[{{t_i}, {t_{i + 1}}} \right) $

    其中

    $ {W_i}\left( u \right) = \int_1^u {\frac{{{\rm{d}}s}}{{{w_i}\left( {{\phi ^{- 1}}\left( s \right)} \right)}}} \;\;\;\;\;\;\;\;i = 1, 2 $
    $ {{\tilde f}_i}\left( {t, s} \right) = \mathop {\max }\limits_{{t_0} \le \tau \le t} {f_i}\left( {\tau, s} \right)\;\;\;\;\;\;\;i = 1, 2 $
    $ {e_1}\left( t \right) = \mathop {\max }\limits_{{t_0} \le \tau \le t} \left| {a\left( \tau \right)} \right| $
    $ {E_i}\left( t \right) = {e_1}\left( t \right) + \sum\limits_{k = 0}^{i- 1} {\sum\limits_{j = 1}^2 {\int_{{t_k}}^{{t_{k + 1}}} {{{\tilde f}_j}\left( {t, s} \right){w_j}\left( {u\left( s \right)} \right){\rm{d}}s} } } + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\begin{array}{*{20}{c}} {{\beta _k}\left( {\phi \left( {u\left( {{t_k}- 0} \right)} \right)} \right)}&{\forall t \in \left[{{t_i}, {t_{i + 1}}} \right)}&{i = 1, 2, \cdots } \end{array} $
    $ {e_2} = {W_1}\left( {{E_1}\left( t \right)} \right) + \int_{{t_0}}^t {{{\tilde f}_1}\left( {t, s} \right){\rm{d}}s} \;\;\;\;\;\;\;\;\;\forall t \in \left[{{t_0}, {t_1}} \right) $
    $ {e_3} = {W_2}\left( {W_1^{- 1}\left( {{e_2}\left( t \right)} \right)} \right) + \int_{{t_0}}^t {{{\tilde f}_2}\left( {t, s} \right){\rm{d}}s} \;\;\;\;\;\;\;\;\;\forall t \in \left[{{t_0}, {t_1}} \right) $
    $ {e_2} = {W_1}\left( {{E_i}\left( t \right)} \right) + \int_{{t_i}}^t {{{\tilde f}_1}\left( {t, s} \right){\rm{d}}s} \;\;\;\;\;\;\;\;\;\forall t \in \left[{{t_i}, {t_{i + 1}}} \right)\;\;\;\;\;\;\;\;\;i = 1, 2, \cdots $
    $ {e_3} = {W_2}\left( {W_1^{- 1}\left( {{e_2}\left( t \right)} \right)} \right) + \int_{{t_i}}^t {{{\tilde f}_2}\left( {t, s} \right){\rm{d}}s} \;\;\;\;\;\;\;\;\;\forall t \in \left[{{t_i}, {t_{i + 1}}} \right)\;\;\;\;\;\;\;\;\;i = 1, 2, \cdots $ (2)

      令

    $ {{\tilde f}_2}\left( {t, s} \right) = \int_s^t {\mathop {\max }\limits_{{t_0} \le \tau \le t} f\left( {t, \tau } \right)g\left( {\tau, s} \right){\rm{d}}\tau } $ (3)

    由于f(ts),g(ts),w(u(t))都是连续函数,得

    $ \begin{array}{l} \int_{{t_0}}^t {f\left( {t, s} \right)} \int_{{t_0}}^s {g\left( {s, \tau } \right)w\left( {u\left( \tau \right)} \right){\rm{d}}\tau {\rm{d}}s} = \\ \int_{{t_0}}^t {w\left( {u\left( \tau \right)} \right)} \int_\tau ^t {f\left( {t, s} \right)g\left( {s, \tau } \right){\rm{d}}s\;{\rm{d}}\tau } = \\ \int_{{t_0}}^t {w\left( {u\left( s \right)} \right)} \int_\tau ^t {f\left( {t, \tau } \right)g\left( {\tau, s} \right){\rm{d}}\tau {\rm{d}}s} \le \\ \int_{{t_0}}^t {{{\tilde f}_2}\left( {t, s} \right)w\left( {u\left( s \right)} \right){\rm{d}}s} \end{array} $ (4)

