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

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

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

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

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

1 主要结论

(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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

2 在脉冲微分方程中的应用

 $\frac{{{\rm{d}}\left( {x\left( t \right)} \right)}}{{{\rm{d}}t}} = F\left( {t, x} \right)\;\;\;\;\;t \ne {t_i}, t \in \left[{{t_0}, \infty } \right)$ (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)

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

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

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

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)

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

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

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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