著名的 Gronwall-Bellman不等式^{[1, 2]}可以表述如下

$$u(t)\leq c+\int^t_a f(s)u(s){\rm d}s,~~~~~t\in[a,b],$$

其中 $c\ge 0$ 为常数,$f$ 是非负连续函数, $u$ 是未知函数. 该不等式广泛应用于微分方程和差分方程解的存在性、唯一性、有界性、振动性、稳定性 及不变流型等性质的研究. Gronwall-Bellman型不等式的推广形式很多, 参见文献 ^{[2, 3, 4, 5, 6, 7, 8]}及其中参考文献.

2008年马庆华 和 Pečarić^[4]讨论了下面的 Volterra-Fredholm 型积分不等式

$\begin{align} & u(t)\le k+\int_{\alpha ({{t}_{0}})}^{\alpha (t)}{{{h}_{1}}}(s)[{{f}_{1}}(s)w(u(s))+\int_{\alpha ({{t}_{0}})}^{s}{{{h}_{2}}}(\tau )w(u(\tau ))\text{d}\tau]\text{d}s \\ & \ \ \ \ \ \ \ \ \ +\int_{\alpha ({{t}_{0}})}^{\alpha (T)}{{{h}_{1}}}(s)[{{f}_{1}}(s)w(u(s))+\int_{\alpha ({{t}_{0}})}^{s}{{{h}_{2}}}(\tau )w(u(\tau ))\text{d}\tau]\text{d}s,~\forall t\in I,\\ \end{align}$

(1.1)

$\begin{align} & u(t)\le k+\int_{\alpha ({{t}_{0}})}^{\alpha (t)}{{{h}_{11}}}(s)[{{f}_{11}}(s)w(u(s))+\int_{\alpha ({{t}_{0}})}^{s}{{{h}_{12}}}(\tau )w(u(\tau ))\text{d}\tau]\text{d}s \\ & +\int_{\alpha ({{t}_{0}})}^{\alpha (T)}{{{h}_{12}}}(s)[{{f}_{21}}(s)w(u(s))+\int_{\alpha ({{t}_{0}})}^{s}{{{h}_{22}}}(\tau )w(u(\tau ))\text{d}\tau]\text{d}s,~\forall t\in I. \\ \end{align}$

(1.2)

2011年 Abdeldaim 等^[6]研究了如下的多重积分不等式

$u(t)\le k+\int_{\alpha ({{t}_{0}})}^{t}{f}(s)u(s)[u(s)+\int_{\alpha ({{t}_{0}})}^{s}{h}(\tau )[u(\tau )+\int_{\alpha ({{t}_{0}})}^{\tau }{g}(\xi )u(\xi )\text{d}\xi]\text{d}\tau]\text{d}s.$

(1.3)

另一方面,随着积分不等式理论及差分方程理论的发展,许多学者更关注 Gronwall-Bellman 型不等式的离散形式,见参考文献[9, 10, 11, 12, 13, 14, 15].

Pachpatte 在文献[11] 中研究了线性迭代不等式

$\begin{equation} u(n)\le u_0 +\sum^{n-1}_{s=n_0}f(s)[u(s)+h(s)]+\sum^{n-1}_{s=n_0}f(s) \bigg(\sum^{s-1}_{\tau=n_0}g(\tau)u(\tau)\bigg),~\forall n\in N_0. \end{equation}$

(1.4)

作者受文献[4, 6, 11]的启发, 研究了一种新的具有多重和的非线性 Volterra-Fredholm 型差分不等式

$\begin{align} & \varphi (u(n))\le a(n)+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{11}}}(n,{{t}_{1}})[{{f}_{11}}(n,{{t}_{1}}){{\phi }_{1}}(u({{t}_{1}}))+\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{12}}}({{t}_{1}},{{t}_{2}})[{{f}_{12}}({{t}_{1}},{{t}_{2}}){{\phi }_{2}}(u({{t}_{2}})) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{3}}={{n}_{0}}}^{{{t}_{2}}-1}{{{h}_{13}}}({{t}_{2}},{{t}_{3}})[{{f}_{13}}({{t}_{2}},{{t}_{3}}){{\phi }_{3}}(u({{t}_{3}}))+\cdots +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{1k-1}}}({{t}_{k-2}},{{t}_{k-1}}) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \times [{{f}_{1k-1}}({{t}_{k-2}},{{t}_{k-1}}){{\phi }_{k-1}}(u({{t}_{k-1}}))+\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{1k}}}({{t}_{k-1}},{{t}_{k}}){{\phi }_{k}}(u({{t}_{k}}))]\cdots]]] \\ & \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{1}}={{n}_{0}}}^{N-1}{{{h}_{21}}}(n,{{t}_{1}})[{{f}_{21}}(n,{{t}_{1}}){{\phi }_{1}}(u({{t}_{1}}))+\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{22}}}({{t}_{1}},{{t}_{2}})[{{f}_{22}}({{t}_{1}},{{t}_{2}}){{\phi }_{2}}(u({{t}_{2}})) \\ & \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{3}}={{n}_{0}}}^{{{t}_{2}}-1}{{{h}_{23}}}({{t}_{2}},{{t}_{3}})[{{f}_{23}}({{t}_{2}},{{t}_{3}}){{\phi }_{3}}(u({{t}_{3}}))+\cdots +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{2k-1}}}({{t}_{k-2}},{{t}_{k-1}}) \\ & \ \ \ \ \ \ \ \ \ \ \ \times [{{f}_{2k-1}}({{t}_{k-2}},{{t}_{k-1}}){{\phi }_{k-1}}(u({{t}_{k-1}}))+\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{2k}}}({{t}_{k-1}},{{t}_{k}}){{\phi }_{k}}(u({{t}_{k}}))]\cdots]]],\\ \end{align}$

(1.5)

其中 $N$ 是正的自然数.

文中,令 ${\mathbb N}_0:=\{0,1,2,\cdots\}$,${\mathbb N}:=\{1,2,\cdots\}$, ${\mathbb N}_{n_0}^{l}=\{n_0,n_0+1,n_0+2,\cdots,l\}$ $(n_0\in{\mathbb N}_0,$ $l\in{\mathbb N})$. 函数 $u(n)$的差分定义为 $\Delta u=u(n+1)-u(n)$. 显然,具有初始条件$u(n_0)=0$的线性差分方程 $\bigtriangleup~ u(n)=b(n)$ 有 $u(n)=\sum\limits_{s=n_0}^{n-1} b(s)$. 为方便起见,我们补充定义 $\sum\limits_{s=n_0}^{n_0-1} b(s)=0$.

为了简化证明过程的叙述,我们给出下面的记号.