    由(2),(4),则(1) 式变为

    $ \phi \left( {u\left( t \right)} \right) \le {e_1}\left( t \right) + \sum\limits_{i = 1}^2 {\int_{{t_0}}^t {{{\tilde f}_i}\left( {t, s} \right){w_i}\left( {u\left( s \right)} \right){\rm{d}}s} } + \sum\limits_{{t_0} < {t_i} < t} {{\beta _i}\phi \left( {u\left( {{t_i}- 0} \right)} \right)} $ (5)

    首先,我们考虑情况t∈[t0t1),任取T∈[t0t1),对任意t∈[t0T],由(5) 式,可得

    $ \phi \left( {u\left( t \right)} \right) \le {e_1}\left( t \right) + \sum\limits_{i = 1}^2 {\int_{{t_0}}^t {{{\tilde f}_i}\left( {T, s} \right){w_i}\left( {u\left( s \right)} \right){\rm{d}}s} } $ (6)

    $ v\left( t \right)={{e}_{1}}\left( t \right)+\sum\limits_{i=1}^{2}{\int_{{{t}_{0}}}^{t}{{{{\tilde{f}}}_{i}}\left( T,s \right){{w}_{i}}\left( u\left( s \right) \right)\text{d}s}} $ (7)

    v(x)为非负不减的连续函数,且

    $ \begin{array}{*{20}{c}} {\phi \left( {u\left( t \right)} \right) \le v\left( t \right)}&{u\left( t \right) \le {\phi ^{ - 1}}\left( {v\left( t \right)} \right)}&{v\left( {{t_0}} \right) = {e_1}\left( {{t_0}} \right)} \end{array} $ (8)

    对式(7) 求导,得

    $ v'\left( t \right) = {{e'}_1}\left( t \right) + \sum\limits_{i = 1}^2 {{{\tilde f}_i}\left( {T, t} \right){w_i}\left( {u\left( t \right)} \right)} $ (9)

    $ {\psi _i}\left( t \right) = \frac{{{w_i}\left( t \right)}}{{{w_1}\left( t \right)}}\;\;\;\;\;i = 1, 2 $ (10)

    由(9) 和(10) 式可得

    $ \begin{array}{l} \frac{{v'\left( t \right)}}{{{w_1}\left( {{\phi ^{ - 1}}\left( {v\left( t \right)} \right)} \right)}} = \frac{{{{e'}_1}\left( t \right) + \sum\limits_{i = 1}^2 {{{\tilde f}_i}\left( {T, t} \right){w_i}\left( {u\left( t \right)} \right)} }}{{{w_1}\left( {{\phi ^{ - 1}}\left( {v\left( t \right)} \right)} \right)}} \le \\ \frac{{{{e'}_1}\left( t \right) + \sum\limits_{i = 1}^2 {{{\tilde f}_i}\left( {T, t} \right){w_i}\left( {{\phi ^{ - 1}}\left( {v\left( t \right)} \right)} \right)} }}{{{w_1}\left( {{\phi ^{ - 1}}\left( {v\left( t \right)} \right)} \right)}} = \\ \frac{{{{e'}_1}\left( t \right)}}{{{w_1}\left( {{\phi ^{ - 1}}\left( {v\left( t \right)} \right)} \right)}} + {f_1}\left( {T, t} \right) + \frac{{{{\tilde f}_2}\left( {t, t} \right){w_2}\left( {{\phi ^{ - 1}}\left( {v\left( t \right)} \right)} \right)}}{{{w_1}\left( {{\phi ^{ - 1}}\left( {v\left( t \right)} \right)} \right)}} = \\ \frac{{{{e'}_1}\left( t \right)}}{{{w_1}\left( {{\phi ^{ - 1}}\left( {v\left( t \right)} \right)} \right)}} + {f_1}\left( {T, t} \right) + {{\tilde f}_2}\left( {T, t} \right){\psi _2}\left( {{\phi ^{ - 1}}\left( {v\left( t \right)} \right)} \right) \end{array} $ (11)

