西南师范大学学报(自然科学版)   2019, Vol. 44 Issue (7): 8-16.  DOI: 10.13718/j.cnki.xsxb.2019.07.002
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  • 一类含有未知导函数的三重积分不等式中未知函数的估计    [PDF全文]
    黄星寿 , 王五生 , 罗日才     
    河池学院 数学与统计学院, 广西 宜州 546300
    摘要:研究了一类非线性三重积分不等式,其中被积函数中含有未知函数及其导函数,积分项外包含了非常数项.利用变量替换技巧、放大技巧和反函数技巧等分析手段,给出了三重积分-微分不等式中未知函数的显上界估计,推广了已有结果.最后举例说明所得结果可以用来研究微分-积分方程解的估计.
    关键词非线性积分不等式    含有未知导函数的三重积分    微分-积分方程    显式估计    

    文献[1-2]在研究微分方程的解对参数的连续依赖性时,建立了下面的积分不等式

    $ u(t) \leqslant c+\int_{a}^{t} f(s) u(s) \mathrm{d} s \quad t \in[a, b] $

    其中c≥0是常数,给出了不等式中未知函数的估计

    $ u(t) \leqslant c \exp \left(\int_{a}^{t} f(s) \mathrm{d} s\right) \quad t \in[a, b] $ (1)

    大部分研究者研究积分号内不含未知函数的导函数的积分不等式[3-12].由于积分号内包含未知函数及其导函数的积分不等式在研究微分-积分方程中具有重要作用,文献[13]定理1.7.3,1.8.1,1.8.2中研究了下面的积分号内含有未知函数及其导函数的线性积分不等式

    $ \dot u\left( t \right) \leqslant a\left( t \right) + b\left( t \right)\int_0^t {c\left( s \right)\left( {u\left( s \right) + \dot u\left( s \right)} \right){\text{d}}s} ,t \in {\mathbb{R}_ + } $ (2)
    $ \dot u\left( t \right) \leqslant u\left( 0 \right) + \int_0^t {a\left( s \right)\left( {u\left( s \right) + \dot u\left( s \right)} \right){\text{d}}s} + \int_0^t {a\left( s \right)\left( {\int_0^s {b\left( \sigma \right)\dot u\left( \sigma \right){\text{d}}\sigma } } \right){\text{d}}s} ,t \in {\mathbb{R}_ + } $ (3)
    $ \begin{array}{l} u\left( t \right) \le k\left( t \right) + p\left( t \right)\left\{ {\int_0^t {f\left( s \right)u\left( s \right){\rm{d}}s} + \int_0^t {f\left( s \right)p\left( s \right)\left( {\int_0^s {g\left( \tau \right)u\left( \tau \right){\rm{d}}\tau } } \right){\rm{d}}s} + } \right.\\ \;\;\;\;\;\;\;\;\;\left. {\int_0^t {f\left( s \right)p\left( s \right)\left[ {\int_0^s {g\left( \tau \right)p\left( \tau \right)\left( {\int_0^\tau {h\left( \sigma \right)u\left( \sigma \right){\rm{d}}\sigma } } \right){\rm{d}}\tau } } \right]{\rm{d}}s} } \right\} \end{array} $ (4)

    文献[14]进一步研究了时标上的线性积分不等式

    $ {u^\varDelta }\left( t \right) \leqslant a\left( t \right) + b\left( t \right)\int_{{t_0}}^t {c\left( s \right)\left( {u\left( s \right) + {u^\varDelta }\left( s \right)} \right)\varDelta s} ,t \in {\mathbb{T}_0} $ (5)
    $ u\left( t \right) \leqslant w\left( t \right) + p\left( t \right)\int_{{t_0}}^t {\left\{ {\left[ {a\left( \tau \right) + b\left( \tau \right)} \right]u\left( \tau \right) + b\left( \tau \right)p\left( \tau \right)\int_{{t_0}}^\tau {\left[ {c\left( s \right)u\left( s \right) + d\left( s \right)} \right]\varDelta s} } \right\}\varDelta \tau } ,t \in {\mathbb{T}_0} $ (6)
    $ u\left( t \right) \leqslant {u_0} + \int_{{t_0}}^t {b\left( s \right)\left\{ {u\left( s \right) + \int_{{t_0}}^s {c\left( \tau \right)\left[ {u\left( \tau \right) + \int_{{t_0}}^\tau {q\left( \gamma \right)u\left( \gamma \right)\varDelta \gamma } } \right]\varDelta \tau } } \right\}\varDelta s} ,t \in {\mathbb{T}_0} $ (7)

    文献[15]研究了积分号内含有未知函数及其导函数的非线性积分不等式

    $ \dot u\left( t \right) \leqslant c + \int_0^t {k\left( s \right)\dot u\left( s \right)\left( {{{\dot u}^p}\left( s \right) + {u^2}\left( s \right)} \right){\text{d}}s} ,t \in {\mathbb{R}_ + } $ (8)

    其中:c是正常数,p≥1.

    受文献[13-15]的启发,本文研究了积分号外具有非常数因子,且积分号内含有未知函数及其导函数的非线性三重积分不等式

    $ \begin{array}{l} \dot u\left( t \right) \le q\left( t \right) + p\left( t \right)\left\{ {u\left( t \right) + \int_{{t_0}}^t {f\left( s \right)\left( {u\left( s \right) + \dot u\left( s \right)} \right){\rm{d}}s} } \right. + \\ \;\;\;\;\;\;\;\;\;\int_{{t_0}}^t {f\left( s \right)p\left( s \right)\left( {\int_{{t_0}}^s {g\left( \tau \right)\left( {u\left( \tau \right) + \dot u\left( \tau \right)} \right){\rm{d}}\tau } } \right){\rm{d}}s} + \int_{{t_0}}^t {f\left( s \right)p\left( s \right)} \times \\ \;\;\;\;\;\;\;\;\;\left. {\left[ {\int_{{t_0}}^s {g\left( \tau \right)p\left( \tau \right)\left( {\int_{{t_0}}^\tau {h\left( \sigma \right)\dot u\left( \sigma \right)\left( {{u^2}\left( \sigma \right) + {{\dot u}^\theta }\left( \sigma \right)} \right){\rm{d}}\sigma } } \right){\rm{d}}\sigma } } \right]{\rm{d}}s} \right\},t \in \left[ {{t_0},\infty } \right) \end{array} $ (9)

    不等式(9)把文献[13]中的不等式(3)推广成非线性积分不等式,把文献[15]中的不等式(8)推广成积分号外具有非常数因子的三重积分不等式.本文利用分析技巧给出了不等式(9)中未知函数的估计.最后举例说明了本文研究结果可用来研究相应类型的微分-积分方程解的估计.