我们用式(1.5)中的函数$\phi_i(u)$ 和 $\varphi(u)$定义一个函数列 $\{w_i(u)\}$,递归定义如下

$\begin{equation} \left\{\begin{array}{l} w_1(u):=\max\limits_{\tau\in[0,u]}\{\phi_1(\varphi^{-1}(\tau))\}, \\ w_{i+1}(u):= \max\limits_{\tau\in[0,u]}\{ {\phi_{i+1}(\varphi^{-1}(\tau))}/{w_i(\varphi^{-1}(\tau))} \}\; w_i(u), ~~~~ i=1,\cdots,k. \end{array}\right. \label{www} \end{equation}$

(2.1)

显然,函数列 $\{w_i(u)\}$ 由非负递减函数组成,且满足 $w_i(u)\ge \phi_i(\varphi^{-1}(u)),i=1,\cdots,k$. 此外函数列 $\{w_i(u)\}$还满足比值 ${w_{i+1}(u)}/{w_i(u)}$,$i=1,\cdots,k-1$也都是不减的函数. 文献 ^[17]中把这种函数单调性的比较记作

$\begin{equation} w_i\propto w_{i+1},~~~~~i=1,2,\cdots,k-1. \end{equation}$

(2.2)

对于给定的常数 $u_i>0$,我们用(2.1)式中的函数$w_i$定义函数$W_i$

$\left\{ \begin{array}{*{35}{l}} {{W}_{1}}(u,{{u}_{1}}):=\int_{{{u}_{1}}}^{u}{\frac{\text{d}s}{{{w}_{1}}(s)}},\\ {{W}_{i}}(u,{{u}_{i}}):=\int_{{{u}_{i}}}^{u}{\frac{{{w}_{i-1}}(W_{1}^{-1}(\cdots W_{i-1}^{-1}(s)\cdots ))\text{d}s}{{{w}_{i}}(W_{1}^{-1}(\cdots W_{i-1}^{-1}(s)\cdots ))}},~~i=2,\cdots ,k. \\ \end{array} \right.$

(2.3)

显然,这些函数都是严格的递增函数. 在不至于产生歧义的情况下, 用简单的记号$W_i(u)$ 表示$W_i(u,u_i)$. $W_i^{-1}$ 表示$W_i(u)$的逆函数. 根据文献[3]中注释2,在函数 $W_i$中选择不同的$u_i$ 不会影响我们的结果. 利用(1.5)式中的函数$h_{ij},f_{i,j}$,我们定义函数 $h_j,f_j$ 如下

$\begin{equation} \left\{\begin{array}{ll} h_j(t_{j-1},t_j):=\max \Big\{\max\limits_{s\in {\mathbb N}_{n_0}^{t_{j-1}}}\{ h_{1j}(s,t_j)\}, \max\limits_{s\in {\mathbb N}_{n_0}^{t_{j-1}}}\{h_{2j}(s,t_j)\}\Big\},&j=1,2,\cdots,k,\\ f_j(t_{j-1},t_j):=\max \Big\{\max\limits_{s\in {\mathbb N}_{n_0}^{t_{j-1}}}\{h_{1j}(s,t_j)\}, \max\limits_{s\in {\mathbb N}_{n_0}^{t_{j-1}}}\{h_{2j}(s,t_j)\}\Big\},&j=1,2,\cdots,k-1, \end{array}\right. \label{Volterra-Fredholhjfjdef} \end{equation}$

(2.4)

其中 $t_0=n$. 显然,$h_1,h_2,\cdots,h_k,f_1,f_2,\cdots,f_{k-1}$ 关于第一个变量都是单调不减的. 利用函数 $h_j,f_j$, 我们定义新的函数 $\{H_i(n)\}~(i=1,2,\cdots,k)$

$\left\{ \begin{array}{*{35}{l}} {{H}_{1}}(n):=\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}(n,{{t}_{1}}){{f}_{1}}(n,{{t}_{1}}),\\ {{H}_{2}}(n):=\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}(n,{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}}){{f}_{2}}({{t}_{1}},{{t}_{2}})],\\ \qquad \qquad \cdots \\ {{H}_{k-1}}(n):=\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}(n,{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}}) \\ \qquad \qquad ~~\times {{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}})]\cdots]],\\ {{H}_{k}}(n):=\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}(n,{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}}) \\ \qquad \qquad [\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})]]\cdots]]. \\ \end{array} \right.$

(2.5)

定义函数

$\begin{align} & G(\xi )={{W}_{k}}\{{{W}_{k-1}}\{\cdots \{{{W}_{2}}\{{{W}_{1}}(2\xi -a(N))\}\}\cdots \}\}-{{W}_{k}}\{{{W}_{k-1}}\{\cdots \{{{W}_{2}}\{{{W}_{1}}(\xi ) \\ & \ \ \ \ \ \ \ \ \ \ \ +{{H}_{1}}(N)\}+{{H}_{2}}(N)\}\cdots \}+{{H}_{k-1}}(N)\}-{{H}_{k}}(N),\forall \xi >a(N). \\ \end{align}$

(2.6)

定理2.1   假设 $\varphi$ 是严格单调递增函数,$h_k(n,m),f_i(n,m), h_i(n,m)\in C({\mathbb N}_{n_0}^{N}\times {\mathbb N}_{n_0}^{N},{\bf R_+}),$ $(i=1,\cdots,k-1)$,$a(n)\in C({\mathbb N}_{n_0}^{N} ,{\bf R_+})$ 是非减的, 所有的 $\phi_i$ 都是连续函数,且对 $u>0$,有$\phi_i(u)>0$ $(i=1,\cdots,n)$, $W_i$ 由$(2.3)$式定义,且满足 $W_i(+\infty)=+\infty,i=1,2,\cdots,n$. 假设$(2.6)$ 式中定义的函数 $G(\xi)$ 是单调递增的且 $G(\xi)=0$ 有解 $\xi=c>a(N)$. 则 不等式 $(1.5)$ 中的未知函数$u(n)$有估计式

$\begin{align} & u(n)\le W_{1}^{-1}\{W_{2}^{-1}\{\cdots \{W_{k}^{-1}\{{{W}_{k}}\{{{W}_{k-1}}\{\cdots \{{{W}_{2}}\{{{W}_{1}}(c(n))+{{H}_{1}}(n)\} \\ & \ \ \ \ \ \ \ +{{H}_{2}}(n)\}\cdots \}+{{H}_{k-1}}(n)\}+{{H}_{k}}(n)\}\}\cdots \}\},~~\forall n\in \mathbb{N}_{{{n}_{0}}}^{N},\\ \end{align}$

(2.7)

其中 $W_i^{-1}(i=1,2,\cdots,n)$ 是 $W_i$的逆函数.

证    利用(2.1) 和 (2.4)式,由 (1.5)式得到

$\begin{align} & u(n)\le a(N)+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[{{f}_{1}}({{N}_{1}},{{t}_{1}}){{w}_{1}}(u({{t}_{1}}))+\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[{{f}_{2}}({{t}_{1}},{{t}_{2}}){{w}_{2}}(u({{t}_{2}})) \\ & \ \ \ \ \ \ +\cdots +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}}){{w}_{k-1}}(u({{t}_{k-1}})) \\ & \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}}){{w}_{k}}(u({{t}_{k}}))]\cdots]]+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{N-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[{{f}_{1}}({{N}_{1}},{{t}_{1}}){{w}_{1}}(u({{t}_{1}})) \\ & \ \ \ \ \ \ \ \ \ \ \ \\ & \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[{{f}_{2}}({{t}_{1}},{{t}_{2}}){{w}_{2}}(u({{t}_{2}}))+\cdots +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}}) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \times [{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}}){{w}_{k-1}}(u({{t}_{k-1}}))+\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}}){{w}_{k}}(u({{t}_{k}}))]\cdots]],\\ \end{align}$

(2.8)

对任意 $n\in{\mathbb N}_{n_0}^{N_1}$成立,其中$N_1\le N$ ,$N_1$ 是任意的.