    对(10) 式两边从t0t同时积分,并利用Wi(t)的定义,我们得到

    $ \begin{array}{l} {W_1}\left( {v\left( t \right)} \right) - {W_1}\left( {v\left( {{t_0}} \right)} \right) \le {W_1}\left( {{e_1}\left( t \right)} \right) - {W_1}\left( {{e_1}\left( {{t_0}} \right)} \right) + \int_{{t_0}}^t {{{\tilde f}_1}\left( {T, s} \right){\rm{d}}s} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\int_{{t_0}}^t {{{\tilde f}_2}\left( {T, s} \right){\psi _2}\left( {{\phi ^{ - 1}}\left( {v\left( s \right)} \right)} \right){\rm{d}}s} \end{array} $ (12)

    由于W1(v(t0))=W1(e1(t0)),则(11) 式可写为

    $ {W_1}\left( {v\left( t \right)} \right) \le {W_1}\left( {{e_1}\left( t \right)} \right) + \int_{{t_0}}^t {{{\tilde f}_1}\left( {T, s} \right){\rm{d}}s} + \int_{{t_0}}^t {{{\tilde f}_2}\left( {T, s} \right){\psi _2}\left( {{\phi ^{ - 1}}\left( {v\left( s \right)} \right)} \right){\rm{d}}s} $ (13)

    $ {\theta _1}\left( t \right) = {W_1}\left( {v\left( t \right)} \right) $ (14)
    $ {e_2}\left( t \right) = {W_1}\left( {{e_1}\left( t \right)} \right) + \int_{{t_0}}^t {{{\tilde f}_1}\left( {T, s} \right){\rm{d}}s} $ (15)

    由(14) 和(15),则(13) 式变为

    $ \begin{array}{l} {\theta _1}\left( t \right) \le {e_2}\left( t \right) + \int_{{t_0}}^t {{{\tilde f}_2}\left( {T, s} \right){\psi _2}\left( {{\phi ^{ - 1}}\left( {v\left( s \right)} \right)} \right){\rm{d}}s} \le \\ \;\;\;\;\;\;\;\;\;\;\;{e_2}\left( t \right) + \int_{{t_0}}^t {{{\tilde f}_2}\left( {T, s} \right){\psi _2}\left( {{\phi ^{ - 1}}\left( {W_1^{ - 1}\left( {{\theta _1}\left( s \right)} \right)} \right)} \right){\rm{d}}s} \end{array} $ (16)

    $ {v_1}\left( t \right) = {e_2}\left( t \right) + \int_{{t_0}}^t {{{\tilde f}_2}\left( {T, s} \right){\psi _2}\left( {{\phi ^{ - 1}}\left( {W_1^{ - 1}\left( {{\theta _1}\left( s \right)} \right)} \right)} \right)} $

    v1(t)在[t0t1)是连续不减的函数,且

    $ {\theta _1}\left( t \right) \le {v_1}\left( t \right)\;\;\;\;\;\;\;{v_1}\left( {{t_0}} \right) = {e_2}\left( {{t_0}} \right) $

    定义函数

    $ {\mathit{\Phi }_2}\left( u \right) = \int_0^u {\frac{{{\rm{d}}s}}{{\left( {{\phi ^{- 1}}\left( {W_1^{- 1}\left( s \right)} \right)} \right)}}} $ (17)