    1 主要结果与证明

    为了使结果的证明过程简单明了,先给出下面的引理.

    引理1[16-17]   令y≥0,pq≥0和p≠0,则对任意K>0有关系式

    $ {y^{\frac{q}{p}}} \le \frac{q}{p}{K^{\frac{{q - p}}{p}}}y + \frac{{p - q}}{p}{K^{\frac{q}{p}}} $ (10)

    引理2   假设函数u(t),a(t),b(t),c(t),d(t)都是定义在[t0,∞)上的非负连续函数,且满足不等式

    $ \begin{array}{l} u\left( t \right) \le u\left( 0 \right) + \int_{{t_0}}^t {a\left( s \right){\rm{d}}s} + \int_{{t_0}}^t {b\left( s \right)u\left( s \right){\rm{d}}s} + \int_{{t_0}}^t {c\left( s \right){u^2}\left( s \right){\rm{d}}s} + \\ \;\;\;\;\;\;\;\;\;\int_{{t_0}}^t {d\left( s \right){u^3}\left( s \right){\rm{d}}s} ,t \in \left[ {{t_0},\infty } \right) \end{array} $ (11)

    如果u(0)>0,(exp(-(ln(u(0)+∫t0ta(s)ds)-∫t0t b(s)ds))- ∫t0tc(s)ds)2-∫t0t2d(s)ds>0,则有未知函数u(t)的估计式

    $ \begin{array}{l} u\left( t \right) \le \left\{ {{{\left[ {\exp \left( { - \ln \left( {u\left( 0 \right) + \int_{{t_0}}^t {a\left( s \right){\rm{d}}s} } \right) - \int_{{t_0}}^t {b\left( s \right){\rm{d}}s} } \right) - \int_0^t {c\left( s \right){\rm{d}}s} } \right]}^2} - } \right.\\ \;\;\;\;\;\;\;\;\;\;{\left. {\int_{{t_0}}^t {2{\rm{d}}\left( s \right){\rm{d}}s} } \right\}^{ - \frac{1}{2}}},t \in \left[ {{t_0},\infty } \right) \end{array} $ (12)

      对于任意非负实数T∈[t0,∞),由(11)式可以看出

    $ \begin{array}{l} u\left( t \right) \le u\left( 0 \right) + \int_{{t_0}}^T {a\left( s \right){\rm{d}}s} + \int_{{t_0}}^t {b\left( s \right)u\left( s \right){\rm{d}}s} + \int_{{t_0}}^t {c\left( s \right){u^2}\left( s \right){\rm{d}}s} + \\ \;\;\;\;\;\;\;\;\;\int_{{t_0}}^t {{\rm{d}}\left( s \right){u^3}\left( s \right){\rm{d}}s} ,t \in \left[ {{t_0},T} \right] \end{array} $ (13)

    由(13)式右端定义函数v(t),即

    $ \begin{array}{l} v\left( t \right) = u\left( 0 \right) + \int_{{t_0}}^T {a\left( s \right){\rm{d}}s} + \int_{{t_0}}^t {b\left( s \right)u\left( s \right){\rm{d}}s} + \int_{{t_0}}^t {c\left( s \right){u^2}\left( s \right){\rm{d}}s} + \\ \;\;\;\;\;\;\;\;\;\int_{{t_0}}^t {d\left( s \right){u^3}\left( s \right){\rm{d}}s} ,t \in \left[ {{t_0},T} \right] \end{array} $ (14)

    由定义式(14)可看出

    $ u\left( t \right) \le v\left( t \right),v\left( {{t_0}} \right) = u\left( 0 \right) + \int_{{t_0}}^T {a\left( s \right){\rm{d}}s} ,t \in \left[ {{t_0},T} \right] $ (15)

    再求函数v(t)的导函数,利用(15)式得

    $ \dot v\left( t \right) = b\left( t \right)u\left( t \right) + c\left( t \right){u^2}\left( t \right) + d\left( t \right){u^3}\left( t \right) \le b\left( t \right)v\left( t \right) + c\left( t \right){v^2}\left( t \right) + d\left( t \right){v^3}\left( t \right), \\ t \in \left[ {{t_0},T} \right] $ (16)

    把不等式(16)两边同时除以v(t)得到

    $ \frac{{\dot v\left( t \right)}}{{v\left( t \right)}} \le b\left( t \right) + c\left( t \right)v\left( t \right) + d\left( t \right){v^2}\left( t \right),t \in \left[ {{t_0},T} \right] $ (17)

    先把(17)式中的t替换成s,然后两边关于st0t积分,得到

    $ \begin{array}{l} \ln v\left( t \right) \le \ln v\left( {{t_0}} \right) + \int_{{t_0}}^t {b\left( s \right){\rm{d}}s} + \int_{{t_0}}^t {c\left( s \right)v\left( s \right){\rm{d}}s} + \int_{{t_0}}^t {d\left( s \right){v^2}\left( s \right){\rm{d}}s} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\ln v\left( {{t_0}} \right) + \int_{{t_0}}^T {b\left( s \right){\rm{d}}s} + \int_{{t_0}}^t {c\left( s \right)v\left( s \right){\rm{d}}s} + \int_{{t_0}}^t {d\left( s \right){v^2}\left( s \right){\rm{d}}s} ,t \in \left[ {{t_0},T} \right] \end{array} $ (18)