用 $z_1(n)$ 表示不等式(2.8)中右端,显然它是 ${\mathbb N}_{n_0}^{N_1}$上非负不减的函数. (2.8) 式等价于

$u(n)\le {{z}_{1}}(n),~~\forall n\in \mathbb{N}_{{{n}_{0}}}^{{{N}_{1}}},$

(2.9)

$\begin{align} & {{z}_{1}}({{n}_{0}})=a(N)+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{N-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[{{f}_{1}}({{N}_{1}},{{t}_{1}}){{w}_{1}}(u({{t}_{1}})) \\ & \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[{{f}_{2}}({{t}_{1}},{{t}_{2}}){{w}_{2}}(u({{t}_{2}}))+\cdots \\ & \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}}){{w}_{k-1}}(u({{t}_{k-1}})) \\ & \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}}){{w}_{k}}(u({{t}_{k}}))]\cdots]]. \\ \end{align}$

(2.10)

利用差分公式及 (2.9)式有

$\begin{align} & \Delta {{z}_{1}}(n)={{h}_{1}}({{N}_{1}},n)[{{f}_{1}}({{N}_{1}},n){{w}_{1}}(u(n))+\sum\limits_{{{t}_{2}}={{n}_{0}}}^{n-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[{{f}_{2}}({{t}_{1}},{{t}_{2}}){{w}_{2}}(u({{t}_{2}}))+\cdots \\ & \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}}){{w}_{k-1}}(u({{t}_{k-1}})) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}}){{w}_{k}}(u({{t}_{k}}))]\cdots]] \\ & \ \ \ \ \ \ \ \ \ \ \le {{h}_{1}}({{N}_{1}},n)[{{f}_{1}}({{N}_{1}},n){{w}_{1}}({{z}_{1}}(n))+\sum\limits_{{{t}_{2}}={{n}_{0}}}^{n-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[{{f}_{2}}({{t}_{1}},{{t}_{2}}){{w}_{2}}({{z}_{1}}({{t}_{2}}))+\cdots \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}}){{w}_{k-1}}({{z}_{1}}({{t}_{k-1}})) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}}){{w}_{k}}({{z}_{1}}({{t}_{k}}))]\cdots]],~\forall n\in \mathbb{N}_{{{n}_{0}}}^{{{N}_{1}}}. \\ \end{align}$

(2.11)

因为$w_1$ 及 $z_1$的单调性. 由(2.11)式推出

$\begin{align} & \frac{\Delta {{z}_{1}}(n)}{{{w}_{1}}({{z}_{1}}(n))}\le {{h}_{1}}({{N}_{1}},n)[{{f}_{1}}({{N}_{1}},n)+\sum\limits_{{{t}_{2}}={{n}_{0}}}^{n-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[{{f}_{2}}({{t}_{1}},{{t}_{2}})\frac{{{w}_{2}}({{z}_{1}}({{t}_{2}}))}{{{w}_{1}}({{z}_{1}}({{t}_{2}}))}+\cdots \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}})\frac{{{w}_{k-1}}({{z}_{1}}({{t}_{k-1}}))}{{{w}_{1}}({{z}_{1}}({{t}_{k-1}}))} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}({{z}_{1}}({{t}_{k}}))}{{{w}_{1}}({{z}_{1}}({{t}_{k}}))}]\cdots]],~\forall n\in \mathbb{N}_{{{n}_{0}}}^{{{N}_{1}}}. \\ \end{align}$

(2.12)

另一方面,由积分中值定理,对任意整数

${{W}_{1}}({{z}_{1}}(n+1))-{{W}_{1}}({{z}_{1}}(n))=\int_{{{z}_{1}}(n)}^{{{z}_{1}}(n+1)}{\frac{\text{d}s}{{{w}_{1}}(s)}}=\frac{\Delta {{z}_{1}}(n)}{{{w}_{1}}(v)}\le \frac{\Delta {{z}_{1}}(n)}{{{w}_{1}}({{z}_{1}}(n))},$

(2.13)

对所有的 $\forall n\in{\mathbb N}_{n_0}^{N_1}$成立,其中 $W_1$由式 (2.3)定义. 由(2.12)式和 (2.13)式,推出

$\begin{align} & {{W}_{1}}({{z}_{1}}(n+1))\le {{W}_{1}}({{z}_{1}}(n))+{{h}_{1}}({{N}_{1}},n)[{{f}_{1}}({{N}_{1}},n)+\sum\limits_{{{t}_{2}}={{n}_{0}}}^{n-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[{{f}_{2}}({{t}_{1}},{{t}_{2}})\frac{{{w}_{2}}({{z}_{1}}({{t}_{2}}))}{{{w}_{1}}({{z}_{1}}({{t}_{2}}))} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\cdots +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}})\frac{{{w}_{k-1}}({{z}_{1}}({{t}_{k-1}}))}{{{w}_{1}}({{z}_{1}}({{t}_{k-1}}))} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}({{z}_{1}}({{t}_{k}}))}{{{w}_{1}}({{z}_{1}}({{t}_{k}}))}]\cdots]],~\forall n\in \mathbb{N}_{{{n}_{0}}}^{{{N}_{1}}}. \\ \end{align}$

(2.14)

在 (2.14)式中令$n=s$ ,再分别令 $s=n_0,n_1,n_2,\cdots,n-1$,然后把所得式子相加,我们有

$\begin{align} & {{W}_{1}}({{z}_{1}}(n))\le {{W}_{1}}({{z}_{1}}({{n}_{0}}))+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}}){{f}_{1}}({{N}_{1}},{{t}_{1}})+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}}) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times [{{f}_{2}}({{t}_{1}},{{t}_{2}})\frac{{{w}_{2}}({{z}_{1}}({{t}_{2}}))}{{{w}_{1}}({{z}_{1}}({{t}_{2}}))}+\cdots +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}}) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \frac{{{w}_{k-1}}({{z}_{1}}({{t}_{k-1}}))}{{{w}_{1}}({{z}_{1}}({{t}_{k-1}}))}+\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}({{z}_{1}}({{t}_{k}}))}{{{w}_{1}}({{z}_{1}}({{t}_{k}}))}]\cdots]] \\ & \ \ \ \ \ \ \ \ \ \ \ \le {{W}_{1}}({{z}_{1}}({{n}_{0}}))+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{{{N}_{1}}-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}}){{f}_{1}}({{N}_{1}},{{t}_{1}})+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}}) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times [{{f}_{2}}({{t}_{1}},{{t}_{2}})\frac{{{w}_{2}}({{z}_{1}}({{t}_{2}}))}{{{w}_{1}}({{z}_{1}}({{t}_{2}}))}+\cdots +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}}) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \frac{{{w}_{k-1}}({{z}_{1}}({{t}_{k-1}}))}{{{w}_{1}}({{z}_{1}}({{t}_{k-1}}))}+\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}({{z}_{1}}({{t}_{k}}))}{{{w}_{1}}({{z}_{1}}({{t}_{k}}))}]\cdots]],~\forall n\in \mathbb{N}_{{{n}_{0}}}^{{{N}_{1}}}. \\ \end{align}$

(2.15)

令 $z_2(n)$表示不等式(2.15)的右端,则它在 ${\mathbb N}_{n_0}^{N_1}$上是非负不减的函数. (2.15) 式等价于

$\begin{equation}\label{Volterra-Fredholm2z2} z_1(n)\le W_1^{-1}(z_2(n)),~~\forall n\in{\mathbb N}_{n_0}^{N_1}, \end{equation}$

(2.16)