    $ \begin{array}{l} \frac{{{{v'}_1}\left( t \right)}}{{{\psi _2}\left( {{\phi ^{- 1}}\left( {W_1^{ - 1}\left( {{v_1}\left( t \right)} \right)} \right)} \right)}} = \frac{{{{e'}_2}\left( t \right) + {{\tilde f}_2}\left( {T, t} \right){\psi _2}\left( {{\phi ^{ - 1}}\left( {W_1^{ - 1}\left( {{\theta _1}\left( t \right)} \right)} \right)} \right)}}{{{\psi _2}\left( {{\phi ^{ - 1}}\left( {W_1^{ - 1}\left( {{v_1}\left( t \right)} \right)} \right)} \right)}} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{{{e'}_2}\left( t \right) + {{\tilde f}_2}\left( {T, t} \right){\psi _2}\left( {{\phi ^{ - 1}}\left( {W_1^{ - 1}\left( {{v_1}\left( t \right)} \right)} \right)} \right)}}{{{\psi _2}\left( {{\phi ^{ - 1}}\left( {W_1^{ - 1}\left( {{v_1}\left( t \right)} \right)} \right)} \right)}} \end{array} $ (18)

    对(18) 式的两边,从t0t积分,我们得到

    $ \begin{array}{l} \int_{{t_0}}^t {\frac{{{{v'}_1}\left( s \right){\rm{d}}s}}{{{\psi _2}\left( {{\phi ^{ - 1}}\left( {W_1^{ - 1}\left( {{v_1}\left( t \right)} \right)} \right)} \right)}}} \le \\ \int_{{t_0}}^t {\frac{{{{e'}_2}\left( s \right) + {{\tilde f}_2}\left( {T, s} \right){\psi _2}\left( {{\phi ^{ - 1}}\left( {W_1^{ - 1}\left( {{v_1}\left( s \right)} \right)} \right)} \right)}}{{{\psi _2}\left( {{\phi ^{ - 1}}\left( {W_1^{ - 1}\left( {{v_1}\left( s \right)} \right)} \right)} \right)}}{\rm{d}}s} \le \\ \int_{{t_0}}^t {\frac{{{{e'}_2}\left( s \right){\rm{d}}s}}{{{\psi _2}\left( {{\phi ^{ - 1}}\left( {W_1^{ - 1}\left( {{v_1}\left( s \right)} \right)} \right)} \right)}}} + \int_{{t_0}}^t {{{\tilde f}_2}\left( {T, s} \right){\rm{d}}s} \end{array} $ (19)

    由(10),(17),(19) 式可得

    $ \begin{array}{l} {\mathit{\Phi }_2}\left( {{v_1}\left( t \right)} \right) - {\mathit{\Phi }_2}\left( {{v_1}\left( {{t_0}} \right)} \right) \le \int_{{t_0}}^t {\frac{{{{e'}_2}\left( s \right){\rm{d}}s}}{{{\psi _2}\left( {{\phi ^{ - 1}}\left( {W_1^{ - 1}\left( {{e_2}\left( s \right)} \right)} \right)} \right)}}} + \int_{{t_0}}^t {{{\tilde f}_2}\left( {T, s} \right){\rm{d}}s} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\mathit{\Phi }_2}\left( {{e_2}\left( t \right)} \right) - {\mathit{\Phi }_2}\left( {{e_2}\left( {{t_0}} \right)} \right) + \int_{{t_0}}^t {{{\tilde f}_2}\left( {T, s} \right){\rm{d}}s} \end{array} $ (20)

    由(20) 式可得

    $ {\mathit{\Phi }_2}\left( {{v_1}\left( t \right)} \right) \le {\mathit{\Phi }_2}\left( {{e_2}\left( t \right)} \right) + \int_{{t_0}}^t {{{\tilde f}_2}\left( {T, s} \right){\rm{d}}s} $ (21)

    由(17) 式,我们可以推出

    $ \begin{array}{l} {\mathit{\Phi }_2}\left( u \right) = \int_0^u {\frac{{{\rm{d}}s}}{{{\psi _2}\left( {{\phi ^{ - 1}}\left( {W_1^{ - 1}\left( s \right)} \right)} \right)}}} = \\ \int_0^u {\frac{{{w_1}\left( {{\phi ^{ - 1}}\left( {W_1^{ - 1}\left( s \right)} \right)} \right){\rm{d}}s}}{{{w_2}\left( {{\phi ^{ - 1}}\left( {W_1^{ - 1}\left( s \right)} \right)} \right)}}} = \\ \int_1^{W_1^{ - 1}\left( u \right)} {\frac{{{\rm{d}}s}}{{{w_2}\left( {{\phi ^{ - 1}}\left( {\left( s \right)} \right)} \right)}}} = \\ {W_2}\left( {W_1^{ - 1}\left( u \right)} \right) \end{array} $ (22)

