1 Introduction

Integral inequalities and their various linear and nonlinear generalizations involving continuous or discontinuous functions play very important roles in investigating different qualitative characteristics of solutions for differential equations, partial differential equations and impulsive differential equations such as existence, uniqueness, continuation, boundedness, continuous dependence of parameters, stability, and attraction. The literature on inequalities for continuous functions and their applications is vast (see [111]). In the one-dimensional case, all the main results in the theory of integral inequalities for continuous functions are almost based on the solvability of Chaplygin’s problem [6] for the integral inequality

$u\left(x\right)\le \phi \left(x\right)+{\int }_{{x}_{0}}^{x}\mathrm{\Gamma }\left(x,s,u\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds.$
(1.1)

Recently, more attention has been paid to generalizations of Gronwall-Bellman’s results for discontinuous functions and their applications (see [1227]). One of the important things is that Samoilenko and Perestyuk [26] studied the following inequality:

$u\left(x\right)\le c+{\int }_{{x}_{0}}^{x}f\left(s\right)u\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\sum _{{x}_{0}<{x}_{i}
(1.2)

for the nonnegative piecewise continuous function $u\left(x\right)$, where c, ${\beta }_{i}$ are nonnegative constants, $f\left(x\right)$ is a positive function, and ${x}_{i}$ are the first kind discontinuity points of the function $u\left(x\right)$. Then Borysenko [14] investigated integral inequalities with two independent variables,

$\begin{array}{rl}u\left(x,y\right)\le & a\left(x,y\right)+{\int }_{{x}_{0}}^{x}{\int }_{{y}_{0}}^{y}\tau \left(s,t\right)u\left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +\sum _{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}{\beta }_{i}u\left({x}_{i}-0,{y}_{i}-0\right).\end{array}$
(1.3)

Here $u\left(x,y\right)$ is an unknown nonnegative continuous function with the exception of the points $\left({x}_{i},{y}_{i}\right)$ where there is a finite jump: $u\left({x}_{i}-0,{y}_{i}-0\right)\ne u\left({x}_{i}+0,{y}_{i}+0\right)$, $i=1,2,\dots$ .

In 2007, Borysenko and Iovane [16] considered the following inequalities:

$\begin{array}{rl}u\left(x,y\right)\le & a\left(x,y\right)+{\int }_{{x}_{0}}^{x}{\int }_{{y}_{o}}^{y}\tau \left(s,t\right)u\left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +\sum _{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}{\beta }_{i}{u}^{m}\left({x}_{i}-0,{y}_{i}-0\right),\phantom{\rule{1em}{0ex}}m>0,\end{array}$
(1.4)
$\begin{array}{rl}u\left(x,y\right)\le & a\left(x,y\right)+{\int }_{{x}_{0}}^{x}{\int }_{{y}_{o}}^{y}\tau \left(s,t\right){u}^{m}\left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +\sum _{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}{\beta }_{i}{u}^{m}\left({x}_{i}-0,{y}_{i}-0\right),\phantom{\rule{1em}{0ex}}m>0,\end{array}$
(1.5)
$\begin{array}{rl}u\left(x,y\right)\le & a\left(x,y\right)+q\left(x,y\right){\int }_{{x}_{0}}^{x}{\int }_{{y}_{o}}^{y}\tau \left(s,t\right){u}^{m}\left(\sigma \left(s\right),\sigma \left(t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +\sum _{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}{\beta }_{i}{u}^{m}\left({x}_{i}-0,{y}_{i}-0\right),\phantom{\rule{1em}{0ex}}m>0.\end{array}$
(1.6)

Later, Gallo and Piccirllo [24] studied the following inequalities:

$\begin{array}{rl}u\left(x,y\right)\le & a\left(x,y\right)+q\left(x,y\right){\int }_{{x}_{0}}^{x}{\int }_{{y}_{o}}^{y}\tau \left(s,t\right)\omega \left(u\left(\sigma \left(s\right),\tau \left(t\right)\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +\sum _{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}{\beta }_{i}{u}^{m}\left({x}_{i}-0,{y}_{i}-0\right),\phantom{\rule{1em}{0ex}}m>0.\end{array}$
(1.7)

In this paper, motivated by the work above, we will establish the following much more general integral inequality:

$\begin{array}{rl}u\left(x,y\right)\le & a\left(x,y\right)+\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{0}\right)}^{{b}_{n}\left(x\right)}{\int }_{{c}_{n}\left({y}_{o}\right)}^{{c}_{n}\left(y\right)}{f}_{n}\left(x,y,s,t\right){\omega }_{n}\left(u\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +g\left(x,y\right)\sum _{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}{\beta }_{i}{u}^{m}\left({x}_{i}-0,{y}_{i}-0\right),\phantom{\rule{1em}{0ex}}m>0,\end{array}$
(1.8)

with two independent variables involving two nonlinear terms ${\omega }_{1}\left(u\right)$ and ${\omega }_{2}\left(u\right)$ where we do not restrict ${\omega }_{1}$ and ${\omega }_{2}$ to the class ℘ or the class ȷ. Moreover, ${f}_{n}\left(x,y,s,t\right)$ ($n=1,2$) has a more general form. We also show that many integral inequalities for discontinuous functions such as (1.3)-(1.6) can be reduced to the form of (1.8). Finally, our main result is applied to an estimation of the bounds of the solutions of a partial differential equation with impulsive terms.

2 Main results

Let

$\mathrm{\Omega }=\bigcup _{i,j\ge 1}{\mathrm{\Omega }}_{ij},\phantom{\rule{2em}{0ex}}{\mathrm{\Omega }}_{ij}=\left\{\left(x,y\right):{x}_{i-1}\le x<{x}_{i},{y}_{j-1}\le y<{y}_{j}\right\},$

for $i,j=1,2,\dots$ , ${x}_{0}>0$ and ${y}_{0}>0$, and let ${D}_{1}z\left(x,y\right)$ denote the first-order partial derivative of $z\left(x,y\right)$ with respect to x and ${\sum }_{k=0}^{1}{u}_{k}\left(x,y\right)=0$.

Consider (1.8) and assume that

(H1) $a\left(x,y\right)$ is defined on Ω and $a\left({x}_{0},{y}_{0}\right)\ne 0$; ${\beta }_{i}$ is a nonnegative constant for any positive integer i;

(H2) ${f}_{n}\left(x,y,s,t\right)$ ($n=1,2$) are continuous and nonnegative functions on $\mathrm{\Omega }×\mathrm{\Omega }$ and satisfy a certain condition: ${f}_{n}\left(x,y,s,t\right)=0$ ($n=1,2$) if $\left(s,t\right)\in {\mathrm{\Omega }}_{ij}$, $i\ne j$ for arbitrary $i,j=1,2,\dots$ ;

(H3) ${\omega }_{1}\left(u\right)$ and ${\omega }_{2}\left(u\right)$ are continuous and nonnegative functions on $\left[0,\mathrm{\infty }\right)$ and are positive on $\left(0,\mathrm{\infty }\right)$ such that $\frac{{\omega }_{2}\left(u\right)}{{\omega }_{1}\left(u\right)}$ is nondecreasing;

(H4) $g\left(x,y\right)$ is continuous and nonnegative on Ω;

(H5) $u\left(x,y\right)$ is nonnegative and continuous on Ω with the exception of the points $\left({x}_{i},{y}_{i}\right)$ where there is a finite jump: $u\left({x}_{i}-0,{y}_{i}-0\right)\ne u\left({x}_{i}+0,{y}_{i}+0\right)$, $i=1,2,\dots$ . Here $\left({x}_{i},{y}_{i}\right)<\left({x}_{i+1},{y}_{i+1}\right)$ if ${x}_{i}<{x}_{i+1}$, ${y}_{i}<{y}_{i+1}$, $i=0,1,2,\dots$ , and ${lim}_{i\to \mathrm{\infty }}{x}_{i}=\mathrm{\infty }$, ${lim}_{i\to \mathrm{\infty }}{y}_{i}=\mathrm{\infty }$;

(H6) ${b}_{n}\left(x\right)$ and ${c}_{n}\left(y\right)$ ($n=1,2$) are continuously differentiable and nondecreasing such that ${x}_{0}\le {b}_{n}\left(x\right)\le x$ on $\left[{x}_{0},\mathrm{\infty }\right)$ and ${y}_{0}\le {c}_{n}\left(y\right)\le y$ on $\left[{y}_{0},\mathrm{\infty }\right)$.