$\begin{equation} \label{Volterra-Fredholm2z20} z_2(n_0)= W_1(z_1(n_0)) + \sum^{N_1-1}_{t_{1}=n_0}h_1(N_1,t_1)f_1(N_1,t_1). \end{equation}$

(2.17)

由 $z_2$的定义,利用差分公式及 (2.16) 式 我们有

$\begin{align} & \Delta {{z}_{2}}(n)={{h}_{1}}({{N}_{1}},n)[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{n-1}{{{h}_{2}}}(n,{{t}_{2}})[{{f}_{2}}(n,{{t}_{2}})\frac{{{w}_{2}}({{z}_{1}}({{t}_{2}}))}{{{w}_{1}}({{z}_{1}}({{t}_{2}}))}+\cdots \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}})\frac{{{w}_{k-1}}({{z}_{1}}({{t}_{k-1}}))}{{{w}_{1}}({{z}_{1}}({{t}_{k-1}}))} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}({{z}_{1}}({{t}_{k}}))}{{{w}_{1}}({{z}_{1}}({{t}_{k}}))}]\cdots]] \\ & \ \ \ \ \ \ \ \ \ \ \ \le {{h}_{1}}({{N}_{1}},n)[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{n-1}{{{h}_{2}}}(n,{{t}_{2}})[{{f}_{2}}(n,{{t}_{2}})\frac{{{w}_{2}}(W_{1}^{-1}({{z}_{2}}({{t}_{2}})))}{{{w}_{1}}(W_{1}^{-1}({{z}_{2}}({{t}_{2}})))}+\cdots \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}})\frac{{{w}_{k-1}}(W_{1}^{-1}({{z}_{2}}({{t}_{k-1}})))}{{{w}_{1}}(W_{1}^{-1}({{z}_{2}}({{t}_{k-1}})))} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}(W_{1}^{-1}({{z}_{2}}({{t}_{k}})))}{{{w}_{1}}(W_{1}^{-1}({{z}_{2}}({{t}_{k}})))}]\cdots]],~\forall n\in \mathbb{N}_{{{n}_{0}}}^{{{N}_{1}}}. \\ \end{align}$

(2.18)

利用 函数$w_i/w_1$ $(i=2,\cdots,k)$,$W_1^{-1}$和 $z_2$ 的单调性, 由(2.18)式可以推出

$\begin{align} & \frac{\Delta {{z}_{2}}(n){{w}_{1}}(W_{1}^{-1}({{z}_{2}}(n)))}{{{w}_{2}}(W_{1}^{-1}({{z}_{2}}(n)))} \\ & \le {{h}_{1}}({{N}_{1}},n)\sum\limits_{{{t}_{2}}={{n}_{0}}}^{n-1}{{{h}_{2}}}(n,{{t}_{2}}){{f}_{2}}(n,{{t}_{2}})+{{h}_{1}}({{N}_{1}},n) \\ & \times [\sum\limits_{{{t}_{2}}={{n}_{0}}}^{n-1}{{{h}_{2}}}(n,{{t}_{2}})[\sum\limits_{{{t}_{3}}={{n}_{0}}}^{{{t}_{2}}-1}{{{h}_{3}}}({{t}_{2}},{{t}_{3}})[{{f}_{3}}({{t}_{2}},{{t}_{3}})\frac{{{w}_{3}}(W_{1}^{-1}({{z}_{2}}({{t}_{3}})))}{{{w}_{2}}(W_{1}^{-1}({{z}_{2}}({{t}_{3}})))}+\cdots \\ & \\ & +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}})\frac{{{w}_{k-1}}(W_{1}^{-1}({{z}_{2}}({{t}_{k-1}})))}{{{w}_{2}}(W_{1}^{-1}({{z}_{2}}({{t}_{k-1}})))} \\ & +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}(W_{1}^{-1}({{z}_{2}}({{t}_{k}})))}{{{w}_{2}}(W_{1}^{-1}({{z}_{2}}({{t}_{k}})))}]\cdots]]],~\forall n\in \mathbb{N}_{{{n}_{0}}}^{{{N}_{1}}}. \\ \end{align}$

(2.19)

进行(2.13) 式及 (2.14)式的类似推导,由(2.19)式推出

$\begin{align} & {{W}_{2}}({{z}_{2}}(n))\le {{W}_{2}}({{z}_{2}}({{n}_{0}}))+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}}){{f}_{2}}({{t}_{1}},{{t}_{2}})] \\ & \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\sum\limits_{{{t}_{3}}={{n}_{0}}}^{{{t}_{2}}-1}{{{h}_{3}}}({{t}_{2}},{{t}_{3}})[{{f}_{3}}({{t}_{2}},{{t}_{3}})\frac{{{w}_{3}}(W_{1}^{-1}({{z}_{2}}({{t}_{3}})))}{{{w}_{2}}(W_{1}^{-1}({{z}_{2}}({{t}_{3}})))} \\ & \ \ \ \ \ \ \ \ +\cdots +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}})\frac{{{w}_{k-1}}(W_{1}^{-1}({{z}_{2}}({{t}_{k-1}})))}{{{w}_{2}}(W_{1}^{-1}({{z}_{2}}({{t}_{k-1}})))} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}(W_{1}^{-1}({{z}_{2}}({{t}_{k}})))}{{{w}_{2}}(W_{1}^{-1}({{z}_{2}}({{t}_{k}})))}]\cdots]]] \\ & \ \ \ \ \ \ \ \ \ \ \ \le {{W}_{2}}({{z}_{2}}({{n}_{0}}))+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{{{N}_{1}}-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}}){{f}_{2}}({{t}_{1}},{{t}_{2}})] \\ & \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\sum\limits_{{{t}_{3}}={{n}_{0}}}^{{{t}_{2}}-1}{{{h}_{3}}}({{t}_{2}},{{t}_{3}})[{{f}_{3}}({{t}_{2}},{{t}_{3}})\frac{{{w}_{3}}(W_{1}^{-1}({{z}_{2}}({{t}_{3}})))}{{{w}_{2}}(W_{1}^{-1}({{z}_{2}}({{t}_{3}})))} \\ & \ \ \ \ \ \ \ \ +\cdots +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}})\frac{{{w}_{k-1}}(W_{1}^{-1}({{z}_{2}}({{t}_{k-1}})))}{{{w}_{2}}(W_{1}^{-1}({{z}_{2}}({{t}_{k-1}})))} \\ & \ \ \ \ \ +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}(W_{1}^{-1}({{z}_{2}}({{t}_{k}})))}{{{w}_{2}}(W_{1}^{-1}({{z}_{2}}({{t}_{k}})))}]\cdots]]],~~\forall n\in \mathbb{N}_{{{n}_{0}}}^{{{N}_{1}}},\\ \end{align}$

(2.20)