    将不等式(18)的右端定义为函数w(t),即

    $ w\left( t \right) = \ln v\left( {{t_0}} \right) + \int_{{t_0}}^T {b\left( s \right){\rm{d}}s} + \int_{{t_0}}^t {c\left( s \right)v\left( s \right){\rm{d}}s} + \int_{{t_0}}^t {d\left( s \right){v^2}\left( s \right){\rm{d}}s} ,t \in \left[ {{t_0},T} \right] $ (19)

    由(18)式和(19)式可以看出w(t)是非负连续增函数,且满足

    $ w\left( {{t_0}} \right) = \ln v\left( {{t_0}} \right) + \int_{{t_0}}^T {b\left( s \right){\rm{d}}s} ,v\left( t \right) \le \exp \left( {w\left( t \right)} \right),t \in \left[ {{t_0},T} \right] $ (20)

    求函数w(t)的导函数得到

    $ \dot w\left( t \right) = c\left( t \right)v\left( t \right) + d\left( t \right){v^2}\left( t \right) \le c\left( t \right)\exp \left( {w\left( t \right)} \right) + d\left( t \right)\exp \left( {2w\left( t \right)} \right),t \in \left[ {{t_0},T} \right] $ (21)

    不等式(21)两边同除以-exp(w(t))得到

    $ - \exp \left( { - w\left( t \right)} \right)\dot w\left( t \right) \ge - c\left( t \right) - d\left( t \right)\exp \left( {w\left( t \right)} \right),t \in \left[ {{t_0},T} \right] $ (22)

    然后把不等式(22)中的t改写成s,两边再关于s从0到t积分,得

    $ \begin{array}{l} \exp \left( { - w\left( t \right)} \right) \ge \exp \left( { - w\left( {{t_0}} \right)} \right) - \int_{{t_0}}^t {c\left( s \right){\rm{d}}s} - \int_{{t_0}}^t {d\left( s \right)\exp \left( {w\left( s \right)} \right){\rm{d}}s} \ge \\ \exp \left( { - w\left( {{t_0}} \right)} \right) - \int_{{t_0}}^T {c\left( s \right){\rm{d}}s} - \int_{{t_0}}^t {d\left( s \right)\exp \left( {w\left( s \right)} \right){\rm{d}}s} \;t \in \left[ {{t_0},T} \right] \end{array} $ (23)

    将不等式(23)的右端定义为函数z(t),即

    $ z\left( t \right) = \exp \left( { - w\left( {{t_0}} \right)} \right) - \int_{{t_0}}^T {c\left( s \right){\rm{d}}s} - \int_{{t_0}}^t {d\left( s \right)\exp \left( {w\left( s \right)} \right){\rm{d}}s} \;t \in \left[ {{t_0},T} \right] $ (24)

    由(23)式和(24)式可以看出z(t)是连续减函数,且满足

    $ z\left( {{t_0}} \right) = \exp \left( { - w\left( {{t_0}} \right)} \right) - \int_{{t_0}}^T {c\left( s \right){\rm{d}}s} ,z\left( t \right) \le \exp \left( { - w\left( t \right)} \right),t \in \left[ {{t_0},T} \right] $ (25)

    求函数z(t)的导函数得到

    $ \dot z\left( t \right) = - d\left( t \right)\exp \left( {w\left( t \right)} \right) \ge - \frac{{d\left( t \right)}}{{z\left( t \right)}},t \in \left[ {{t_0},T} \right] $ (26)

    不等式(26)两边同乘z(t)得到

    $ z\left( t \right)\dot z\left( t \right) \ge - d\left( t \right),t \in \left[ {{t_0},T} \right] $ (27)

    然后把不等式(27)中的t改写成s,两边再关于s从0到t积分,得

    $ {z^2}\left( t \right) \ge {z^2}\left( {{t_0}} \right) - \int_{{t_0}}^t {2d\left( s \right){\rm{d}}s} ,t \in \left[ {{t_0},T} \right] $ (28)

    综合(15),(20),(25)和(28)式推出

    $ \begin{array}{l} u\left( t \right) \le v\left( t \right) \le \exp \left( {w\left( t \right)} \right) \le \frac{1}{{z\left( t \right)}} \le {\left( {{z^2}\left( {{t_0}} \right) - \int_{{t_0}}^t {2d\left( s \right){\rm{d}}s} } \right)^{ - \frac{1}{2}}} = \\ {\left[ {{{\left( {\exp \left( { - w\left( {{t_0}} \right)} \right) - \int_0^T {c\left( s \right){\rm{d}}s} } \right)}^2} - \int_{{t_0}}^t {2d\left( s \right){\rm{d}}s} } \right]^{ - \frac{1}{2}}} = \\ {\left\{ {{{\left[ {\exp - \ln v\left( {{t_0}} \right) - \int_0^T {b\left( s \right){\rm{d}}s} - \int_0^T {c\left( s \right){\rm{d}}s} } \right]}^2} - \int_{{t_0}}^t {2d\left( s \right){\rm{d}}s} } \right\}^{ - \frac{1}{2}}} = \\ \left\{ {{{\left[ {\exp - \ln \left( {u\left( 0 \right) + \int_0^T {a\left( s \right){\rm{d}}s} } \right) - \int_0^T {b\left( s \right){\rm{d}}s} - \int_0^T {c\left( s \right){\rm{d}}s} } \right]}^2}} \right. - \\ {\left. {\int_{{t_0}}^t {2d\left( s \right){\rm{d}}s} } \right\}^{ - \frac{1}{2}}},t \in \left[ {0,T} \right] \end{array} $ (29)

    在(29)式中令t=T,得到

    $ \begin{array}{l} u\left( T \right) \le \left\{ {{{\left[ {\exp \left( { - \ln \left( {u\left( 0 \right) + \int_{{t_0}}^T {a\left( s \right){\rm{d}}s} } \right) - \int_{{t_0}}^T {b\left( s \right){\rm{d}}s} } \right) - \int_{{t_0}}^T {c\left( s \right){\rm{d}}s} } \right]}^2} - } \right.\\ \;\;\;\;\;\;\;\;\;\;\;{\left. {\int_{{t_0}}^T {2d\left( s \right){\rm{d}}s} } \right\}^{ - \frac{1}{2}}} \end{array} $ (30)

    因为T是任意的,可以把(30)式写成

    $ \begin{array}{l} u\left( t \right) \le \left\{ {{{\left[ {\exp \left( { - \ln \left( {u\left( 0 \right) + \int_{{t_0}}^t {a\left( s \right){\rm{d}}s} } \right) - \int_{{t_0}}^t {b\left( s \right){\rm{d}}s} } \right) - \int_{{t_0}}^t {c\left( s \right){\rm{d}}s} } \right]}^2} - } \right.\\ {\left. {\int_{{t_0}}^t {2d\left( s \right){\rm{d}}s} } \right\}^{ - \frac{1}{2}}},t \in \left[ {{t_0},\infty } \right) \end{array} $ (31)

    这正是所要证明的估计式(12).