Let ${W}_{j}\left(u\right)={\int }_{{\stackrel{˜}{u}}_{j}}^{u}\frac{dz}{{\omega }_{j}\left(z\right)}$ for $u\ge {\stackrel{˜}{u}}_{j}$ and $j=1,2$ where ${\stackrel{˜}{u}}_{j}$ is a given positive constant. Clearly, ${W}_{j}$ is strictly increasing so its inverse ${W}_{j}^{-1}$ is well defined, continuous, and increasing in its corresponding domain.

Theorem 2.1 Suppose that (H k ) ($k=1,\dots ,6$) hold and $u\left(x,y\right)$ satisfies (1.8) for a positive constant m. If we let ${u}_{i}\left(x,y\right)=u\left(x,y\right)$ for $\left(x,y\right)\in {\mathrm{\Omega }}_{ii}$, then the estimate of $u\left(x,y\right)$ is recursively given, for $\left(x,y\right)\in {\mathrm{\Omega }}_{ii}$, $i=1,2,\dots$ , by

$\begin{array}{rl}{u}_{i}\left(x,y\right)\le & {W}_{2}^{-1}\left\{{W}_{2}\circ {W}_{1}^{-1}\left[{W}_{1}\left({r}_{i}\left(x,y\right)\right)+{\int }_{{b}_{1}\left({x}_{i-1}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{i-1}\right)}^{{c}_{1}\left(y\right)}\stackrel{˜}{{f}_{1}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right]\\ +{\int }_{{b}_{2}\left({x}_{i-1}\right)}^{{b}_{2}\left(x\right)}{\int }_{{c}_{2}\left({y}_{i-1}\right)}^{{c}_{2}\left(y\right)}\stackrel{˜}{{f}_{2}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right\},\end{array}$
(2.1)

where

$\begin{array}{c}{r}_{1}\left(x,y\right)=\underset{{x}_{0}\le \xi \le x,{y}_{0}\le \eta \le y}{max}|a\left(\xi ,\eta \right)|,\phantom{\rule{2em}{0ex}}{\stackrel{˜}{f}}_{n}\left(x,y,s,t\right)=\underset{{x}_{0}\le \xi \le x,{y}_{0}\le \eta \le y}{max}{f}_{n}\left(\xi ,\eta ,s,t\right),\hfill \\ \begin{array}{rl}{r}_{i}\left(x,y\right)=& {r}_{1}\left(x,y\right)+\sum _{k=1}^{i-1}\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{k-1}\right)}^{{b}_{n}\left({x}_{k}\right)}{\int }_{{c}_{n}\left({y}_{k-1}\right)}^{{c}_{n}\left({y}_{k}\right)}{f}_{n}\left(x,y,s,t\right){\omega }_{n}\left({u}_{k}\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +g\left(x,y\right)\sum _{k=1}^{i-1}{\beta }_{k}{u}_{k}^{m}\left({x}_{k}-0,{y}_{k}-0\right),\end{array}\hfill \end{array}$
(2.2)

provided that

$\begin{array}{c}{W}_{1}\left({r}_{i}\left(x,y\right)\right)+{\int }_{{b}_{1}\left({x}_{i-1}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{i-1}\right)}^{{c}_{1}\left(y\right)}\stackrel{˜}{{f}_{1}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\le {\int }_{{\stackrel{˜}{u}}_{1}}^{\mathrm{\infty }}\frac{dz}{{\omega }_{1}\left(z\right)},\hfill \\ \begin{array}{r}{W}_{2}\circ {W}_{1}^{-1}\left[{W}_{1}\left({r}_{i}\left(x,y\right)\right)+{\int }_{{b}_{1}\left({x}_{i-1}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{i-1}\right)}^{{c}_{1}\left(y\right)}\stackrel{˜}{{f}_{1}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right]\\ \phantom{\rule{1em}{0ex}}+{\int }_{{b}_{2}\left({x}_{i-1}\right)}^{{b}_{2}\left(x\right)}{\int }_{{c}_{2}\left({y}_{i-1}\right)}^{{c}_{2}\left(y\right)}\stackrel{˜}{{f}_{2}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\le {\int }_{{\stackrel{˜}{u}}_{2}}^{\mathrm{\infty }}\frac{dz}{{\omega }_{2}\left(z\right)}.\end{array}\hfill \end{array}$
(2.3)

The proof is given in Section 3.

Remark 2.1 If ${\omega }_{j}$ satisfies ${\int }_{{\stackrel{˜}{u}}_{j}}^{u}\frac{dz}{{\omega }_{j}\left(z\right)}=\mathrm{\infty }$ for $j=1,2$, then i in Theorem 2.1 can be any nonzero integer. Reference [4] pointed out that different choices of ${\stackrel{˜}{u}}_{j}$ in ${W}_{j}$ do not affect our results for $j=1,2$. If $a\left(x,y\right)\equiv 0$, then define ${W}_{1}\left(0\right)=0$ and (2.1) is still true.

Remark 2.2 If $a\left(x,y\right)$ is nondecreasing, Theorem 2.1 generalizes many known results. For example:

1. (1)

If we take ${f}_{1}\left(x,y,s,t\right)=\tau \left(s,t\right)$, ${f}_{2}\left(x,y,s,t\right)=0$, ${\omega }_{1}\left(u\right)=u$, $m=1$, ${b}_{1}\left(x\right)=x$, ${c}_{1}\left(y\right)=y$ and $g\left(x,y\right)=1$, then (1.8) reduces to (1.3). It is easy to check that ${W}_{1}\left(u\right)=ln\frac{u}{{\stackrel{˜}{u}}_{1}}$ and ${W}_{1}^{-1}\left(u\right)={\stackrel{˜}{u}}_{1}{e}^{u}$. From Theorem 2.1, we know that for $\left(x,y\right)\in {\mathrm{\Omega }}_{ii}$

${u}_{i}\left(x,y\right)\le {r}_{i}\left(x,y\right){e}^{{\int }_{{x}_{i-1}}^{x}{\int }_{{y}_{i-1}}^{y}\tau \left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt}$

with

${r}_{i}\left(x,y\right)=a\left(x,y\right)+\sum _{k=1}^{i-1}{\int }_{{x}_{k-1}}^{{x}_{k}}{\int }_{{y}_{k-1}}^{{y}_{k}}\tau \left(s,t\right){u}_{k}\left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt+\sum _{k=1}^{i-1}{\beta }_{k}{u}_{k}\left({x}_{k}-0,{y}_{k}-0\right).$

Hence

$\begin{array}{r}{r}_{1}\left(x,y\right)=a\left(x,y\right),\phantom{\rule{2em}{0ex}}{u}_{1}\left(x,y\right)=a\left(x,y\right){e}^{{\int }_{{x}_{0}}^{x}{\int }_{{y}_{0}}^{y}\tau \left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt},\\ {r}_{2}\left(x,y\right)=a\left(x,y\right)\left(1+{\beta }_{1}\right){e}^{{\int }_{{x}_{0}}^{{x}_{1}}{\int }_{{y}_{0}}^{{y}_{1}}\tau \left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt},\\ {u}_{2}\left(x,y\right)=a\left(x,y\right)\left(1+{\beta }_{1}\right){e}^{{\int }_{{x}_{0}}^{x}{\int }_{{y}_{0}}^{y}\tau \left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt}.\end{array}$

After recursive calculations, we have

$u\left(x,y\right)\le a\left(x,y\right){\mathrm{\Pi }}_{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}\left(1+{\beta }_{i}\right){e}^{{\int }_{{x}_{0}}^{x}{\int }_{{y}_{0}}^{y}\tau \left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt},$

which is the same as the expression in [14];

1. (2)

If we take ${f}_{1}\left(x,y,s,t\right)=\tau \left(s,t\right)$, ${f}_{2}\left(x,y,s,t\right)=0$, ${\omega }_{1}\left(u\right)=u$, $m>0$, ${b}_{1}\left(x\right)=x$, ${c}_{1}\left(y\right)=y$ and $g\left(x,y\right)=1$, then (1.8) reduces to (1.4) and Theorem 2.1 becomes Theorem 2.1 in [16];

2. (3)