这里函数 $W_2$ 由 (2.3)式定义. 重复 (2.16)式到 (2.20)式的类似推导,我们有

$\begin{align} & {{W}_{k-2}}({{z}_{k-2}}(n)) \\ & \le {{W}_{k-2}}({{z}_{k-2}}({{n}_{0}}))+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})\cdots \\ & \ \ \ [\sum\limits_{{{t}_{k-2}}={{n}_{0}}}^{{{t}_{k-3}}-1}{{{h}_{k-2}}}({{t}_{k-3}},{{t}_{k-2}}){{f}_{k-2}}({{t}_{k-3}},{{t}_{k-2}})]\cdots] \\ & \ +\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}}) \\ & \ \ [{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}})\frac{{{w}_{k-1}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-3}^{-1}({{z}_{k-2}}({{t}_{k-1}})))\cdots )))}{{{w}_{k-2}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-3}^{-1}({{z}_{k-2}}({{t}_{k-1}})))\cdots )))} \\ & \ \ +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-3}^{-1}({{z}_{k-2}}({{t}_{k}})))\cdots )))}{{{w}_{k-2}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-3}^{-1}({{z}_{k-2}}({{t}_{k}})))\cdots )))}]]\cdots]] \\ & \le {{W}_{k-2}}({{z}_{k-2}}({{n}_{0}}))+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{{{N}_{1}}-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})\cdots \\ & \ \ [\sum\limits_{{{t}_{k-2}}={{n}_{0}}}^{{{t}_{k-3}}-1}{{{h}_{k-2}}}({{t}_{k-3}},{{t}_{k-2}}){{f}_{k-2}}({{t}_{k-3}},{{t}_{k-2}})]\cdots] \\ & +\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}}) \\ & \ [{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}})\frac{{{w}_{k-1}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-3}^{-1}({{z}_{k-2}}({{t}_{k-1}})))\cdots )))}{{{w}_{k-2}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-3}^{-1}({{z}_{k-2}}({{t}_{k-1}})))\cdots )))} \\ & +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-3}^{-1}({{z}_{k-2}}({{t}_{k}})))\cdots )))}{{{w}_{k-2}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-3}^{-1}({{z}_{k-2}}({{t}_{k}})))\cdots )))}]]\cdots]],~~~~~ \\ \end{align}$

(2.21)

对所有的 $n\in{\mathbb N}_{n_0}^{N_1}$成立,其中 $W_{k-2}$ 由 (2.3)式定义. 令 $z_{k-1}(n)$为不等式 (2.21)中右端的函数,则它在 ${\mathbb N}_{n_0}^{N_1}$上是非负不减的. (2.21) 式等价于

$\begin{equation} \label{Volterra-Fredholm2z3} z_{k-2}(n)\le W_{k-2}^{-1}(z_{k-1}(n)),~~\forall n\in{\mathbb N}_{n_0}^{N_1}, \end{equation}$

(2.22)

$\begin{align} & {{z}_{k-1}}({{n}_{0}})={{W}_{k-2}}({{z}_{k-2}}({{n}_{0}}))+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{{{N}_{1}}-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})\cdots \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ [\sum\limits_{{{t}_{k-2}}={{n}_{0}}}^{{{t}_{k-3}}-1}{{{h}_{k-2}}}({{t}_{k-3}},{{t}_{k-2}}){{f}_{k-2}}({{t}_{k-3}},{{t}_{k-2}})]\cdots]. \\ \end{align}$

(2.23)

由$z_{k-1}$的定义,利用差分公式及 (2.22)式我们有

$\begin{align} & \Delta {{z}_{k-1}}(n)={{h}_{1}}({{N}_{1}},n)[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{n-1}{{{h}_{2}}}(n,{{t}_{2}})[\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}}) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\times \frac{{{w}_{k-1}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-3}^{-1}({{z}_{k-2}}({{t}_{k-1}})))\cdots )))}{{{w}_{k-2}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-3}^{-1}({{z}_{k-2}}({{t}_{k-1}})))\cdots )))} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-3}^{-1}({{z}_{k-2}}({{t}_{k}})))\cdots )))}{{{w}_{k-2}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-3}^{-1}({{z}_{k-2}}({{t}_{k}})))\cdots )))}]]\cdots]] \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \le {{h}_{1}}({{N}_{1}},n)[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{n-1}{{{h}_{2}}}(n,{{t}_{2}})[\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}}) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \frac{{{w}_{k-1}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-2}^{-1}({{z}_{k-1}}({{t}_{k-1}})))\cdots )))}{{{w}_{k-2}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-2}^{-1}({{z}_{k-1}}({{t}_{k-1}})))\cdots )))} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-2}^{-1}({{z}_{k-1}}({{t}_{k}})))\cdots )))}{{{w}_{k-2}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-2}^{-1}({{z}_{k-1}}({{t}_{k}})))\cdots )))}]]\cdots]],~~~~~ \\ \end{align}$

(2.24)

对所有 $n\in{\mathbb N}_{n_0}^{N_1}$成立. 从(2.24)式我们推出

$\begin{align} & \frac{\Delta {{z}_{k-1}}(n){{w}_{k-2}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-2}^{-1}({{z}_{k-1}}(n)))\cdots )))}{{{w}_{k-1}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-2}^{-1}({{z}_{k-1}}(n)))\cdots )))} \\ & \le {{h}_{1}}({{N}_{1}},n)[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{n-1}{{{h}_{2}}}(n,{{t}_{2}})[\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}}) \\ & +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-2}^{-1}({{z}_{k-1}}({{t}_{k}})))\cdots )))}{{{w}_{k-1}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-2}^{-1}({{z}_{k-1}}({{t}_{k}})))\cdots )))}]]\cdots]],\\ \end{align}$

(2.25)

对所有 $n\in{\mathbb N}_{n_0}^{N_1}$成立,这里利用了函数 $z_{k-1}, W_1^{-1},\cdots,W_{k-2}^{-1}$ 及 $w_{k-2}/w_{k-1}$的单调性.

进行由(2.19)式推出(2.21)式的推导,由(2.25) 式推出

$\begin{align} & {{W}_{k-1}}({{z}_{k-1}}(n)) \\ & \le {{W}_{k-1}}({{z}_{k-1}}({{n}_{0}}))+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\cdots \\ & \ \ \ \ \ [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}}){{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}})]\cdots]] \\ & \ \ \ +\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}}) \\ & \ \ \ \ \ \ [\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-2}^{-1}({{z}_{k-1}}({{t}_{k}})))\cdots )))}{{{w}_{k-1}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-2}^{-1}({{z}_{k-1}}({{t}_{k}})))\cdots )))}]]\cdots]] \\ & \le {{W}_{k-1}}({{z}_{k-1}}({{n}_{0}}))+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{{{N}_{1}}-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\cdots \\ & \ \ [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}}){{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}})]\cdots]] \\ & \ \ +\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}}) \\ & [\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-2}^{-1}({{z}_{k-1}}({{t}_{k}})))\cdots )))}{{{w}_{k-1}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-2}^{-1}({{z}_{k-1}}({{t}_{k}})))\cdots )))}]]\cdots]] \\ \end{align}$

(2.26)

对所有的 $n\in{\mathbb N}_{n_0}^{N_1}$成立.