    定理1   假设q(t),p(t),f(t),g(t),h(t)都是定义在[t0,∞)上的非负连续已知函数,$\theta=\frac{\theta_{2}}{\theta_{1}} <1$是正常数,u(t)和$\dot{u}(t)$是定义在[t0,∞)上的满足不等式(9)的非负未知函数,u(t0)>0.对于任意t∈[t0,∞),如果

    $ {\left( {\exp \left( { - \left( {\ln \left( {u\left( {{t_0}} \right) + \int_{{t_0}}^t {A\left( s \right){\rm{d}}s} } \right) - \int_{{t_0}}^t {B\left( s \right){\rm{d}}s} } \right)} \right) - \int_{{t_0}}^t {C\left( s \right){\rm{d}}s} } \right)^2} - \int_{{t_0}}^t {2D\left( s \right){\rm{d}}s} > 0 $ (32)

    则对于任意K>0,有未知函数u(t)的估计式

    $ u(t) \leqslant u\left(t_{0}\right)+\int_{t_{0}}^{t}(q(s)+p(s) Z(s)) \mathrm{d} s, t \in\left[t_{0}, \infty\right) $ (33)

    其中

    $ \begin{aligned} Z(t) :=& u\left(t_{0}\right) \exp \left(\int_{t_{0}}^{t}(p(s)+f(s)) \mathrm{d} s\right)+\int_{t_{0}}^{t}[q(s)(1+f(s))+f(s) p(s) R(s)] \\ & \exp \left(\int_{s}^{t}(p(\tau)+f(\tau)) \mathrm{d} \tau\right) \mathrm{d} s \end{aligned} $ (34)
    $ \begin{array}{l} R\left( t \right): = u\left( {{t_0}} \right) + \int_{{t_0}}^t {\left\{ {q\left( s \right)\left( {1 + f\left( s \right) + g\left( s \right)} \right) + \left[ {p\left( s \right)\left( {1 + f\left( s \right) + g\left( s \right)} \right) + } \right.} \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\left. {\left. {f\left( s \right) + g\left( s \right)} \right]M\left( s \right)} \right\}{\rm{d}}s \end{array} $ (35)
    $ \begin{array}{l} M\left( t \right): = \left( {\left( {\exp \left( { - \left( {\ln \left( {u\left( {{t_0}} \right) + \int_{{t_0}}^t {A\left( s \right){\rm{d}}s} } \right) - \int_{{t_0}}^t {B\left( s \right){\rm{d}}s} } \right)} \right) - } \right.} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;{\left. {{{\left. {\int_{{t_0}}^t {C\left( s \right){\rm{d}}s} } \right)}^2} - \int_{{t_0}}^t {2D\left( s \right){\rm{d}}s} } \right)^{ - \frac{1}{2}}} \end{array} $ (36)
    $ A\left( t \right): = q\left( t \right)\left( {1 + f\left( t \right) + g\left( t \right) + \frac{{{\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1} - {\theta _2}}}{{{\theta _2}}}}}q\left( t \right)h\left( t \right) + \frac{{{\theta _2} - {\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1}}}{{{\theta _2}}}}}h\left( t \right)} \right) $ (37)
    $ \begin{array}{l} B\left( t \right): = p\left( t \right) + f\left( t \right) + g\left( t \right) + f\left( t \right)p\left( t \right) + g\left( t \right)p\left( t \right) + h\left( t \right)q\left( t \right)\left( {\frac{{{\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1} - {\theta _2}}}{{{\theta _2}}}}}} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;h\left( t \right)p\left( t \right)\left( {\frac{{{\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1} - {\theta _2}}}{{{\theta _2}}}}}q\left( t \right) + \frac{{{\theta _2} - {\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1}}}{{{\theta _2}}}}}} \right) \end{array} $ (38)
    $ C\left( t \right): = h\left( t \right)p\left( t \right)\left( {\frac{{{\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1} - {\theta _2}}}{{{\theta _2}}}}}} \right) + h\left( t \right)q\left( t \right) $ (39)
    $ D\left( t \right): = h\left( t \right)p\left( t \right) $ (40)

      由不等式(9)定义函数z(t)

    $ \begin{array}{l} z(t): = u(t) + \int_{{t_0}}^t f (s)(u(s) + \dot u(s)){\rm{d}}s + \\ \;\;\;\;\;\;\;\;\;\int_{{t_0}}^t f (s)p(s)\left( {\int_{{t_0}}^s g (\tau )(u(\tau ) + \dot u(\tau )){\rm{d}}\tau } \right){\rm{d}}s + \int_{{t_0}}^t f (s)p(s) \times \\ \;\;\;\;\;\;\;\;\;\left[ {\int_{{t_0}}^s g (\tau )p(\tau )\left( {\int_{{t_0}}^\tau h (\sigma )\dot u(\sigma )\left( {{u^2}(\sigma ) + {{\dot u}^\theta }(\sigma )} \right){\rm{d}}\sigma } \right){\rm{d}}\tau } \right]{\rm{d}}s,t \in \left[ {{t_0},\infty } \right) \end{array} $ (41)

    由(9)式和(41)式可看出z(t)是非减函数,且有

    $ z\left(t_{0}\right)=u\left(t_{0}\right), u(t) \leqslant z(t), \dot{u}(t) \leqslant q(t)+p(t) z(t), t \in\left[t_{0}, \infty\right) $ (42)