If we take ${f}_{1}\left(x,y,s,t\right)=\tau \left(s,t\right)$, ${f}_{2}\left(x,y,s,t\right)=0$, ${\omega }_{1}\left(u\right)={u}^{m}$, $m>0$, ${b}_{1}\left(x\right)=x$, ${c}_{1}\left(y\right)=y$ and $g\left(x,y\right)=1$, then (1.8) reduces to (1.5) and Theorem 2.1 becomes Theorem 2.2 in [16];

3. (4)

If ${\sigma }^{\prime }\left(t\right)>0$ on $\left[{t}_{0},\mathrm{\infty }\right)$ where ${t}_{0}=min\left\{{x}_{0},{y}_{0}\right\}$, then (1.6) can be rewritten as

$\begin{array}{rl}u\left(x,y\right)\le & a\left(x,y\right)+q\left(x,y\right){\int }_{\sigma \left({x}_{0}\right)}^{\sigma \left(x\right)}{\int }_{\sigma \left({y}_{o}\right)}^{\sigma \left(y\right)}\frac{\tau \left({\sigma }^{-1}\left(s\right),{\sigma }^{-1}\left(t\right)\right)}{{\sigma }^{\prime }\left({\sigma }^{-1}\left(s\right)\right){\sigma }^{\prime }\left({\sigma }^{-1}\left(t\right)\right)}{u}^{m}\left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +\sum _{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}{\beta }_{i}{u}^{m}\left({x}_{i}-0,{y}_{i}-0\right).\end{array}$
(2.4)

If we let ${f}_{1}\left(x,y,s,t\right)=q\left(x,y\right)\frac{\tau \left({\sigma }^{-1}\left(s\right),{\sigma }^{-1}\left(t\right)\right)}{{\sigma }^{\prime }\left({\sigma }^{-1}\left(s\right)\right){\sigma }^{\prime }\left({\sigma }^{-1}\left(t\right)\right)}$, ${f}_{2}\left(x,y,s,t\right)=0$, and ${\omega }_{1}\left(u\right)={u}^{m}$, the above inequality is the same as (1.8).

Consider the inequality

$\begin{array}{rl}\phi \left(u\left(x,y\right)\right)\le & a\left(x,y\right)+\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{0}\right)}^{{b}_{n}\left(x\right)}{\int }_{{c}_{n}\left({y}_{o}\right)}^{{c}_{n}\left(y\right)}{f}_{n}\left(x,y,s,t\right){\omega }_{n}\left(u\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +g\left(x,y\right)\sum _{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}{\beta }_{i}\psi \left(u\left({x}_{i}-0,{y}_{i}-0\right)\right),\end{array}$
(2.5)

which looks much more complicated than (1.8).

Corollary 2.1 Suppose that (H k ) ($k=1,\dots ,6$) hold, $\psi \left(u\right)$ is positive on $\left(0,\mathrm{\infty }\right)$, $\phi \left(u\right)$ is positive and strictly increasing on $\left(0,\mathrm{\infty }\right)$ and $u\left(x,y\right)$ satisfies (2.5). If we let ${u}_{i}\left(x,y\right)=u\left(x,y\right)$ for $\left(x,y\right)\in {\mathrm{\Omega }}_{ii}$, then the estimate of $u\left(x,y\right)$ is recursively given, for $\left(x,y\right)\in {\mathrm{\Omega }}_{ii}$, $i=1,2,\dots$ , by

$\begin{array}{rl}{u}_{i}\left(x,y\right)\le & {\phi }^{-1}\left\{{W}_{2}^{-1}\left[{W}_{2}\circ {W}_{1}^{-1}\left({W}_{1}\left({r}_{i}\left(x,y\right)\right)+{\int }_{{b}_{1}\left({x}_{i-1}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{i-1}\right)}^{{c}_{1}\left(y\right)}\stackrel{˜}{{f}_{1}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right)\\ +{\int }_{{b}_{2}\left({x}_{i-1}\right)}^{{b}_{2}\left(x\right)}{\int }_{{c}_{2}\left({y}_{i-1}\right)}^{{c}_{2}\left(y\right)}\stackrel{˜}{{f}_{2}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right]\right\},\end{array}$
(2.6)

where ${W}_{j}\left(u\right)={\int }_{{\stackrel{˜}{u}}_{j}}^{u}\frac{dz}{{\omega }_{j}\left({\phi }^{-1}\left(z\right)\right)}$, ${r}_{1}\left(x,y\right)$ and ${\stackrel{˜}{f}}_{n}\left(x,y,s,t\right)$ are given in Theorem  2.1, ${r}_{i}\left(x,y\right)$ is defined as follows:

$\begin{array}{rl}{r}_{i}\left(x,y\right)=& {r}_{1}\left(x,y\right)+\sum _{k=1}^{i-1}\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{k-1}\right)}^{{b}_{n}\left({x}_{k}\right)}{\int }_{{c}_{n}\left({y}_{k-1}\right)}^{{c}_{n}\left({y}_{k}\right)}{f}_{n}\left(x,y,s,t\right){\omega }_{n}\left({u}_{k}\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +g\left(x,y\right)\sum _{k=1}^{i-1}{\beta }_{k}\psi \left({u}_{k}\left({x}_{k}-0,{y}_{k}-0\right)\right).\end{array}$

Proof Let $\phi \left(u\left(x,y\right)\right)=h\left(x,y\right)$. Since the function φ is strictly increasing on $\left[0,\mathrm{\infty }\right)$, its inverse ${\phi }^{-1}$ is well defined. Equation (2.5) becomes

$\begin{array}{rl}h\left(x,y\right)\le & a\left(x,y\right)+\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{0}\right)}^{{b}_{n}\left(x\right)}{\int }_{{c}_{n}\left({y}_{o}\right)}^{{c}_{n}\left(y\right)}{f}_{n}\left(x,y,s,t\right){\omega }_{n}\left({\phi }^{-1}\left(h\left(s,t\right)\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +g\left(x,y\right)\sum _{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}{\beta }_{i}\psi \left({\phi }^{-1}\left(h\left({x}_{i}-0,{y}_{i}-0\right)\right)\right).\end{array}$
(2.7)

Let ${\stackrel{˜}{\omega }}_{n}={\omega }_{n}\circ {\phi }^{-1}$ and $\stackrel{˜}{\psi }=\psi \circ {\phi }^{-1}$. Equation (2.7) becomes

$\begin{array}{rl}h\left(x,y\right)\le & a\left(x,y\right)+\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{0}\right)}^{{b}_{n}\left(x\right)}{\int }_{{c}_{n}\left({y}_{o}\right)}^{{c}_{n}\left(y\right)}{f}_{n}\left(x,y,s,t\right){\stackrel{˜}{\omega }}_{n}\left(h\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +g\left(x,y\right)\sum _{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}{\beta }_{i}\stackrel{˜}{\psi }\left(h\left({x}_{i}-0,{y}_{i}-0\right)\right).\end{array}$
(2.8)

It is easy to see that $\stackrel{˜}{\psi }\left(u\right)>0$, ${\stackrel{˜}{\omega }}_{1}\left(u\right)$ and ${\stackrel{˜}{\omega }}_{2}\left(u\right)$ are continuous and nonnegative functions on $\left[0,\mathrm{\infty }\right)$, and $\frac{{\stackrel{˜}{\omega }}_{2}\left(u\right)}{{\stackrel{˜}{\omega }}_{1}\left(u\right)}$ is nondecreasing on $\left(0,\mathrm{\infty }\right)$. Even though $\stackrel{˜}{\psi }\left(u\right)$ is much more general, in the same way as in Theorem 2.1, for $\left(x,y\right)\in {\mathrm{\Omega }}_{ii}$, $i=1,2,\dots$ , we can obtain the estimate of $u\left(x,y\right)$,

$\begin{array}{rl}{u}_{i}\left(x,y\right)\le & {\phi }^{-1}\left\{{W}_{2}^{-1}\left[{W}_{2}\circ {W}_{1}^{-1}\left({W}_{1}\left({r}_{i}\left(x,y\right)\right)+{\int }_{{b}_{1}\left({x}_{i-1}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{i-1}\right)}^{{c}_{1}\left(y\right)}\stackrel{˜}{{f}_{1}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right)\\ +{\int }_{{b}_{2}\left({x}_{i-1}\right)}^{{b}_{2}\left(x\right)}{\int }_{{c}_{2}\left({y}_{i-1}\right)}^{{c}_{2}\left(y\right)}\stackrel{˜}{{f}_{2}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right]\right\}.\end{array}$
(2.9)

This completes the proof of Corollary 2.1. □

If $\phi \left(u\right)={u}^{\lambda }$ where $\lambda >0$ is a constant, we can study the inequality

$\begin{array}{rl}{u}^{\lambda }\left(x,y\right)\le & a\left(x,y\right)+\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{0}\right)}^{{b}_{n}\left(x\right)}{\int }_{{c}_{n}\left({y}_{o}\right)}^{{c}_{n}\left(y\right)}{f}_{n}\left(x,y,s,t\right){\omega }_{n}\left(u\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +g\left(x,y\right)\sum _{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}{\beta }_{i}\psi \left(u\left({x}_{i}-0,{y}_{i}-0\right)\right).\end{array}$
(2.10)

According to Corollary 2.1, we have the following result.