令 $z_{k}(n)$ 为不等式 (2.26)右边的函数,则它在 ${\mathbb N}_{n_0}^{N_1}$上是非负不减的函数. (2.26)式等价于

$\begin{equation}\label{Volterra-Fredholmzn-13} z_{k-1}(n)\le W_{k-1}^{-1}(z_k(n)),~~\forall n\in{\mathbb N}_{n_0}^{N_1}, \end{equation}$

(2.27)

$\begin{eqnarray} \label{Volterra-Fredholmzn-130} z_{k}(n_0)&=& W_{k-1}( z_{k-1}(n_0))+\sum^{N_1-1}_{t_{1}=n_0}h_1(N_1,t_1)\Big[\sum\limits_{t_2=n_0}^{t_1-1}h_2(t_1,t_2)\Big[\cdots\nonumber\\ &&\Big[\sum^{t_{k-2}-1}_{t_{k-1}=n_0}h_{k-1}(t_{k-2},t_{k-1})f_{k-1}(t_{k-2},t_{k-1})\Big] \cdots\Big]\Big]. \end{eqnarray}$

(2.28)

利用差分公式及(2.27)式我们有

$\begin{align} & \Delta {{z}_{k}}(n)={{h}_{1}}({{N}_{1}},n)[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{n-1}{{{h}_{2}}}(n,{{t}_{2}})[\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}}) \\ & \ \ \ \ \ \ \ \ \ \ \ \ [\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-2}^{-1}({{z}_{k-1}}({{t}_{k}})))\cdots )))}{{{w}_{k-1}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-2}^{-1}({{z}_{k-1}}({{t}_{k}})))\cdots )))}]]\cdots]] \\ & \ \ \ \ \ \ \ \ \le {{h}_{1}}({{N}_{1}},n)[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{n-1}{{{h}_{2}}}(n,{{t}_{2}})[\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}}) \\ & \ \ \ \ \ \ \ \ \ [\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})\frac{{{w}_{k}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-1}^{-1}({{z}_{k}}({{t}_{k}})))\cdots )))}{{{w}_{k-1}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-1}^{-1}({{z}_{k}}({{t}_{k}})))\cdots )))}]]\cdots]],\\ \end{align}$

(2.29)

对所有的 $n\in{\mathbb N}_{n_0}^{N_1}$成立. 由 (2.29)式我们有

$\begin{align} & \frac{\Delta {{z}_{k}}(n){{w}_{k-1}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-1}^{-1}({{z}_{k}}(n)))\cdots )))}{{{w}_{k}}(W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-1}^{-1}({{z}_{k}}(n)))\cdots )))} \\ & \le {{h}_{1}}({{N}_{1}},n)[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{n-1}{{{h}_{2}}}(n,{{t}_{2}})[\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})]]\cdots]],\\ \end{align}$

(2.30)

对所有的 $n\in{\mathbb N}_{n_0}^{N_1}$.

重复(2.14) 式及 (2.15)式类似的推导,我们有

$\begin{align} & {{W}_{k}}({{z}_{k}}(n))-{{W}_{k}}({{z}_{k}}({{n}_{0}}))\le \sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}}) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})]]\cdots]],~\forall n\in \mathbb{N}_{{{n}_{0}}}^{{{N}_{1}}}. \\ \end{align}$

(2.31)

从(2.16),(2.22),(2.27)式及 (2.31)式我们有

$\begin{align} & {{z}_{1}}(n)\le W_{1}^{-1}(W_{2}^{-1}(\cdots (W_{k-1}^{-1}({{z}_{k}}(n)))\cdots )) \\ & \ \ \ \ \ \ \ \le W_{1}^{-1}\{W_{2}^{-1}\{\cdots \{W_{k}^{-1}\{{{W}_{k}}({{z}_{k}}({{n}_{0}}))+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}}) \\ & \ \ \ \ \ \ \ \ \ \ [\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})]]\cdots]]\}\}\cdots \}\},\\ \end{align}$

(2.32)

对所有的 $n\in{\mathbb N}_{n_0}^{N_1}$成立. 将 (2.17),(2.23)式及 (2.28) 式带入 (2.32) 式,我们有

$\begin{align} & {{z}_{1}}(n)\le W_{1}^{-1}\{W_{2}^{-1}\{\cdots \{W_{k}^{-1}\{{{W}_{k}}\{{{W}_{k-1}}\{{{W}_{k-2}}\{\cdots \{{{W}_{2}}\{{{W}_{1}}({{z}_{1}}({{n}_{0}})) \\ & \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{1}}={{n}_{0}}}^{{{N}_{1}}-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}}){{f}_{1}}(n,{{t}_{1}})\}+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{{{N}_{1}}-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}}){{f}_{2}}({{t}_{1}},{{t}_{2}})]\}\cdots \} \\ & \ \ \ \ \ \ +\sum\limits_{{{t}_{1}}={{n}_{0}}}^{{{N}_{1}}-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\cdots ~~ \\ & \ \ \ \ \ \ \ \ \ \ \ [\int_{\alpha ({{n}_{0}})}^{{{t}_{k-3}}}{{{h}_{k-2}}}({{t}_{k-3}},{{t}_{k-2}}){{f}_{k-2}}({{t}_{k-3}},{{t}_{k-2}})]\cdots]]\} \\ & \ \ \ \ \ \ +\sum\limits_{{{t}_{1}}={{n}_{0}}}^{{{N}_{1}}-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\cdots ~~ \\ & \ \ \ \ \ \ \ \ \ \ \ ~[\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}}){{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}})]\cdots]]\} \\ & \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})~~ \\ & \ \ \ \ \ \ \ \ \ \ \ \ ~[\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})]]\cdots]]\}\}\cdots \}\},~~\forall t\in \mathbb{N}_{{{n}_{0}}}^{{{N}_{1}}}. \\ \end{align}$

(2.33)

因为 $N_1$ 是任意选取的,我们有

$\begin{align} & {{z}_{1}}(n)\le W_{1}^{-1}\{W_{2}^{-1}\{\cdots \{W_{k}^{-1}\{{{W}_{k}}\{{{W}_{k-1}}\{{{W}_{k-2}}\{\cdots \{{{W}_{2}}\{{{W}_{1}}({{z}_{1}}({{n}_{0}})) \\ & \ \ \ \ \ \ \ +\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}(n,{{t}_{1}}){{f}_{1}}(n,{{t}_{1}})\}+\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}}){{f}_{2}}({{t}_{1}},{{t}_{2}})]\}\cdots \} \\ & \ \ \ \ \ +\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}(n,{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\cdots ~~~ \\ & \ \ \ \ \ \ \ \ \ [\int_{\alpha ({{n}_{0}})}^{{{t}_{k-3}}}{{{h}_{k-2}}}({{t}_{k-3}},{{t}_{k-2}}){{f}_{k-2}}({{t}_{k-3}},{{t}_{k-2}})]\cdots]]\} \\ & \ \ \ \ \ +\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}(n,{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\cdots ~~ \\ & \ \ \ \ \ \ \ ~[\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}}){{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}})]\cdots]]\} \\ & \ \ \ \ +\sum\limits_{{{t}_{1}}={{n}_{0}}}^{n-1}{{{h}_{1}}}(n,{{t}_{1}})[\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[\cdots [\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})~~~ \\ & \ \ \ \ \ \ \ [\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}})]]\cdots]]\}\}\cdots \}\} \\ & \ \ \ \ =W_{1}^{-1}\{W_{2}^{-1}\{\cdots \{W_{k}^{-1}\{{{W}_{k}}\{{{W}_{k-1}}\{\cdots \{{{W}_{2}}\{{{W}_{1}}({{z}_{1}}({{n}_{0}}))+{{H}_{1}}(n)\} \\ & \ \ \ \ \ \ +{{H}_{2}}(n)\}\ \cdots \}+{{H}_{k-1}}(n)\}+{{H}_{k}}(n)\}\}\cdots \}\},\forall n\in \mathbb{N}_{{{n}_{0}}}^{{{N}_{1}}}. \\ \end{align}$

(2.34)