    求(41)式定义的函数z(t)的导函数

    $ \begin{array}{l} \dot z(t) = \dot u(t) + f(t)(u(t) + \dot u(t)) + f(t)p(t)\left[ {\int_{{t_0}}^t g (\tau )(u(\tau ) + \dot u(\tau )){\rm{d}}\tau } \right] + \\ \;\;\;\;\;\;\;\;f(t)p(t)\left[ {\int_{{t_0}}^t g (\tau )p(\tau )\left( {\int_{{t_0}}^\tau h (\sigma )\dot u(\sigma )\left( {{u^2}(\sigma ) + {{\dot u}^\theta }(\sigma )} \right){\rm{d}}\sigma } \right){\rm{d}}\tau } \right],t \in \left[ {{t_0},\infty } \right) \end{array} $ (43)

    把(42)式代入(43)式得到

    $ \begin{array}{l} \dot z(t) \le q(t) + p(t)z(t) + f(t)(z(t) + q(t) + p(t)z(t)) + f(t)p(t)\int_{{t_0}}^t g (\tau )(z(\tau ) + \\ \;\;\;\;\;\;\;\;q(\tau ) + p(\tau )z(\tau ){\rm{d}}\tau + f(t)p(t)\int_{{t_0}}^t g (\tau )p(\tau )\left( {\int_{{t_0}}^\tau h (\sigma )(q(\sigma ) + } \right.\\ \;\;\;\;\;\;\;\;p(\sigma )z(\sigma ))\left( {{z^2}(\sigma ) + {{(q(\sigma ) + p(\sigma )z(\sigma ))}^\theta }} \right){\rm{d}}\sigma ){\rm{d}}\tau \le \\ \;\;\;\;\;\;\;\;q(t)(1 + f(t)) + (p(t) + f(t))z(t) + f(t)p(t)\left\{ {z(t) + \int_{{t_0}}^t g (\tau )(z(\tau ) + } \right.\\ \;\;\;\;\;\;\;\;q(\tau ) + p(\tau )z(\tau ){\rm{d}}\tau + \int_{{t_0}}^t g (\tau )p(\tau )\left[ {\int_{{t_0}}^\tau h (\sigma )(q(\sigma ) + } \right.\\ \;\;\;\;\;\;\;\;p(\sigma )z(\sigma ))\left( {{z^2}(\sigma ) + {{(q(\sigma ) + p(\sigma )z(\sigma ))}^\theta }} \right){\rm{d}}\sigma ]{\rm{d}}\tau \} ,t \in \left[ {{t_0},\infty } \right) \end{array} $ (44)

    再定义函数r(t)

    $ \begin{array}{l} r(t): = z(t) + \int_{{t_0}}^t g (\tau )(z(\tau ) + q(\tau ) + p(\tau )z(\tau )){\rm{d}}\tau + \int_{{t_0}}^t g (\tau )p(\tau )\left( {\int_{{t_0}}^\tau h (\sigma )(q(\sigma ) + } \right.\\ \;\;\;\;\;\;\;\;\;p(\sigma )z(\sigma ))\left( {{z^2}(\sigma ) + {{(q(\sigma ) + p(\sigma )z(\sigma ))}^\theta }} \right){\rm{d}}\sigma ){\rm{d}}\tau ,t \in \left[ {{t_0},\infty } \right) \end{array} $ (45)

    从定义式(45)可以看出r(t)是非减函数,且有

    $ r\left(t_{0}\right)=z\left(t_{0}\right), z(t) \leqslant r(t), t \in\left[t_{0}, \infty\right) $ (46)

    把(45)式和(46)式代入(44)式得

    $ \dot{z}(t) \leqslant q(t)(1+f(t))+[p(t)+f(t)+f(t) p(t)] r(t), t \in\left[t_{0}, \infty\right) $ (47)

    再求函数r(t)的导函数得

    $ \begin{array}{l} \dot r(t) = \dot z(t) + g(t)(z(t) + q(t) + p(t)z(t)) + g(t)p(t)\int_{{t_0}}^t h (\sigma )(q(\sigma ) + \\ \;\;\;\;\;\;\;\;p(\sigma )z(\sigma ))\left( {{z^2}(\sigma ) + {{(q(\sigma ) + p(\sigma )z(\sigma ))}^\theta }} \right){\rm{d}}\sigma ,t \in \left[ {{t_0},\infty } \right) \end{array} $ (48)

    再把(46)式和(47)式代入(48)式得

    $ \begin{array}{l} \dot r(t) \le q(t)(1 + f(t)) + [p(t) + f(t) + f(t)p(t)]r(t) + g(t)(r(t) + q(t) + p(t)r(t)) + \\ \;\;\;\;\;\;\;\;g(t)p(t)\int_{{t_0}}^t h (\sigma )(q(\sigma ) + p(\sigma )r(\sigma ))\left( {{r^2}(\sigma ) + {{(q(\sigma ) + p(\sigma )r(\sigma ))}^\theta }} \right){\rm{d}}\sigma = \\ \;\;\;\;\;\;\;\;q(t)(1 + f(t) + g(t)) + [p(t) + f(t) + g(t) + f\left( t \right)p(t)]r(t) + g(t){\rm{ }}p(t)\{ r(t) + \\ \;\;\;\;\;\;\;\;\int_{{t_0}}^t h (\sigma )(q(\sigma ) + p(\sigma )r(\sigma ))\left( {{r^2}(\sigma ) + {{(q(\sigma ) + p(\sigma )r(\sigma ))}^\theta }} \right){\rm{d}}\sigma \} ,t \in \left[ {{t_0},\infty } \right) \end{array} $ (49)

    为了进一步简化,再定义函数m(t)

    $ m(t)=r(t)+\int_{t_{0}}^{t} h(\sigma)(q(\sigma)+p(\sigma) r(\sigma))\left(r^{2}(\sigma)+(q(\sigma)+p(\sigma) r(\sigma))^{\theta}\right) \mathrm{d} \sigma, t \in\left[t_{0}, \infty\right) $ (50)