Corollary 2.2 Suppose that (H k ) ($k=1,\dots ,6$) hold, $\psi \left(u\right)>0$, and $u\left(x,y\right)$ satisfies (2.10). If we let ${u}_{i}\left(x,y\right)=u\left(x,y\right)$ for $\left(x,y\right)\in {\mathrm{\Omega }}_{ii}$, then the estimate of $u\left(x,y\right)$ is recursively given, for $\left(x,y\right)\in {\mathrm{\Omega }}_{ii}$, $i=1,2,\dots$ , by

$\begin{array}{rl}{u}_{i}\left(x,y\right)\le & \left\{{W}_{2}^{-1}\left[{W}_{2}\circ {W}_{1}^{-1}\left({W}_{1}\left({r}_{i}\left(x,y\right)\right)+{\int }_{{b}_{1}\left({x}_{i-1}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{i-1}\right)}^{{c}_{1}\left(y\right)}\stackrel{˜}{{f}_{1}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right)\\ +{\int }_{{b}_{2}\left({x}_{i-1}\right)}^{{b}_{2}\left(x\right)}{\int }_{{c}_{2}\left({y}_{i-1}\right)}^{{c}_{2}\left(y\right)}\stackrel{˜}{{f}_{2}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right]{\right\}}^{\frac{1}{\lambda }},\end{array}$
(2.11)

where ${W}_{j}\left(u\right)={\int }_{{\stackrel{˜}{u}}_{j}}^{u}\frac{dz}{\omega \left({z}^{\frac{1}{\lambda }}\right)}$, ${r}_{1}\left(x,y\right)$, ${r}_{i}\left(x,y\right)$ and ${\stackrel{˜}{f}}_{n}\left(x,y,s,t\right)$ are given in Corollary  2.1.

3 Proof of Theorem 2.1

Obviously, for any $\left(x,y\right)\in \mathrm{\Omega }$, ${r}_{1}\left(x,y\right)$ is positive and nondecreasing with respect to x and y, ${\stackrel{˜}{f}}_{n}\left(x,y,s,t\right)$ ($n=1,2$) is nonnegative and nondecreasing with respect to x and y for each fixed s and t. They satisfy $a\left(x,y\right)\le {r}_{1}\left(x,y\right)$ and ${f}_{n}\left(x,y,s,t\right)\le {\stackrel{˜}{f}}_{n}\left(x,y,s,t\right)$.

We first consider $\left(x,y\right)\in {\mathrm{\Omega }}_{11}=\left\{\left(x,y\right):{x}_{0}\le x<{x}_{1},{y}_{0}\le y<{y}_{1}\right\}$ and have from (1.8)

$\begin{array}{rl}u\left(x,y\right)& \le a\left(x,y\right)+\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{0}\right)}^{{b}_{n}\left(x\right)}{\int }_{{c}_{n}\left({y}_{o}\right)}^{{c}_{n}\left(y\right)}{f}_{n}\left(x,y,s,t\right){\omega }_{n}\left(u\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ \le {r}_{1}\left(x,y\right)+\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{0}\right)}^{{b}_{n}\left(x\right)}{\int }_{{c}_{n}\left({y}_{o}\right)}^{{c}_{n}\left(y\right)}{\stackrel{˜}{f}}_{n}\left(x,y,s,t\right){\omega }_{n}\left(u\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt.\end{array}$
(3.1)

Take any fixed $\stackrel{˜}{x}\in \left({x}_{0},{x}_{1}\right)$, $\stackrel{˜}{y}\in \left({y}_{0},{y}_{1}\right)$, and for arbitrary $x\in \left[{x}_{0},\stackrel{˜}{x}\right]$, $y\in \left[{y}_{0},\stackrel{˜}{y}\right]$ we have

$u\left(x,y\right)\le {r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)+\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{0}\right)}^{{b}_{n}\left(x\right)}{\int }_{{c}_{n}\left({y}_{o}\right)}^{{c}_{n}\left(y\right)}{\stackrel{˜}{f}}_{n}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right){\omega }_{n}\left(u\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt.$
(3.2)

Let

$z\left(x,y\right)={r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)+\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{0}\right)}^{{b}_{n}\left(x\right)}{\int }_{{c}_{n}\left({y}_{o}\right)}^{{c}_{n}\left(y\right)}{\stackrel{˜}{f}}_{n}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right){\omega }_{n}\left(u\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt$
(3.3)

and $z\left({x}_{0},y\right)={r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)$. Hence, $u\left(x,y\right)\le z\left(x,y\right)$. Clearly, $z\left(x,y\right)$ is a nonnegative, nondecreasing and differentiable function for $x\in \left[{x}_{0},\stackrel{˜}{x}\right]$ and $y\in \left[{y}_{0},\stackrel{˜}{y}\right]$. Moreover, ${b}_{n}\left(x\right)$ (or ${c}_{n}\left(y\right)$) is differentiable and nondecreasing in $x\in \left[{x}_{0},\stackrel{˜}{x}\right]$ (or $y\in \left[{y}_{0},\stackrel{˜}{y}\right]$) for $n=1,2$. Thus, ${b}_{n}^{\prime }\left(x\right)\ge 0$ (or ${c}_{n}^{\prime }\left(y\right)\ge 0$) for $x\in \left[{x}_{0},\stackrel{˜}{x}\right]$ (or $y\in \left[{y}_{0},\stackrel{˜}{y}\right]$). Since ${r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)>0$ and ${\omega }_{1}\left(z\left(x,y\right)\right)>0$, we have

$\begin{array}{rl}\frac{{D}_{1}z\left(x,y\right)}{{\omega }_{1}\left(z\left(x,y\right)\right)}\le & \frac{{\int }_{{c}_{1}\left({y}_{o}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{1}\left(x\right),t\right){b}_{1}^{\prime }\left(x\right){\omega }_{1}\left(u\left({b}_{1}\left(x\right),t\right)\right)\phantom{\rule{0.2em}{0ex}}dt}{{\omega }_{1}\left(z\left(x,y\right)\right)}\\ +\frac{{\int }_{{c}_{2}\left({y}_{o}\right)}^{{c}_{2}\left(y\right)}{\stackrel{˜}{f}}_{2}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{2}\left(x\right),t\right){b}_{2}^{\prime }\left(x\right){\omega }_{2}\left(u\left({b}_{2}\left(x\right),t\right)\right)\phantom{\rule{0.2em}{0ex}}dt}{{\omega }_{1}\left(z\left(x,y\right)\right)}\\ \le & \frac{{\int }_{{c}_{1}\left({y}_{o}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{1}\left(x\right),t\right){b}_{1}^{\prime }\left(x\right){\omega }_{1}\left(z\left({b}_{1}\left(x\right),t\right)\right)\phantom{\rule{0.2em}{0ex}}dt}{{\omega }_{1}\left(z\left(x,y\right)\right)}\\ +\frac{{\int }_{{c}_{2}\left({y}_{o}\right)}^{{c}_{2}\left(y\right)}{\stackrel{˜}{f}}_{2}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{2}\left(x\right),t\right){b}_{2}^{\prime }\left(x\right){\omega }_{2}\left(z\left({b}_{2}\left(x\right),t\right)\right)\phantom{\rule{0.2em}{0ex}}dt}{{\omega }_{1}\left(z\left(x,y\right)\right)}\\ \le & \frac{{\int }_{{c}_{1}\left({y}_{o}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{1}\left(x\right),t\right){b}_{1}^{\prime }\left(x\right){\omega }_{1}\left(z\left(x,t\right)\right)\phantom{\rule{0.2em}{0ex}}dt}{{\omega }_{1}\left(z\left(x,y\right)\right)}\\ +\frac{{\int }_{{c}_{2}\left({y}_{o}\right)}^{{c}_{2}\left(y\right)}{\stackrel{˜}{f}}_{2}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{2}\left(x\right),t\right){b}_{2}^{\prime }\left(x\right){\omega }_{2}\left(z\left({b}_{2}\left(x\right),t\right)\right)\phantom{\rule{0.2em}{0ex}}dt}{{\omega }_{1}\left(z\left(x,y\right)\right)}\\ \le & {\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{1}\left(x\right),t\right){b}_{1}^{\prime }\left(x\right)\phantom{\rule{0.2em}{0ex}}dt\\ +{\int }_{{c}_{2}\left({y}_{0}\right)}^{{c}_{2}\left(y\right)}{\stackrel{˜}{f}}_{2}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{2}\left(x\right),t\right){b}_{2}^{\prime }\left(x\right)\frac{{\omega }_{2}\left(z\left({b}_{2}\left(x\right),t\right)\right)}{{\omega }_{1}\left(z\left({b}_{2}\left(x\right),t\right)\right)}\phantom{\rule{0.2em}{0ex}}dt.\end{array}$
(3.4)