由 $z_1$ 的定义及 (2.10)式我们推出

$\begin{align} & 2{{z}_{1}}({{n}_{0}})-a(N)=a(N)+2\sum\limits_{{{t}_{1}}={{n}_{0}}}^{N-1}{{{h}_{1}}}({{N}_{1}},{{t}_{1}})[{{f}_{1}}({{N}_{1}},{{t}_{1}}){{w}_{1}}(u({{t}_{1}})) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{2}}={{n}_{0}}}^{{{t}_{1}}-1}{{{h}_{2}}}({{t}_{1}},{{t}_{2}})[{{f}_{2}}({{t}_{1}},{{t}_{2}}){{w}_{2}}(u({{t}_{2}}))+\cdots \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k-1}}={{n}_{0}}}^{{{t}_{k-2}}-1}{{{h}_{k-1}}}({{t}_{k-2}},{{t}_{k-1}})[{{f}_{k-1}}({{t}_{k-2}},{{t}_{k-1}}){{w}_{k-1}}(u({{t}_{k-1}})) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum\limits_{{{t}_{k}}={{n}_{0}}}^{{{t}_{k-1}}-1}{{{h}_{k}}}({{t}_{k-1}},{{t}_{k}}){{w}_{k}}(u({{t}_{k}}))]\cdots]] \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ ={{z}_{1}}(N). \\ \end{align}$

(2.35)

因为 $N_1$ 是任意选取的,从 (2.34) 式及 (2.35)式,我们推出

$$\begin{align} & 2{{z}_{1}}({{n}_{0}})-a(N)\le W_{1}^{-1}\{W_{2}^{-1}\{\cdots \{W_{k}^{-1}\{{{W}_{k}}\{{{W}_{k-1}}\{\cdots \{{{W}_{2}}\{{{W}_{1}}({{z}_{1}}({{n}_{0}})) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +{{H}_{1}}(N)\}+{{H}_{2}}(N)\}\cdots \}+{{H}_{k-1}}(N)\}+{{H}_{k}}(N)\}\}\cdots \}\},\\ \end{align}$$

或者

$\begin{align} & {{W}_{k}}\{{{W}_{k-1}}\{\cdots \{{{W}_{2}}\{{{W}_{1}}(2{{z}_{1}}({{n}_{0}})-a(N))\}\}\cdots \}\}-{{W}_{k}}\{{{W}_{k-1}}\{\cdots \{{{W}_{2}}\{{{W}_{1}}({{z}_{1}}({{n}_{0}})) \\ & +{{H}_{1}}(N)\}+{{H}_{2}}(N)\}\cdots \}+{{H}_{k-1}}(N)\}-{{H}_{k}}(N)\le 0. \\ \end{align}$

(2.36)

从函数 $G$的定义,定理 2.1的假设 及(2.36)式,我们得出

$$G(z_1(n_0),t)\le 0 =G(c(n),t),$$

其中$G$ 关于第一个变量是递增的,从最后的不等式及 (2.9) 式, 我们得到所求的估计 (2.7)式.

我们定义下面的函数

$\begin{equation} H_1(n)=\sum^{n-1}_{t_{1}=n_0}h_1(n,t_1)f_1(n,t_1), \end{equation}$

(2.37)

$\begin{equation} H_2(n)= \sum^{n-1}_{t_{1}=n_0} h_1(n,t_1)\Big[\sum\limits_{t_2=n_0}^{t_1-1}h_2(t_1,t_2)\Big], \end{equation}$

(2.38)

$\begin{equation} \label{Volterra-FredholmucoroGdefine} E(\xi)=W_{2}\Big\{W_1(2\xi-a(N))\Big\}- W_2\Big\{W_1(\xi) +H_1(N)\Big\}-H_{2}(N), \end{equation}$

(2.39)

对所有的 $u> k$,其中 $W_i,i=1,2$ 由 (2.3)式定义.

推论2.1   假设$(1.5)$式中 $k=2$,$\varphi,a,f_1,h_i,\phi_i,W_i,i=1,2$的定义同定理 $2.1$一样. 假设函数 $E(\xi)$ 是递增的且 当$\xi>k$时方程$E(\xi)=0$有一个解 $\xi=c$ . 如果 $u(n)$ 满足 (1.5)式,则

$\begin{equation}\label{Volterra-Fredholmcoro1f} u(n)\le W_1^{-1}\Big\{W_2^{-1}\Big\{W_2\Big\{W_1(c) +H_1(n)\Big\}+H_2(n)\Big\}\Big\},~~\forall n\in {\mathbb N}_{n_0}^{N}, \end{equation}$

(2.40)

其中 $W_i^{-1}~(i=1,2)$ 是 $W_i$的逆函数.

注   如果 在推论2.1中$\phi_1=\phi_2$, 则推论 2.1 的结果就是文献[4]的定理 2.1$'$. 因为如果 $\phi_1=\phi_2$,则

$$W_2(u)=u-u_2,\qquad W_2^{-1}(u)=u+u_2,$$

由(2.40)式推出 $u(n)\le W_1^{-1}\big\{W_1(c) +H_1(n)+H_2(n)\big\}$ 对所有的 $t\in {\mathbb N}_{n_0}^{N}$成立.

最后,我们用推论2.1中的结果来研究 Volterra-Fredholm 差分方程

$\begin{equation}\label{Gapp1} x(n)=x_0+\sum\limits_{s=n_0}^{N-1}G\Big(n,s,x(s),\sum\limits_{\tau=n_0}^{s-1} F_2(s,\tau,x(\tau)) \Big),~\forall n\in {\mathbb N}_{n_0}^{N}, \end{equation}$

(3.1)

式中

$\begin{equation} |G(n,s,u,y)|\le\left\{ \begin{array}{ll} 2F_1(n,s,u,y),&s

(3.2)

下面我们研究方程(3.1)解的界.

推论3.1    假设 $(3.1)$式及 $(3.2)$ 式中的$F_1,F_2$ 满足条件

$\begin{equation}\label{F1d} |F_1(n,s,u,y)| \le h_1(n,s)[f_1(n,s)|u|^{p_1}+|y|], \end{equation}$

(3.3)

$\begin{equation} \label{F2d} |F_2(n,s,u)|\le h_2(n,s)|u|^{p_2}, \end{equation}$

(3.4)

其中 $f_1(n,s),h_1(n,s),h_2(n,s)$的定义和推论 2.1相同. 假设 $p_1,p_2$ 是常数且 $0

$\begin{align} & |x(n)|\le W_{1}^{-1}\{W_{2}^{-1}[{{W}_{2}}[{{W}_{1}}(c)+\sum\limits_{n={{n}_{0}}}^{n-1}{{{h}_{1}}}(n,s){{f}_{1}}(n,s)] \\ & \ \ \ \ \ +\sum\limits_{n={{n}_{0}}}^{n-1}{{{h}_{1}}}(n,s)[\sum\limits_{n={{n}_{0}}}^{s-1}{{{h}_{2}}}(s,\tau )]]\},~~\forall n\in \mathbb{N}_{{{n}_{0}}}^{N},\\ \end{align}$

(3.5)

其中

$\begin{equation}\label{widef1} W_1(u)=\frac{u^{1-p_1}}{1-p_1}-\frac{u_1^{1-p_1}}{1-p_1}, \end{equation}$

(3.6)

$\begin{equation} \label{widef2} W_1^{-1}(u)=\Big((1-p_1)u+u_1^{1-p_1}\Big)^{\frac{1}{1-p_1}}, \end{equation}$