    由(22),(21) 式可变为

    $ {W_2}\left( {W_1^{ - 1}\left( {{v_1}\left( t \right)} \right)} \right) \le {W_2}\left( {W_1^{ - 1}\left( {{e_2}\left( t \right)} \right)} \right) + \int_{{t_0}}^t {{{\tilde f}_2}\left( {T, s} \right){\rm{d}}s} $ (23)

    由(23) 式可推出

    $ {v_1}\left( t \right) \le {W_1}\left( {W_2^{ - 1}\left( {{W_2}\left( {W_1^{ - 1}\left( {{e_2}\left( t \right)} \right)} \right) + \int_{{t_0}}^t {{{\tilde f}_2}\left( {T, s} \right){\rm{d}}s} } \right)} \right) $ (24)

    由(8),(14) 和(24) 式可得

    $ \begin{array}{l} u\left( t \right) \le {\phi ^{ - 1}}\left( {W_2^{ - 1}\left( {{W_2}\left( {W_1^{ - 1}\left( {{e_2}\left( t \right)} \right)} \right) + \int_{{t_0}}^t {{{\tilde f}_2}\left( {T, s} \right){\rm{d}}s} } \right)} \right) \le \\ \;\;\;\;\;\;\;\;\;{\phi ^{ - 1}}\left( {W_2^{ - 1}\left( {{e_3}\left( t \right)} \right)} \right)\;\;\;\;\;\;\;t \in \left[{{t_0}, T} \right] \end{array} $

    其中

    $ {e_3}\left( t \right) = {W_2}\left( {W_1^{ - 1}\left( {{e_2}\left( t \right)} \right)} \right) + \int_{{t_0}}^t {{{\tilde f}_2}\left( {T, s} \right){\rm{d}}s} \;\;\;\;\;\;\;\;t \in \left[{{t_0}, T} \right) $

    T的任意性可得

    $ \begin{array}{l} u\left( t \right) \le {\phi ^{- 1}}\left( {W_2^{- 1}\left( {{W_2}\left( {W_1^{- 1}\left( {{e_2}\left( t \right)} \right)} \right) + \int_{{t_0}}^t {{{\tilde f}_2}\left( {t, s} \right){\rm{d}}s} } \right)} \right) \le \\ \;\;\;\;\;\;\;\;\;{\phi ^{ - 1}}\left( {W_2^{ - 1}\left( {{e_3}\left( t \right)} \right)} \right)\;\;\;\;\;\;\;t \in \left[{{t_0}, {t_1}} \right) \end{array} $

    t∈[t0t1)时我们证明了估计式.

    t∈[t1t2)时,任意确定T1∈[t1t2),对于任意的t∈[t1T1],不等式(4) 变为

    $ \begin{array}{l} \phi \left( {u\left( t \right)} \right) \le {e_1}\left( t \right) + \sum\limits_{i = 1}^2 {\int_{{t_0}}^t {{{\tilde f}_i}\left( {{T_1}, s} \right){w_i}\left( {u\left( s \right)} \right){\rm{d}}s + \beta_1 \phi \left( {u\left( {{t_1} - 0} \right)} \right)} } \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{e_1}\left( t \right) + \sum\limits_{i = 1}^2 {\int_{{t_0}}^{{t_1}} {{{\tilde f}_i}\left( {{T_1}, s} \right){w_i}\left( {u\left( s \right)} \right){\rm{d}}s} } + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{i = 1}^2 {\int_{{t_1}}^t {{{\tilde f}_i}\left( {{T_1}, s} \right){w_i}\left( {u\left( s \right)} \right){\rm{d}}s + {\beta _1}\phi \left( {u\left( {{t_1} - 0} \right)} \right)} } \end{array} $ (25)