Integrating both sides of the above inequality from ${x}_{0}$ to x, we obtain

$\begin{array}{rl}{W}_{1}\left(z\left(x,y\right)\right)-{W}_{1}\left(z\left({x}_{0},y\right)\right)\le & {\int }_{{x}_{0}}^{x}{\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{1}\left(s\right),t\right){b}_{1}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +{\int }_{{x}_{0}}^{x}{\int }_{{c}_{2}\left({y}_{0}\right)}^{{c}_{2}\left(y\right)}{\stackrel{˜}{f}}_{2}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{2}\left(s\right),t\right){b}_{2}^{\prime }\left(s\right)\frac{{\omega }_{2}\left(z\left({b}_{2}\left(s\right),t\right)\right)}{{\omega }_{1}\left(z\left({b}_{2}\left(s\right),t\right)\right)}\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt.\end{array}$

Thus,

$\begin{array}{rl}{W}_{1}\left(z\left(x,y\right)\right)\le & {W}_{1}\left({r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\right)+{\int }_{{b}_{1}\left({x}_{0}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +{\int }_{{b}_{2}\left({x}_{0}\right)}^{{b}_{2}\left(x\right)}{\int }_{{c}_{2}\left({y}_{0}\right)}^{{c}_{2}\left(y\right)}{\stackrel{˜}{f}}_{2}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right)\varphi \left(z\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\end{array}$

for ${x}_{0}\le x\le \stackrel{˜}{x}$ and ${y}_{0}\le y\le \stackrel{˜}{y}$, where $\varphi \left(u\right)=\frac{{\omega }_{2}\left(u\right)}{{\omega }_{1}\left(u\right)}$, or equivalently

$\begin{array}{rl}\xi \left(x,y\right)\le & {W}_{1}\left({r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\right)+{\int }_{{b}_{1}\left({x}_{0}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +{\int }_{{b}_{2}\left({x}_{0}\right)}^{{b}_{2}\left(x\right)}{\int }_{{c}_{2}\left({y}_{0}\right)}^{{c}_{2}\left(y\right)}{\stackrel{˜}{f}}_{2}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right)\varphi \left({W}_{1}^{-1}\left(\xi \left(s,t\right)\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\triangleq {z}_{1}\left(x,y\right),\end{array}$
(3.5)

where

$\xi \left(x,y\right)={W}_{1}\left(z\left(x,y\right)\right).$

It is easy to check that $\xi \left(x,y\right)\le {z}_{1}\left(x,y\right)$, ${z}_{1}\left({x}_{0},y\right)={W}_{1}\left({r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\right)$ and ${z}_{1}\left(x,y\right)$ is differentiable, positive and nondecreasing on $\left({x}_{0},\stackrel{˜}{x}\right]$ and $\left({y}_{0},\stackrel{˜}{y}\right]$. Since $\varphi \left({W}_{1}^{-1}\left(u\right)\right)$ is nondecreasing from the assumption (H3), we have

$\begin{array}{rl}\frac{{D}_{1}{z}_{1}\left(x,y\right)}{\varphi \left({W}_{1}^{-1}\left({z}_{1}\left(x,y\right)\right)\right)}\le & \frac{{\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{1}\left(x\right),t\right){b}_{1}^{\prime }\left(x\right)\phantom{\rule{0.2em}{0ex}}dt}{\varphi \left({W}_{1}^{-1}\left({z}_{1}\left(x,y\right)\right)\right)}\\ +\frac{{\int }_{{c}_{2}\left({y}_{0}\right)}^{{c}_{2}\left(y\right)}{\stackrel{˜}{f}}_{2}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{2}\left(x\right),t\right){b}_{2}^{\prime }\left(x\right)\varphi \left({W}_{1}^{-1}\left(\xi \left({b}_{2}\left(x\right),t\right)\right)\right)\phantom{\rule{0.2em}{0ex}}dt}{\varphi \left({W}_{1}^{-1}\left({z}_{1}\left(x,y\right)\right)\right)}\\ \le & \frac{{\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{1}\left(x\right),t\right){b}_{1}^{\prime }\left(x\right)\phantom{\rule{0.2em}{0ex}}dt}{\varphi \left[{W}_{1}^{-1}\left({W}_{1}\left({r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\right)+{\int }_{{b}_{1}\left({x}_{0}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right)\right]}\\ +\frac{{\int }_{{c}_{2}\left({y}_{0}\right)}^{{c}_{2}\left(y\right)}{\stackrel{˜}{f}}_{2}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{2}\left(x\right),t\right){b}_{2}^{\prime }\left(x\right)\varphi \left({W}_{1}^{-1}\left({z}_{1}\left(x,t\right)\right)\right)\phantom{\rule{0.2em}{0ex}}dt}{\varphi \left({W}_{1}^{-1}\left({z}_{1}\left(x,y\right)\right)\right)}\\ \le & \frac{{\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{1}\left(x\right),t\right){b}_{1}^{\prime }\left(x\right)\phantom{\rule{0.2em}{0ex}}dt}{\varphi \left[{W}_{1}^{-1}\left({W}_{1}\left({r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\right)+{\int }_{{b}_{1}\left({x}_{0}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right)\right]}\\ +{\int }_{{c}_{2}\left({y}_{0}\right)}^{{c}_{2}\left(y\right)}{\stackrel{˜}{f}}_{2}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{2}\left(x\right),t\right){b}_{2}^{\prime }\left(x\right)\phantom{\rule{0.2em}{0ex}}dt.\end{array}$
(3.6)

Note that

$\begin{array}{rl}{\int }_{{x}_{0}}^{x}\frac{{D}_{1}{z}_{1}\left(s,y\right)}{\varphi \left({W}_{1}^{-1}\left({z}_{1}\left(s,y\right)\right)\right)}\phantom{\rule{0.2em}{0ex}}ds& ={\int }_{{x}_{0}}^{x}\frac{{D}_{1}{z}_{1}\left(s,y\right){\omega }_{1}\left({W}_{1}^{-1}\left({z}_{1}\left(s,y\right)\right)\right)}{{\omega }_{2}\left({W}_{1}^{-1}\left({z}_{1}\left(s,y\right)\right)\right)}\phantom{\rule{0.2em}{0ex}}ds={\int }_{{W}_{1}^{-1}\left({z}_{1}\left({x}_{0},y\right)\right)}^{{W}_{1}^{-1}\left({z}_{1}\left(x,y\right)\right)}\frac{du}{{\omega }_{2}\left(u\right)}\\ ={W}_{2}\circ {W}_{1}^{-1}\left({z}_{1}\left(x,y\right)\right)-{W}_{2}\circ {W}_{1}^{-1}\left({z}_{1}\left({x}_{0},y\right)\right)\\ ={W}_{2}\circ {W}_{1}^{-1}\left({z}_{1}\left(x,y\right)\right)-{W}_{2}\circ {W}_{1}^{-1}\left({W}_{1}\left({r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\right)\right)\\ ={W}_{2}\circ {W}_{1}^{-1}\left({z}_{1}\left(x,y\right)\right)-{W}_{2}\left({r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\right).\end{array}$