(3.7)

$\begin{equation} \label{widef3} W_2(u)=\frac{1-p_1}{1-p_2}\Big((1-p_1)u+u_1^{1-p_1}\Big)^{\frac{1-p_2}{1-p_1}}-\frac{1-p_1}{1-p_2}\Big((1-p_1)u_2+u_1^{1-p_1}\Big)^{\frac{1-p_2}{1-p_1}}, \end{equation}$

(3.8)

$\begin{equation} \label{widef4} W_2^{-1}(u)=\frac{1}{1-p_1}\Big\{\Big[\frac{1-p_2}{1-p_1}u+\frac{1-p_1}{1-p_2}\Big((1-p_1)u_2+u_1^{1-p_1}\Big)^{\frac{1-p_2}{1-p_1}}\Big]^{\frac{1-p_1}{1-p_2}}+u_1^{1-p_1}\Big\}, \end{equation}$

(3.9)

$u_1,u_2$ 是非负常数,$W_1^{-1}$ ,$W_2^{-1}$ 分别是 $W_1$和 $W_2$的逆函数, $c$ 是下面方程的一个解

$\begin{align} & {{W}_{2}}[{{W}_{1}}(2u-|{{x}_{0}}|)]-{{W}_{2}}[{{W}_{1}}(u)+\sum\limits_{s={{n}_{0}}}^{N-1}{{{h}_{1}}}(N,s){{f}_{1}}(N,s)] \\ & =\sum\limits_{s={{n}_{0}}}^{N-1}{{{h}_{1}}}(N,s)[\sum\limits_{\tau ={{n}_{0}}}^{s-1}{{{h}_{2}}}(s,\tau )].~~ \\ \end{align}$

(3.10)

证   由定理 2.1中函数 $W_i$ 的定义,我们推出

$\begin{equation} W_1(u,u_1):=\int^{u}_{u_1} \frac{{\rm d}s}{s^{p_1}}, W_2(u,u_2):=\int^{u}_{u_2} \frac{(W^{-1}(s))^{p_1}{\rm d}s}{(W^{-1}(s))^{p_2}}. \label{w1def} \end{equation}$

(3.11)

由(3.11)式,我们得到 (3.6),(3.7),(3.8) 式及 (3.9)式. 我们令

$\begin{align} & E(u):={{W}_{2}}[{{W}_{1}}(2u-|{{x}_{0}}|)]-{{W}_{2}}[{{W}_{1}}(u)+\sum\limits_{s={{n}_{0}}}^{N-1}{{{h}_{1}}}(N,s){{f}_{1}}(N,s)] \\ & \ \ \ \ \ \ \ -\sum\limits_{s={{n}_{0}}}^{N-1}{{{h}_{1}}}(N,s)[\sum\limits_{\tau ={{n}_{0}}}^{s-1}{{{h}_{2}}}(s,\tau )]. \\ \end{align}$

(3.12)

从(3.6),(3.8) 式及(3.12)式,以及 $E(|x_0|)<0$. 由(3.12)式,我们推出

$\begin{equation}\label{Edfinp1} ~~E'(u):= 2 W_2'\Big[W_1(2u-|x_0|)\Big]W_1'(2u-|x_0|)-W_2'\Big[W_1(u) + \sum^{N-1}_{s=n_0}h_1(N,s)f_1(N,s)\Big]W_1'(u) .~~ \end{equation}$

(3.13)

从(3.6)式 及 (3.8)式,我们推出

$\begin{align} & \ \ \ 2{{W}_{{{2}'}}}[{{W}_{1}}(2u-|{{x}_{0}}|)]{{W}_{{{1}'}}}(2u-|{{x}_{0}}|) \\ & =2{{((1-{{p}_{1}})(\frac{{{(2u-|{{x}_{0}}|)}^{1-{{p}_{1}}}}}{1-{{p}_{1}}}-\frac{u_{1}^{1-{{p}_{1}}}}{1-{{p}_{1}}})+u_{1}^{1-{{p}_{1}}})}^{\frac{{{p}_{1}}-{{p}_{2}}}{1-{{p}_{1}}}}}{{(2u-|{{x}_{0}}|)}^{-{{p}_{1}}}} \\ & =2{{(2u-|{{x}_{0}}|)}^{{{p}_{1}}-{{p}_{2}}}}{{(2u-|{{x}_{0}}|)}^{-{{p}_{1}}}}\ge 2{{(2u)}^{{{p}_{1}}-{{p}_{2}}}}{{(2u)}^{-{{p}_{1}}}}\ge {{u}^{{{p}_{1}}-{{p}_{2}}}}{{u}^{-{{p}_{1}}}} \\ & ={{W}_{{{2}'}}}[{{W}_{1}}(u)]{{W}_{{{1}'}}}(u)>{{W}_{{{2}'}}}[{{W}_{1}}(u)+\sum\limits_{s={{n}_{0}}}^{N-1}{{{h}_{1}}}(N,s){{f}_{1}}(N,s)]{{W}_{{{1}'}}}(u),~~\forall u>|{{x}_{0}}|. \\ \end{align}$

(3.14)

由(3.13)式及 (3.14)式,以及 $E(u)$ 是单调递增的且方程 $E(n)=0$有一个解 $c$ 满足$u>|x_0|$. 利用(3.2),(3.3) 式及 (3.4)式的条件,从(3.1)式 我们推出

$\begin{align} & |x(n)|\le |{{x}_{0}}|+\sum\limits_{s={{n}_{0}}}^{N-1}{|}G(n,s,x(s),\sum\limits_{\tau ={{n}_{0}}}^{s-1}{{{F}_{2}}}(s,\tau ,x(\tau )))| \\ & \ \ \ \ \ \ \ \ \ \le |{{x}_{0}}|+\sum\limits_{s={{n}_{0}}}^{n-1}{{{F}_{1}}}(n,s,x(s),\sum\limits_{\tau ={{n}_{0}}}^{s-1}{{{F}_{2}}}(s,\tau ,x(\tau ))) \\ & \ \ \ \ \ \ +\sum\limits_{s={{n}_{0}}}^{N-1}{{{F}_{1}}}(n,s,x(s),\sum\limits_{\tau ={{n}_{0}}}^{s-1}{{{F}_{2}}}(s,\tau ,x(\tau ))) \\ & \ \ \ \ \ \ \le |{{x}_{0}}|+\sum\limits_{s={{n}_{0}}}^{n-1}{{{h}_{1}}}(n,s)[{{f}_{1}}(n,s)|x(s){{|}^{{{p}_{1}}}}+\sum\limits_{\tau ={{n}_{0}}}^{s-1}{{{h}_{2}}}(s,\tau )|x(\tau ){{|}^{{{p}_{2}}}}] \\ & \ \ \ \ \ \ +\sum\limits_{s={{n}_{0}}}^{N-1}{{{h}_{1}}}(n,s)[{{f}_{1}}(n,s)|x(s){{|}^{{{p}_{1}}}}+\sum\limits_{\tau ={{n}_{0}}}^{s-1}{{{h}_{2}}}(s,\tau )|x(\tau ){{|}^{{{p}_{2}}}}],~\forall n\in \mathbb{N}_{{{n}_{0}}}^{N}. \\ \end{align}$

(3.15)

利用推论2.1的结果,由上式我们就可以得到所求的估计 (3.5)式.