    Γ(t)表示(25) 式的右边,

    $ {E_1}\left( t \right) = {e_1}\left( t \right) + \sum\limits_{i = 1}^2 {\int_{{t_0}}^{{t_1}} {{{\tilde f}_i}\left( {t, s} \right){w_i}\left( {u\left( s \right)} \right){\rm{d}}s + {\beta _1}\phi \left( {u\left( {{t_1} - 0} \right)} \right)} } $

    Γ(t)是单调不减函数,且有

    $ \phi \left( {u\left( t \right)} \right) \le \mathit{\Gamma }\left( t \right) $
    $ \begin{array}{l} \phi \left( {u\left( {{t_1}} \right)} \right) \le \mathit{\Gamma }\left( {{t_1}} \right) = {E_1}\left( {{t_1}} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{e_1}\left( {{t_1}} \right) + \sum\limits_{i = 1}^2 {\int_{{t_0}}^{{t_1}} {{{\tilde f}_i}\left( {{T_1}, s} \right){w_i}\left( {u\left( s \right)} \right){\rm{d}}s + {\beta _1}\phi \left( {u\left( {{t_1} - 0} \right)} \right)} } \end{array} $ (26)

    Γ(t)的两边关于t求导得

    $ \begin{array}{l} \mathit{\Gamma '}\left( t \right) \le {{E'}_1}\left( t \right) + \sum\limits_{i = 1}^2 {{{\tilde f}_i}\left( {{T_1}, t} \right){w_i}\left( {u\left( t \right)} \right)} \le \\ \;\;\;\;\;\;\;\;\;\;\;{{E'}_1}\left( t \right) + \sum\limits_{i = 1}^2 {{{\tilde f}_i}\left( {{T_1}, t} \right){w_i}\left( {{\phi ^{ - 1}}\left( {\mathit{\Gamma }\left( t \right)} \right)} \right)} \end{array} $ (27)

    使(27) 式两边同时除以w1(ϕ-1(Γ(t))),可得

    $ \frac{{\mathit{\Gamma '}\left( t \right)}}{{{w_1}\left( {{\phi ^{ - 1}}\left( {\mathit{\Gamma }\left( t \right)} \right)} \right)}} \le \frac{{{{E'}_1}\left( t \right) + \sum\limits_{i = 1}^2 {{{\tilde f}_i}\left( {{T_1}, t} \right){w_i}\left( {{\phi ^{ - 1}}\left( {\mathit{\Gamma }\left( t \right)} \right)} \right)} }}{{{w_1}\left( {{\phi ^{ - 1}}\left( {\mathit{\Gamma }\left( t \right)} \right)} \right)}} $ (28)

    又对(28) 式两边从t1t积分可得

    $ \begin{array}{l} {W_1}\left( {\mathit{\Gamma }\left( t \right)} \right) - {W_1}\left( {\mathit{\Gamma }\left( {{t_1}} \right)} \right) \le \int_{{t_1}}^t {\frac{{{{E'}_1}\left( s \right) + \sum\limits_{i = 1}^2 {{{\tilde f}_i}\left( {{T_1}, s} \right){w_i}\left( {{\phi ^{ - 1}}\left( {\mathit{\Gamma }\left( s \right)} \right)} \right)} }}{{{w_1}\left( {{\phi ^{ - 1}}\left( {\mathit{\Gamma }\left( s \right)} \right)} \right)}}{\rm{d}}s} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{W_1}\left( {{\mathit{E}_1}\left( t \right)} \right) - {W_1}\left( {{\mathit{E}_1}\left( {{t_1}} \right)} \right) + \int_{{t_1}}^t {{{\tilde f}_1}\left( {{T_1}, s} \right){\rm{d}}s} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\int_{{t_1}}^t {{{\tilde f}_2}\left( {T_1, s} \right){\psi _2}\left( {{\phi ^{ - 1}}\left( {\mathit{\Gamma }\left( s \right)} \right)} \right){\rm{d}}s} \end{array} $