Integrating both sides of the inequality (3.6) from ${x}_{0}$ to x, we obtain

$\begin{array}{r}{W}_{2}\circ {W}_{1}^{-1}\left({z}_{1}\left(x,y\right)\right)-{W}_{2}\left({r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\right)\\ \phantom{\rule{1em}{0ex}}={\int }_{{x}_{0}}^{x}\frac{{D}_{1}{z}_{1}\left(s,y\right)}{\varphi \left({W}_{1}^{-1}\left({z}_{1}\left(s,y\right)\right)\right)}\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{1em}{0ex}}\le {\int }_{{x}_{0}}^{x}\frac{{\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{1}\left(s\right),t\right){b}_{1}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}dt}{\varphi \left[{W}_{1}^{-1}\left({W}_{1}\left({r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\right)+{\int }_{{b}_{1}\left({x}_{0}\right)}^{{b}_{1}\left(s\right)}{\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},\tau ,t\right)\phantom{\rule{0.2em}{0ex}}d\tau \phantom{\rule{0.2em}{0ex}}dt\right)\right]}\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+{\int }_{{x}_{0}}^{x}{\int }_{{c}_{2}\left({y}_{0}\right)}^{{c}_{2}\left(y\right)}{\stackrel{˜}{f}}_{2}\left(\stackrel{˜}{x},\stackrel{˜}{y},{b}_{2}\left(s\right),t\right){b}_{2}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ \phantom{\rule{1em}{0ex}}\le {W}_{2}\circ {W}_{1}^{-1}\left[{W}_{1}\left({r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\right)+{\int }_{{b}_{1}\left({x}_{0}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right]-{W}_{2}\left({r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\right)\\ \phantom{\rule{2em}{0ex}}+{\int }_{{b}_{2}\left({x}_{0}\right)}^{{b}_{2}\left(x\right)}{\int }_{{c}_{2}\left({y}_{0}\right)}^{{c}_{2}\left(y\right)}{\stackrel{˜}{f}}_{2}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt.\end{array}$

Thus,

$\begin{array}{rl}{W}_{2}\circ {W}_{1}^{-1}\left({z}_{1}\left(x,y\right)\right)\le & {W}_{2}\circ {W}_{1}^{-1}\left[{W}_{1}\left({r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\right)+{\int }_{{b}_{1}\left({x}_{0}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right]\\ +{\int }_{{b}_{2}\left({x}_{0}\right)}^{{b}_{2}\left(x\right)}{\int }_{{c}_{2}\left({y}_{0}\right)}^{{c}_{2}\left(y\right)}{\stackrel{˜}{f}}_{2}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt.\end{array}$

Hence

$\begin{array}{rl}u\left(x,y\right)\le & z\left(x,y\right)\le {W}_{1}^{-1}\left(\xi \left(x,y\right)\right)\le {W}_{1}^{-1}\left({z}_{1}\left(x,y\right)\right)\\ \le & {W}_{2}^{-1}\left[{W}_{2}\circ {W}_{1}^{-1}\left({W}_{1}\left({r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\right)+{\int }_{{b}_{1}\left({x}_{0}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right)\\ +{\int }_{{b}_{2}\left({x}_{0}\right)}^{{b}_{2}\left(x\right)}{\int }_{{c}_{2}\left({y}_{0}\right)}^{{c}_{2}\left(y\right)}{\stackrel{˜}{f}}_{2}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right].\end{array}$

Since the above inequality is true for any $x\in \left[{x}_{0},\stackrel{˜}{x}\right]$, $y\in \left[{y}_{0},\stackrel{˜}{y}\right]$, we obtain

$\begin{array}{rl}u\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\le & {W}_{2}^{-1}\left[{W}_{2}\circ {W}_{1}^{-1}\left({W}_{1}\left({r}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\right)+{\int }_{{b}_{1}\left({x}_{0}\right)}^{{b}_{1}\left(\stackrel{˜}{x}\right)}{\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(\stackrel{˜}{y}\right)}{\stackrel{˜}{f}}_{1}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right)\\ +{\int }_{{b}_{2}\left({x}_{0}\right)}^{{b}_{2}\left(\stackrel{˜}{x}\right)}{\int }_{{c}_{2}\left({y}_{0}\right)}^{{c}_{2}\left(\stackrel{˜}{y}\right)}{\stackrel{˜}{f}}_{2}\left(\stackrel{˜}{x},\stackrel{˜}{y},s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right].\end{array}$
(3.7)

Replacing $\stackrel{˜}{x}$ by x and $\stackrel{˜}{y}$ by y yields

$\begin{array}{rl}u\left(x,y\right)\le & {W}_{2}^{-1}\left[{W}_{2}\circ {W}_{1}^{-1}\left({W}_{1}\left({r}_{1}\left(x,y\right)\right)+{\int }_{{b}_{1}\left({x}_{0}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{0}\right)}^{{c}_{1}\left(y\right)}{\stackrel{˜}{f}}_{1}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right)\\ +{\int }_{{b}_{2}\left({x}_{0}\right)}^{{b}_{2}\left(x\right)}{\int }_{{c}_{2}\left({y}_{0}\right)}^{{c}_{2}\left(y\right)}{\stackrel{˜}{f}}_{2}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right].\end{array}$
(3.8)

This means that (2.1) is true for $\left(x,y\right)\in {\mathrm{\Omega }}_{11}$ and $i=1$ if replace $u\left(x,y\right)$ with ${u}_{1}\left(x,y\right)$.

For $i=2$ and $\left(x,y\right)\in {\mathrm{\Omega }}_{22}=\left\{\left(x,y\right):{x}_{1}\le x<{x}_{2},{y}_{1}\le y<{y}_{2}\right\}$, (1.8) becomes

$\begin{array}{rl}u\left(x,y\right)\le & {r}_{1}\left(x,y\right)+\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{0}\right)}^{{b}_{n}\left({x}_{1}\right)}{\int }_{{c}_{n}\left({y}_{o}\right)}^{{c}_{n}\left({y}_{1}\right)}{f}_{n}\left(x,y,s,t\right){\omega }_{n}\left({u}_{1}\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +g\left(x,y\right){\beta }_{1}{u}_{1}^{m}\left({x}_{1}-0,{y}_{1}-0\right)\\ +\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{1}\right)}^{{b}_{n}\left(x\right)}{\int }_{{c}_{n}\left({y}_{1}\right)}^{{c}_{n}\left(y\right)}{f}_{n}\left(x,y,s,t\right){\omega }_{n}\left(u\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ \le & {r}_{2}\left(x,y\right)+\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{1}\right)}^{{b}_{n}\left(x\right)}{\int }_{{c}_{n}\left({y}_{1}\right)}^{{c}_{n}\left(y\right)}{f}_{n}\left(x,y,s,t\right){\omega }_{n}\left(u\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt,\end{array}$
(3.9)

where the definition of ${r}_{2}\left(x,y\right)$ is given in (2.2). Note that the estimate of ${u}_{1}\left(x,y\right)$ is known. Clearly, (3.9) is the same as (3.1) if replace ${r}_{1}\left(x,y\right)$ and $\left({x}_{0},{y}_{0}\right)$ by ${r}_{2}\left(x,y\right)$ and $\left({x}_{1},{y}_{1}\right)$. Thus, by (3.8) we have

$\begin{array}{rl}u\left(x,y\right)\le & {W}_{2}^{-1}\left[{W}_{2}\circ {W}_{1}^{-1}\left({W}_{1}\left({r}_{2}\left(x,y\right)\right)+{\int }_{{b}_{1}\left({x}_{1}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{1}\right)}^{{c}_{1}\left(y\right)}\stackrel{˜}{{f}_{1}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right)\\ +{\int }_{{b}_{2}\left({x}_{1}\right)}^{{b}_{2}\left(x\right)}{\int }_{{c}_{2}\left({y}_{1}\right)}^{{c}_{2}\left(y\right)}\stackrel{˜}{{f}_{2}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right].\end{array}$

This implies that (2.1) is true for $\left(x,y\right)\in {\mathrm{\Omega }}_{22}$ and $i=2$ if replace $u\left(x,y\right)$ by ${u}_{2}\left(x,y\right)$.