    从定义式(50)可以看出m(t)是非减函数,且

    $ m\left(t_{0}\right)=r\left(t_{0}\right), r(t) \leqslant m(t), t \in\left[t_{0}, \infty\right) $ (51)

    求函数m(t)的导函数,利用(49),(50)和(51)式得

    $ \begin{array}{l} \dot m(t) = \dot r(t) + h(t)(q(t) + p(t)r(t))\left( {{r^2}(t) + {{(q(t) + p(t)r(t))}^\theta }} \right) \le \\ \;\;\;\;\;\;\;\;\;\;q(t)(1 + f(t) + g(t)) + [p(t) + f(t) + g(t) + f(t)p(t)]r(t) + g(t)p(t)m(t) + \\ \;\;\;\;\;\;\;\;\;\;\begin{array}{*{20}{l}} {h(t)(q(t) + p(t)r(t))\left( {{r^2}(t) + {{(q(t) + p(t)r(t))}^\theta }} \right) \le }\\ {q(t)(1 + f(t) + g(t)) + [p(t) + f(t) + g(t) + f(t)p(t)]m(t) + g(t)p(t)m(t) + } \end{array}\\ \;\;\;\;\;\;\;\;\;\;h(t)(q(t) + p(t)m(t))\left( {{m^2}(t) + {{(q(t) + p(t)m(t))}^\theta }} \right),t \in \left[ {{t_0},\infty } \right) \end{array} $ (52)

    对于任意K>0,利用引理1可以推出

    $ \begin{array}{l} {(q(t) + p(t)m(t))^\theta } \le \frac{{{\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1} - {\theta _2}}}{{{\theta _2}}}}}(q(t) + p(t)m(t)) + \frac{{{\theta _2} - {\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1}}}{{{\theta _2}}}}} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{{\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1} - {\theta _2}}}{{{\theta _2}}}}}q(t) + \frac{{{\theta _2} - {\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1}}}{{{\theta _2}}}}} + \frac{{{\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1} - {\theta _2}}}{{{\theta _2}}}}}p(t)m(t) \end{array} $ (53)

    把(53)式代入(52)式可得

    $ \begin{array}{l} \dot m(t) \le q(t)\left( {1 + f(t) + g(t) + \frac{{{\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1} - {\theta _2}}}{{{\theta _2}}}}}q(t)h(t) + \frac{{{\theta _2} - {\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1}}}{{{\theta _2}}}}}h(t)} \right) + \\ \;\;\;\;\;\;\;\;\;\left[ {p(t) + f(t) + g(t) + f(t)p(t) + g(t)p(t) + h(t)q(t)\left( {\frac{{{\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1} - {\theta _2}}}{{{\theta _2}}}}}} \right) + } \right.\\ \;\;\;\;\;\;\;\;\;\left. {h(t)p(t)\left( {\frac{{{\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1} - {\theta _2}}}{{{\theta _2}}}}}q(t) + \frac{{{\theta _2} - {\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1}}}{{{\theta _2}}}}}} \right)} \right]m(t) + \\ \;\;\;\;\;\;\;\;\;\left( {h(t)p(t)\left( {\frac{{{\theta _1}}}{{{\theta _2}}}{K^{\frac{{{\theta _1} - {\theta _2}}}{{{\theta _2}}}}}} \right) + h(t)q(t)} \right){m^2}(t) + h(t)p(t){m^3}(t) = \\ \;\;\;\;\;\;\;\;\;A(t) + B(t)m(t) + C(t){m^2}(t) + D(t){m^3}(t),t \in \left[ {{t_0},\infty } \right) \end{array} $ (54)

    其中:A(t),B(t),C(t),D(t)由定理1中的(37),(38),(39)和(40)式定义.先把不等式(54)中的t改写成s,然后对不等式两边关于st0t积分,得到

    $ \begin{gathered} m(t) \leqslant m\left( {{t_0}} \right) + \int_{{t_0}}^t A (s){\text{d}}s + \int_{{t_0}}^t B (s)m(s){\text{d}}s + \int_{{t_0}}^t C (s){m^2}(s){\text{d}}s + \hfill \\ \;\;\;\;\;\;\;\;\;\int_{{t_0}}^t D (s){m^3}(s){\text{d}}s,t \in {\mathbb{R}_ + } \hfill \\ \end{gathered} $ (55)

    利用(42),(46),(51)式将(55)式改写成

    $ \begin{array}{c}{m(t) \leqslant u\left(t_{0}\right)+\int_{t_{0}}^{t} A(s) \mathrm{d} s+\int_{t_{0}}^{t} B(s) m(s) \mathrm{d} s+\int_{t_{0}}^{t} C(s) m^{2}(s) \mathrm{d} s+} \\{\int_{t_{0}}^{t} D(s) m^{3}(s) \mathrm{d} s, t \in \mathbb{R}_{+}}\end{array} $ (56)

    由于(56)式具有引理2中不等式(11)的形式,且相关函数满足引理2中的相应条件,我们利用引理2就可以得到不等式(56)中m的估计

    $ \begin{aligned} m(t) \leqslant &\left(\left(\exp \left(-\left(\ln \left(u\left(t_{0}\right)+\int_{t_{0}}^{t} A(s) \mathrm{d} s\right)-\int_{t_{0}}^{t} B(s) \mathrm{d} s\right)\right)-\right.\right.\\ & \int_{t_{0}}^{t} C(s) \mathrm{d} s )^{2}-\int_{t_{0}}^{t} 2 D(s) \mathrm{d} s )^{-\frac{1}{2}}=M(t), t \in\left[t_{0}, \infty\right) \end{aligned} $ (57)

    其中M(t)由定理1中的(36)式定义.把(51)式和(57)式代入(49)式可得

    $ \begin{aligned} \dot{r}(t) \leqslant & q(t)(1+f(t)+g(t))+[p(t)+f(t)+g(t)+\\ & f(t) p(t)+g(t) p(t) ] M(t), t \in\left[t_{0}, \infty\right) \end{aligned} $ (58)