    $ {W_1}\left( {\mathit{\Gamma }\left( t \right)} \right) \le {W_1}\left( {{\mathit{E}_1}\left( t \right)} \right) + \int_{{t_1}}^t {{{\tilde f}_1}\left( {{T_1}, s} \right){\rm{d}}s} + \int_{{t_1}}^t {{{\tilde f}_2}\left( {{T_1}, s} \right){\psi _2}\left( {{\phi ^{ - 1}}\left( {\mathit{\Gamma }\left( s \right)} \right)} \right){\rm{d}}s} $ (29)

    从而(28) 式变为了(11) 式的形式,利用相同的方法可以得到估计式

    $ u\left( t \right) \le {\phi ^{ - 1}}\left( {W_2^{ - 1}\left( {{e_3}\left( t \right)} \right)} \right)\;\;\;\;\;\;\;\;\forall t \in \left[{{t_1}, t} \right) $

    同理,对任意自然数k,当t∈[tktk+1)时,我们可以得到未知函数的估计式

    $ u\left( t \right) \le {\phi ^{- 1}}\left( {W_2^{- 1}\left( {{e_3}\left( t \right)} \right)} \right)\;\;\;\;\;\;\;\;\forall t \in \left[{{t_k}, {t_{k + 1}}} \right) $

    综上定理被证明.

    2 在脉冲微分方程中的应用

    本节我们用得到的结果给出脉冲微分系统解的上界估计.考虑脉冲微分系统

    $ \frac{{{\rm{d}}\left( {x\left( t \right)} \right)}}{{{\rm{d}}t}} = F\left( {t, x} \right)\;\;\;\;\;t \ne {t_i}, t \in \left[{{t_0}, \infty } \right) $ (30)
    $ \begin{array}{l} \Delta \left( x \right)\left| {_{t = {t_i}} = {\beta _i}x\left( {{t_i}- 0} \right)} \right.\\ x\left( {{t_0}} \right) = c \end{array} $ (31)

    其中:$0\le {{t}_{0}} < {{t}_{1}} < {{t}_{2}} < \cdots, \mathop {\lim }\limits_{i \to \infty } {\mkern 1mu} {t_i}=\infty, c>1 $是常数,F(tx)关于tx在[t0,∞)×s(-∞,+∞)上连续.假设(30) 式中F(tx)满足

    $ \left| {F\left( {t, x} \right)} \right| \le {f_1}\left( t \right)\left| {{x^{\frac{1}{2}}}} \right| + {f_2}\left( x \right){e^{\left| x \right|}} $ (32)

    其中f1(t),f2(t)是[t0,∞)上连续的非负函数.

    推论1  在条件(32) 式成立的情况下,系统(30),(31) 式所有的解x(t)满足估计式:

    $ u\left( t \right) \le W_2^{- 1}\left( {{e_3}\left( t \right)} \right)\;\;\;\;\;\;\forall t \in \left[{{t_i}, {t_{i + 1}}} \right) $ (33)