Assume that (2.1) is true for $\left(x,y\right)\in {\mathrm{\Omega }}_{ii}=\left\{\left(x,y\right):{x}_{i-1}\le x<{x}_{i},{y}_{i-1}\le y<{y}_{i}\right\}$, i.e.,

$\begin{array}{rl}{u}_{i}\left(x,y\right)\le & {W}_{2}^{-1}\left[{W}_{2}\circ {W}_{1}^{-1}\left({W}_{1}\left({r}_{i}\left(x,y\right)\right)+{\int }_{{b}_{1}\left({x}_{i-1}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{i-1}\right)}^{{c}_{1}\left(y\right)}\stackrel{˜}{{f}_{1}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right)\\ +{\int }_{{b}_{2}\left({x}_{i-1}\right)}^{{b}_{2}\left(x\right)}{\int }_{{c}_{2}\left({y}_{i-1}\right)}^{{c}_{2}\left(y\right)}\stackrel{˜}{{f}_{2}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right].\end{array}$
(3.10)

For $\left(x,y\right)\in {\mathrm{\Omega }}_{i+1,i+1}=\left\{\left(x,y\right):{x}_{i}\le x<{x}_{i+1},{y}_{i}\le y<{y}_{i+1}\right\}$, (1.8) becomes

$\begin{array}{rl}u\left(x,y\right)\le & a\left(x,y\right)+\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{0}\right)}^{{b}_{n}\left(x\right)}{\int }_{{c}_{n}\left({y}_{o}\right)}^{{c}_{n}\left(y\right)}{f}_{n}\left(x,y,s,t\right){\omega }_{n}\left(u\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +g\left(x,y\right)\sum _{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}{\beta }_{i}{u}^{m}\left({x}_{i}-0,{y}_{i}-0\right)\\ \le & {r}_{1}\left(x,y\right)+\sum _{k=1}^{i}\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{k-1}\right)}^{{b}_{n}\left({x}_{k}\right)}{\int }_{{c}_{n}\left({y}_{k-1}\right)}^{{c}_{n}\left({y}_{k\right)}}{f}_{n}\left(x,y,s,t\right){\omega }_{n}\left({u}_{k}\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +g\left(x,y\right)\sum _{k=1}^{i}{\beta }_{k}{u}_{k}^{m}\left({x}_{k}-0,{y}_{k}-0\right)\\ +\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{i}\right)}^{{b}_{n}\left(x\right)}{\int }_{{c}_{n}\left({y}_{i}\right)}^{{c}_{n}\left(y\right)}{f}_{n}\left(x,y,s,t\right){\omega }_{n}\left(u\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ \le & {r}_{i+1}\left(x,y\right)+\sum _{n=1}^{2}{\int }_{{b}_{n}\left({x}_{i}\right)}^{{b}_{n}\left(x\right)}{\int }_{{c}_{n}\left({y}_{i}\right)}^{{c}_{n}\left(y\right)}{f}_{n}\left(x,y,s,t\right){\omega }_{n}\left(u\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt,\end{array}$
(3.11)

where we use the fact that the estimate of $u\left(x,y\right)$ is already known for $\left(x,y\right)\in {\mathrm{\Omega }}_{ii}$ ($i=1,2,\dots$). Again, (3.11) is the same as (3.1) if replace ${r}_{1}\left(x,y\right)$ and $\left({x}_{0},{y}_{0}\right)$ by ${r}_{i+1}\left(x,y\right)$ and $\left({x}_{i},{y}_{i}\right)$. Thus, by (3.8) we have

$\begin{array}{rl}u\left(x,y\right)\le & {W}_{2}^{-1}\left[{W}_{2}\circ {W}_{1}^{-1}\left({W}_{1}\left({r}_{i+1}\left(x,y\right)\right)+{\int }_{{b}_{1}\left({x}_{i}\right)}^{{b}_{1}\left(x\right)}{\int }_{{c}_{1}\left({y}_{i}\right)}^{{c}_{1}\left(y\right)}\stackrel{˜}{{f}_{1}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right)\\ +{\int }_{{b}_{2}\left({x}_{i}\right)}^{{b}_{2}\left(x\right)}{\int }_{{c}_{2}\left({y}_{i}\right)}^{{c}_{2}\left(y\right)}\stackrel{˜}{{f}_{2}}\left(x,y,s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right].\end{array}$

This shows that (2.1) is true for $\left(x,y\right)\in {\mathrm{\Omega }}_{i+1,i+1}$ if replace $u\left(x,y\right)$ by ${u}_{i+1}\left(x,y\right)$. By induction, we know that (2.1) holds for $\left(x,y\right)\in {\mathrm{\Omega }}_{i+1,i+1}$ for any nonnegative integer i. This completes the proof of Theorem 2.1.

4 Applications

Consider the following partial differential equation with an impulsive term:

$\left\{\begin{array}{l}\frac{{\partial }^{2}v\left(x,y\right)}{\partial x\partial y}=H\left(x,y,v\left(x,y\right)\right),\phantom{\rule{1em}{0ex}}\left(x,y\right)\in {\mathrm{\Omega }}_{ii},x\ne {x}_{i},y\ne {y}_{i},\\ △v{|}_{x={x}_{i},y={y}_{i}}={I}_{i}\left(v\right),\\ v\left(x,{y}_{0}\right)={\varphi }_{1}\left(x\right),\phantom{\rule{2em}{0ex}}v\left({x}_{0},y\right)={\varphi }_{2}\left(y\right),\phantom{\rule{2em}{0ex}}{\varphi }_{1}\left({x}_{0}\right)={\varphi }_{2}\left({y}_{0}\right)\ne 0,\end{array}$
(4.1)

where $v\in \mathbf{R}$, $H\in \mathbf{R}$, ${I}_{i}\in \mathbf{R}$, and $i=1,2,\dots$ .

Assume that

(C1) $|H\left(x,y,v\left(x,y\right)\right)|\le {h}_{1}\left(x,y\right){e}^{|v\left(x,y\right)|}+{h}_{2}\left(x,y\right){e}^{2|v\left(x,y\right)|}$ where ${h}_{1}$, ${h}_{2}$ are nonnegative and continuous on Ω, ${h}_{1}\left(x,y\right)=0$, ${h}_{2}\left(x,y\right)=0$ for $\left(x,y\right)\in {\mathrm{\Omega }}_{ij}$, $i\ne j$, $i,j=1,2,\dots$ ;

(C2) $|{I}_{i}\left(v\right)|\le {\beta }_{i}{|v|}^{m}$ where ${\beta }_{i}$ and m are nonnegative constants.

Corollary 4.1 Suppose that (C1) and (C2) hold. If we let ${v}_{i}\left(x,y\right)=v\left(x,y\right)$ for $\left(x,y\right)\in {\mathrm{\Omega }}_{ii}$, then the solution of system (4.1) has an estimate for $\left(x,y\right)\in {\mathrm{\Omega }}_{ii}$

$\begin{array}{rl}|{v}_{i}\left(x,y\right)|\le & -\frac{1}{2}ln\left[{\left({e}^{-{r}_{i}\left(x,y\right)}-{\int }_{{x}_{i-1}}^{x}{\int }_{{y}_{i-1}}^{y}{h}_{1}\left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right)}^{2}\\ -{\int }_{{x}_{i-1}}^{x}{\int }_{{y}_{i-1}}^{y}{h}_{2}\left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right],\end{array}$
(4.2)

where

$\begin{array}{c}{r}_{1}\left(x,y\right)=\underset{{x}_{0}\le \xi \le x,{y}_{0}\le \eta \le y}{max}|{\varphi }_{1}\left(\xi \right)+{\varphi }_{2}\left(\eta \right)-{\varphi }_{1}\left({x}_{0}\right)|>0,\hfill \\ \begin{array}{rl}{r}_{i}\left(x,y\right)=& {r}_{1}\left(x,y\right)+\sum _{k=1}^{i-1}{\int }_{{x}_{k-1}}^{{x}_{k}}{\int }_{{y}_{k-1}}^{{y}_{k}}{h}_{1}\left(s,t\right){e}^{|{v}_{k}\left(s,t\right)|}\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +\sum _{k=1}^{i-1}{\int }_{{x}_{k-1}}^{{x}_{k}}{\int }_{{y}_{k-1}}^{{y}_{k}}{h}_{2}\left(s,t\right){e}^{2|{v}_{k}\left(s,t\right)|}\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt+\sum _{k=1}^{i-1}{\beta }_{k}{|{v}_{k}\left({x}_{k}-0,{y}_{k}-0\right)|}^{m},\end{array}\hfill \\ {\left({e}^{-{r}_{i}\left(x,y\right)}-{\int }_{{x}_{i-1}}^{x}{\int }_{{y}_{i-1}}^{y}{h}_{1}\left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right)}^{2}-{\int }_{{x}_{i-1}}^{x}{\int }_{{y}_{i-1}}^{y}{h}_{2}\left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt>0.\hfill \end{array}$