    由(42),(46)和(58)式得到

    $ \begin{array}{c}{r(t) \leqslant u\left(t_{0}\right)+\int_{t_{0}}^{t}\{q(s)(1+f(s)+g(s))+[p(s)(1+f(s)+g(s))+} \\ {f(s)+g(s) ] M(s) \} \mathrm{d} s=R(t), t \in\left[t_{0}, \infty\right)}\end{array} $ (59)

    其中R(t)由定理1中(35)式定义.把(59)式代入(44)式可得

    $ \dot{z}(t) \leqslant q(t)(1+f(t))+(p(t)+f(t)) z(t)+f(t) p(t) R(t), t \in\left[t_{0}, \infty\right) $ (60)

    $ \dot z(t) - (p(t) + f(t))z(t) \leqslant q(t)(1 + f(t)) + f(t)p(t)R(t),t \in \left[ {{t_0},\infty } \right) $ (61)

    (61) 式两边同乘exp (-∫t0t(p(s)+f(s))ds

    $ \begin{aligned} \frac{\mathrm{d}\left\{z(t) \exp \left(-\int_{t_{0}}^{t}(p(s)+f(s)) \mathrm{d} s\right)\right\}}{\mathrm{d} t} \leqslant &(q(t)(1+f(t))+f(t) p(t) R(t)) \times \\ & \exp \left(-\int_{t_{0}}^{t}(p(s)+f(s)) \mathrm{d} s\right), t \in\left[t_{0}, \infty\right) \end{aligned} $ (62)

    对(62)式两边积分,利用(42)式得到

    $ \begin{aligned} z(t) \leqslant u\left(t_{0}\right) \exp \left(\int_{t_{0}}^{t}(p(s)+f(s)) \mathrm{d} s\right)+\int_{t_{0}}^{t}[q(s)(1+f(s))+f(s) p(s) R(s)] \\ \exp \left(\int_{s}^{t}(p(\tau)+f(\tau)) \mathrm{d} \tau\right) \mathrm{d} s=Z(t), t \in\left[t_{0}, \infty\right) \\ \dot{u}(t) \leqslant q(t)+p(t) Z(t), t \in\left[t_{0}, \infty\right) & \end{aligned} $ (63)

    其中Z(t)由定理(1)中(34)式定义.由(63)式得到定理(1)所要求的u(t)估计式(33).

    2 应用

    本文结果可以用来研究相应类型的微分-积分方程解的性质.考虑微分-积分方程

    $ \dot{x}(t)=q(t)+p(t)\left(x(t)+\int_{t_{0}}^{t} F(s, x(s), \dot{x}(s)) \mathrm{d} s\right), x(0)=c $ (64)

    推论1   假设方程(64)中|c|是正常数,q(t),p(t)和定理1中q(t),p(t)的定义相同. FC($\mathbb{R} \times \mathbb{R} \times \mathbb{R}, \mathbb{R}$)满足下列条件

    $ \begin{array}{l}{|F(t, x, y)| \leqslant f(t)\left\{(|x|+|y|)+p(t) \int_{t_{0}}^{t}|G(s, x, y)| \mathrm{d} s\right\}} \\ {|G(t, x, y)| \leqslant g(t)\left\{(|x|+|y|)+g(t) p(t) \int_{t_{0}}^{t}|H(s, x, y)| \mathrm{d} s\right\}} \\ {|H(t, x, y)| \leqslant h(t)|y|\left(|x|^{2}+|y|^{\theta}\right)}\end{array} $ (65)

    其中:f(t),g(t),h(t)和θ如定理1中的定义.假设|c|,q(t),p(t),f(t),g(t),h(t)和θ满足

    $ \left(\exp \left(-\left(\ln \left(|c|+\int_{t_{0}}^{t} A(s) \mathrm{d} s\right)+\int_{t_{0}}^{t} B(s) \mathrm{d} s\right)\right)-\int_{t_{0}}^{t} C(s) \mathrm{d} s\right)^{2}-\int_{t_{0}}^{t} 2 D(s) \mathrm{d} s>0, \\ t \in\left[t_{0}, \infty\right) $

    如果x(t)是方程(64)的解,则对于任意K>0,方程(64)解的模的估计式

    $ |x(t)| \leqslant|c|+\int_{t_{0}}^{t}(q(s)+p(s) \widetilde{Z}(s)) \mathrm{d} s, t \in\left[t_{0}, \infty\right) $ (66)

    其中

    $ \widetilde{Z}(t) :=|c| \exp \left(\int_{t_{0}}^{t}(p(s)+f(s)) \mathrm{d} s\right)+\int_{t_{0}}^{t}[q(s)(1+f(s))+f(s) p(s) \widetilde{R}(s)]\\ \exp \left(\int_{s}^{t}(p(\tau)+f(\tau)) \mathrm{d} \tau\right) \mathrm{d} s $
    $ \begin{aligned} \widetilde{R}(t) :=&|c|+\int_{t_{0}}^{t}\{q(s)(1+f(s)+g(s))+[p(s)(1+f(s)+g(s))+\\ & f(s)+g(s) ] \widetilde{M}(s) \} \mathrm{d} s \end{aligned} $
    $ \begin{array}{l}{\widetilde{M}(t) :=\left(\left(\exp \left(-\left(\ln \left(|c|+\int_{t_{0}}^{t} A(s) \mathrm{d} s\right)+\int_{t_{0}}^{t} B(s) \mathrm{d} s\right)\right)-\right.\right.} \\ {\int_{t_{0}}^{t} C(s) \mathrm{d} s )^{2}-\int_{t_{0}}^{t} 2 D(s) \mathrm{d} s )^{-\frac{1}{2}}}\end{array} $

    A(t),B(t),C(t),D(t)由定理1中的(37),(38),(39)和(40)式定义.