    其中

    $ {W_1}\left( u \right) = \int_0^u {\frac{{{\rm{d}}s}}{{{s^{\frac{1}{2}}}}} = 2{u^{\frac{1}{2}}}} \;\;\;\;\;\;W_1^{- 1}\left( u \right) = \frac{{{u^2}}}{4} $
    $ {W_2}\left( u \right) = \int_0^u {\frac{{{\rm{d}}s}}{{{e^s}}} = 1- {e^{- u}}} \;\;\;\;\;\;\;W_2^{ - 1}\left( u \right) = - \ln \left( {1 - u} \right) $
    $ {{\tilde f}_1}\left( {t, s} \right) = {f_1}\left( s \right)\;\;\;\;\;\;{{\tilde f}_2}\left( {t, s} \right) = {f_2}\left( s \right) $
    $ {e_1}\left( t \right) = c $
    $ \begin{array}{l} {E_i}\left( t \right) = c + \sum\limits_{k = 0}^i {\sum\limits_{j = 1}^2 {\int_{{{t}_{k-1}}} {{f_j}\left( s \right){w_j}\left( {u\left( s \right)} \right){\rm{d}}s} } } + \\ \;\;\;\;\;\;\;\;\;\;\;{\beta _k}\left( {\phi \left( {u\left( {{t_i} - 1} \right)} \right)} \right)\;\;\;\;\;\;\;\forall t \in \left[{{t_i}, {t_{i + 1}}} \right)\;\;\;\;\;\;i = 1, 2, \cdots \end{array} $
    $ {e_2} = {W_1}\left( {{e_1}\left( t \right)} \right) + \int_{{t_0}}^t {{f_1}\left( s \right){\rm{d}}s} \;\;\;\;\;\;\;\;\;\forall t \in \left[{{t_0}, {t_1}} \right) $
    $ {e_3} = {W_2}\left( {W_1^{- 1}\left( {{e_2}\left( t \right)} \right)} \right) + \int_{{t_0}}^t {{f_2}\left( s \right){\rm{d}}s} \;\;\;\;\;\;\;\;\;\forall t \in \left[{{t_0}, {t_1}} \right) $
    $ {e_2} = {W_1}\left( {{E_i}\left( t \right)} \right) + \int_{{t_i}}^t {{f_1}\left( s \right){\rm{d}}s} \;\;\;\;\;\;\;\;\forall t \in \left[{{t_i}, {t_{i + 1}}} \right)\;\;\;\;\;\;\;i = 1, 2, \cdots $
    $ {e_3} = {W_2}\left( {W_1^{- 1}\left( {{e_2}\left( t \right)} \right)} \right) + \int_{{t_i}}^t {{f_2}\left( s \right){\rm{d}}s} \;\;\;\;\;\;\;\;\;\forall t \in \left[{{t_i}, {t_{i + 1}}} \right) $

      脉冲微分方程(30) 与(31) 式等价于积分方程

    $ x\left( t \right) = c + \int_{{t_0}}^t {F\left( {s, x\left( s \right)} \right){\rm{d}}s} + \sum\limits_{{t_0} < {t_i} < t} {{\beta _i}x\left( {{t_i}- 0} \right)}, t \in \left[{{t_0}, \infty } \right) $ (34)

    利用条件(32),由(34) 式,可得

    $ \left| {x\left( t \right)} \right| \le c + \int_{{t_0}}^t {{f_1}\left( s \right)\left| {{x^{\frac{1}{2}}}\left( s \right)} \right|{\rm{d}}s} + \int_{{t_0}}^t {{f_2}\left( s \right){e^{\left| {x\left( s \right)} \right|}}{\rm{d}}s} \sum\limits_{{t_0} < {t_i} < t} {{\beta _i}\left| {x\left( {{t_i}- 0} \right)} \right|} $ (35)

    u(t)=|x(t)|,由(35) 式,我们可得不等式

    $ u\left( t \right) \le c + \int_{{t_0}}^t {{f_1}\left( s \right){u^{\frac{1}{2}}}\left( s \right){\rm{d}}s} + \int_{{t_0}}^t {{f_2}\left( s \right){e^{u\left( s \right)}}} + \sum\limits_{{t_0} < {t_i} < t} {{\beta _i}u\left( {{t_i}- 0} \right)} $ (36)

    $ {{w}_{1}}\left( u \right)={{u}^{\frac{1}{2}}}\ \ \ \ \ {{w}_{2}}\left( u \right)={{e}^{u}} $

    我们看出(36) 式是(5) 式的特殊形式.且(36) 式中的函数满足定理1的条件,由定理1,我们可以推出x(t)的估计式(33) 式.

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    Generalization of a Class of Integral Inequalities with Gronwall-Bellman Type for Discontinuous Functions
    LI Zi-zun, LIU Chang-qing     
    School of Mathematics, Baise University, Baise Guangxi 533000, China
    Abstract: In this paper, we give the upper bound estimation of an unknown function containing three nonlinear terms of integral inequality for discontinuous functions. The result is used to estimate the upper bounds of impulsive differential equations.
    Key words: integral inequality for discontinuous functions    estimation of unknown functions    impulsive differential system    
    X