Proof The solution of (4.1) with an initial value is given by

$\begin{array}{rl}v\left(x,y\right)=& v\left(x,{y}_{0}\right)+v\left({x}_{0},y\right)-v\left({x}_{0},{y}_{0}\right)+{\int }_{{x}_{0}}^{x}{\int }_{{y}_{0}}^{y}H\left(s,t,v\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +\sum _{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}{I}_{i}\left(v\left({x}_{i}-0,{y}_{i}-0\right)\right)\\ =& {\varphi }_{1}\left(x\right)+{\varphi }_{2}\left(y\right)-{\varphi }_{1}\left({x}_{0}\right)+{\int }_{{x}_{0}}^{x}{\int }_{{y}_{0}}^{y}H\left(s,t,v\left(s,t\right)\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +\sum _{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}{I}_{i}\left(v\left({x}_{i}-0,{y}_{i}-0\right)\right),\end{array}$
(4.3)

which implies

$\begin{array}{rl}|v\left(x,y\right)|\le & |{\varphi }_{1}\left(x\right)+{\varphi }_{2}\left(y\right)-{\varphi }_{1}\left({x}_{0}\right)|+{\int }_{{x}_{0}}^{x}{\int }_{{y}_{0}}^{y}{h}_{1}\left(s,t\right){e}^{|v\left(s,t\right)|}\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +{\int }_{{x}_{0}}^{x}{\int }_{{y}_{0}}^{y}{h}_{2}\left(s,t\right){e}^{2|v\left(s,t\right)|}\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt+\sum _{\left({x}_{0},{y}_{0}\right)<\left({x}_{i},{y}_{i}\right)<\left(x,y\right)}{\beta }_{i}{|v\left({x}_{i}-0,{y}_{i}-0\right)|}^{m}.\end{array}$
(4.4)

Let

$\begin{array}{c}u\left(x,y\right)=|v\left(x,y\right)|,\phantom{\rule{2em}{0ex}}a\left(x,y\right)=|{\varphi }_{1}\left(x\right)+{\varphi }_{2}\left(y\right)-{\varphi }_{1}\left({x}_{0}\right)|,\phantom{\rule{2em}{0ex}}{b}_{1}\left(x\right)={b}_{2}\left(x\right)=x,\hfill \\ {c}_{1}\left(y\right)={c}_{2}\left(y\right)=y,\phantom{\rule{2em}{0ex}}g\left(x,y\right)=1,\phantom{\rule{2em}{0ex}}{\omega }_{1}\left(u\right)={e}^{u},\phantom{\rule{2em}{0ex}}{\omega }_{2}\left(u\right)={e}^{2u},\hfill \\ {f}_{1}\left(x,y,s,t\right)={h}_{1}\left(s,t\right),\phantom{\rule{2em}{0ex}}{f}_{2}\left(x,y,s,t\right)={h}_{2}\left(s,t\right).\hfill \end{array}$

Thus, (4.4) is the same as (1.8). It is easy to see that for any positive constants ${\stackrel{˜}{u}}_{1}$ and ${\stackrel{˜}{u}}_{2}$

$\begin{array}{c}{r}_{1}\left(x,y\right)=\underset{{x}_{0}\le \xi \le x,{y}_{0}\le \eta \le y}{max}|a\left(\xi ,\eta \right)|>0,\hfill \\ {\stackrel{˜}{f}}_{1}\left(x,y,s,t\right)={h}_{1}\left(s,t\right),\phantom{\rule{2em}{0ex}}{\stackrel{˜}{f}}_{2}\left(x,y,s,t\right)={h}_{2}\left(s,t\right),\hfill \\ {W}_{1}\left(u\right)={\int }_{{\stackrel{˜}{u}}_{1}}^{u}\frac{dz}{{\omega }_{1}\left(z\right)}={\int }_{{\stackrel{˜}{u}}_{1}}^{u}{e}^{-z}\phantom{\rule{0.2em}{0ex}}dz={e}^{-{\stackrel{˜}{u}}_{1}}-{e}^{-u},\phantom{\rule{2em}{0ex}}{W}_{1}^{-1}\left(u\right)=-ln\left({e}^{-{\stackrel{˜}{u}}_{1}}-u\right),\hfill \\ {W}_{2}\left(u\right)={\int }_{{\stackrel{˜}{u}}_{2}}^{u}\frac{dz}{{\omega }_{2}\left(z\right)}={\int }_{{\stackrel{˜}{u}}_{2}}^{u}{e}^{-2z}\phantom{\rule{0.2em}{0ex}}dz=\frac{1}{2}\left({e}^{-2{\stackrel{˜}{u}}_{2}}-{e}^{-2u}\right),\hfill \\ {W}_{2}^{-1}\left(u\right)=-\frac{1}{2}ln\left({e}^{-2{\stackrel{˜}{u}}_{2}}-2u\right),\hfill \\ \begin{array}{rl}{r}_{i}\left(x,y\right)=& {r}_{1}\left(x,y\right)+\sum _{k=1}^{i-1}{\int }_{{x}_{k-1}}^{{x}_{k}}{\int }_{{y}_{k-1}}^{{y}_{k}}{h}_{1}\left(s,t\right){e}^{|{v}_{k}\left(s,t\right)|}\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\\ +\sum _{k=1}^{i-1}{\int }_{{x}_{k-1}}^{{x}_{k}}{\int }_{{y}_{k-1}}^{{y}_{k}}{h}_{2}\left(s,t\right){e}^{2|{v}_{k}\left(s,t\right)|}\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt+\sum _{k=1}^{i-1}{\beta }_{k}{|{v}_{k}\left({x}_{k}-0,{y}_{k}-0\right)|}^{m}.\end{array}\hfill \end{array}$

Therefore, for any nonnegative i and $\left(x,y\right)\in {\mathrm{\Omega }}_{ii}$

$|{v}_{i}\left(x,y\right)|\le -\frac{1}{2}ln\left[{\left({e}^{-{r}_{i}\left(x,y\right)}-{\int }_{{x}_{i-1}}^{x}{\int }_{{y}_{i-1}}^{y}{h}_{1}\left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right)}^{2}-{\int }_{{x}_{i-1}}^{x}{\int }_{{y}_{i-1}}^{y}{h}_{2}\left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right]$

provided that

${\left({e}^{-{r}_{i}\left(x,y\right)}-{\int }_{{x}_{i-1}}^{x}{\int }_{{y}_{i-1}}^{y}{h}_{1}\left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\right)}^{2}-{\int }_{{x}_{i-1}}^{x}{\int }_{{y}_{i-1}}^{y}{h}_{2}\left(s,t\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt>0.$

□

Remark 4.1 From (4.4), we know ${\omega }_{1}\left(u\right)={e}^{u}$. Clearly, ${\omega }_{1}\left(2u\right)={e}^{2u}\le {\omega }_{1}\left(2\right){\omega }_{1}\left(u\right)={e}^{2}{e}^{u}$ does not hold for large $u>0$. Thus, ${\omega }_{1}\left(u\right)={e}^{u}$ does not belong to the class ℘ in [24]. Again ${\omega }_{1}\left(\frac{u}{2}\right)={e}^{\frac{u}{2}}\ge \frac{1}{2}{\omega }_{1}\left(u\right)=\frac{1}{2}{e}^{u}$ does not hold for large $u>0$ so ${\omega }_{1}\left(u\right)$ does not belong to the class ȷ in [24]. Hence, the results in [24] cannot be applied to the inequality (4.1).