      利用条件(65),由方程(64)推出

    $ \begin{array}{l} \left| {\dot x\left( t \right)} \right| \le q\left( t \right) + p\left( t \right)\left\{ {\left| {x\left( t \right)} \right| + \int_{{t_0}}^t {f\left( s \right)\left( {\left| {x\left( s \right)} \right| + \left| {\dot x\left( s \right)} \right|} \right){\rm{d}}s} + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\int_{{t_0}}^t {f\left( s \right)p\left( s \right)\left( {\int_{{t_0}}^s {g\left( \tau \right)\left( {\left| {x\left( \tau \right)} \right| + \left| {\dot x\left( \tau \right)} \right|} \right){\rm{d}}\tau } } \right){\rm{d}}s} + \int_{{t_0}}^t {f\left( s \right)p\left( s \right)} \times \\ \;\;\;\;\;\;\;\;\;\;\;\left. {\left[ {\int_{{t_0}}^s {g\left( \tau \right)p\left( \tau \right)\left( {\int_{{t_0}}^\tau {h\left( \sigma \right)\left| {\dot x\left( \sigma \right)} \right|\left( {{x^2}\left( \sigma \right) + \left| {{{\dot x}^\theta }\left( \sigma \right)} \right|} \right){\rm{d}}\sigma } } \right){\rm{d}}\tau } } \right]{\rm{d}}s} \right\},t \in \left[ {{t_0},\infty } \right) \end{array} $ (67)

    由于式(67)具有不等式(9)的形式,且满足定理1中的相应条件,利用定理1就可以得到所求的方程解的模的估计式(66).

    参考文献
    [1]
    GRONWALL T H. Note on the Derivatives with Respect to a Parameter of the Solutions of a System of Differential Equations[J]. The Annals of Mathematics, 1919, 20(4): 292. DOI:10.2307/1967124
    [2]
    BELLMAN R. The Stability of Solutions of Linear Differential Equations[J]. Duke Mathematical Journal, 1943, 10(4): 643-647. DOI:10.1215/S0012-7094-43-01059-2
    [3]
    AGARWAL R P, DENG S F, ZHANG W N. Generalization of a Retarded Gronwall-Like Inequality and Its Applications[J]. Applied Mathematics and Computation, 2005, 165(3): 599-612. DOI:10.1016/j.amc.2004.04.067
    [4]
    MA Q H, PECARIC'J. Some New Explicit Bounds for Weakly Singular Integral Inequalities with Applications to Fractional Differential and Integral Equations[J]. Journal of Mathematical Analysis and Applications, 2008, 341(2): 894-905. DOI:10.1016/j.jmaa.2007.10.036
    [5]
    欧阳云, 王五生. 一类非线性弱奇异三重积分不等式中未知函数的估计及其应用[J]. 西南大学学报(自然科学版), 2017, 39(3): 69-74.
    [6]
    侯宗毅, 王五生. 一类变下限非线性Volterra-Fredholm型积分不等式及其应用[J]. 西南师范大学学报(自然科学版), 2016, 41(2): 21-25.
    [7]
    梁英. 一类时滞弱奇异Wendroff型积分不等式[J]. 四川师范大学学报(自然科学版), 2014, 37(4): 493-496. DOI:10.3969/j.issn.1001-8395.2014.04.009
    [8]
    ABDELDAIM A. Nonlinear Retarded Integral Inequalities of Gronwall-Bellman Type and Applications[J]. Journal of Mathematical Inequalities, 2016(1): 285-299. DOI:10.7153/jmi-10-24
    [9]
    XU R, MENG F W. Some New Weakly Singular Integral Inequalities and Their Applications to Fractional Differential Equations[J]. Journal of Inequalities and Applications, 2016, 2016: 78. DOI:10.1186/s13660-016-1015-2
    [10]
    黄春妙, 王五生. 弱奇异迭代积分不等式中未知函数的估计[J]. 华南师范大学学报(自然科学版), 2017, 49(4): 111-114.
    [11]
    EN MI Y. A Generalized Gronwall-Bellman Type Delay Integral Inequality with Two Independent Variables on Time Scales[J]. Journal of Mathematical Inequalities, 2017(4): 1151-1160. DOI:10.7153/jmi-2017-11-85
    [12]
    ZHOU J, SHEN J, ZHANG W N. A Powered Gronwall-Type Inequality and Applications to Stochastic Differential Equations[J]. Discrete and Continuous Dynamical Systems, 2016, 36(12): 7207-7234. DOI:10.3934/dcdsa
    [13]
    PACHPATTE B G. Inequalities for Differential and Integral Equations[M]. New York: Academic Press, 1998.
    [14]
    AKIN-BOHNER E, BOHNER M, AKIN F. Pachpatte Inequalities on Time Scales[J]. Journal of Inequalities in Pure and Applied Mathematics, 2005, 6(1): 1-23.
    [15]
    KHAN Z A. On some Fundamental Integrodifferential Inequalities[J]. Applied Mathematics, 2014, 5(19): 2968-2973. DOI:10.4236/am.2014.519282
    [16]
    JIANG F C, MENG F W. Explicit Bounds on Some New Nonlinear Integral Inequalities with Delay[J]. Journal of Computational and Applied Mathematics, 2007, 205(1): 479-486. DOI:10.1016/j.cam.2006.05.038
    [17]
    MA Q H, PECARIC'J. Some New Explicit Bounds for Weakly Singular Integral Inequalities with Applications to Fractional Differential and Integral Equations[J]. Journal of Mathematical Analysis and Applications, 2008, 341(2): 894-905. DOI:10.1016/j.jmaa.2007.10.036
    Estimation of Unknown Function of a Class of Triple Integral Inequalities with Unknown Derivative Function
    HUANG Xing-Shou , WANG Wu-Sheng , LUO Ri-Cai     
    School of Mathematics and Statistics, Hechi University, Yizhou Guangxi 546300, China
    Abstract: Gronwall type integral inequality is the important tool in the study of existence, uniqueness, boundedness and other qualitative properties of solutions of differential equations, integral equation and integro-differential equations. In this paper, a class ofnonlineartriple integral inequality is studied, which includes an unknown function and its derivative function in integrand function, and a nonconstant factor outside integral sign. The upper bounds of the unknown function in the integro-differential inequality is estimated explicitly using the techniques of change of variable, the method of amplification, and inverse function technique, which generalized some known results. The derived results can be applied in the study ofthe explicit upper bounds of solutions of a class of integro-differential equations.
    Key words: nonlinear integral inequality    triple integral with unknown derivative function    integro-differential equation    explicit estimation    
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