Some weakly singular Volterra integral inequalities with maxima in two variables

Abstract

In this paper, we establish some new weakly singular Volterra type integral inequalities that include the maxima of the unknown function of two variables. We also use the results to research the boundedness of solutions to retarded nonlinear Volterra type integral equations.

1 Introduction

Alongside mathematics development, inequalities have played increasingly important roles in theory and applications. Gronwall–Bellman inequality and Bihari inequality are highly prominent inequalities [1–3], which provided important tools to study the qualitative properties of differential equations, integral equations, and integro-differential equations such as existence, uniqueness, oscillation, stability, boundedness, invariant manifolds, and other properties. Over the past few decades, various researchers have worked on related issues, and a lot of research results have been obtained, including differential system, difference system, time-scale system [4–31]. In [7–12, 29], the Volterra–Fredholm type inequalities were examined. There have also been some results for integral inequalities containing the maxima of the unknown functions [12, 23–29]. In recent years, with the rising of fractional order calculation, the study on weak singular inequalities has become a hot topic [13–16, 31].

In 1997, Medved [4] discussed the following Henry type integral inequalities:

$$\begin{aligned} u(t)\leq a(t)+b(t) \int _{t_{0}}^{t}(t-s)^{\beta -1}s^{\gamma -1}F(s)u(s) \,ds,\quad t\geq 0. \end{aligned}$$

(1.1)

In 2008, Ma and Pec̆airé [13] investigated some new explicit bounds for weakly singular integral inequalities

$$\begin{aligned} \begin{aligned} u^{p}(t)\leq a(t)+b(t) \int _{t_{0}}^{t}\bigl(t^{\alpha}-s^{ \alpha} \bigr)^{\beta -1}s^{\gamma -1}f(s)u^{q}(s)\,ds,\quad t\geq 0. \end{aligned} \end{aligned}$$

(1.2)

In 2010, Wang and Zheng [31] investigated the nonlinear weakly singular integral inequalities with two variables

$$\begin{aligned} \begin{aligned} u(x,y)\leq{}& a(x,y)\\ &{}+ \int _{0}^{x} \int _{0}^{y}\bigl(x^{\alpha}-s^{ \alpha} \bigr)^{\beta -1}s^{\gamma -1}\bigl(y^{\alpha}-t^{\alpha} \bigr)^{\beta -1}t^{ \gamma -1}f(x,y,s,t)\omega \bigl(u(s,t)\bigr)\,ds\,dt. \end{aligned} \end{aligned}$$

(1.3)

In 2013, Yan [27] investigated the nonlinear Gronwall–Bellman type integral inequalities with maxima of two variables

$$\begin{aligned} &\begin{aligned}&\varphi \bigl(u(x,y)\bigr)\leq a(x,y)+ \sum^{n}_{i=1} \int _{ \alpha _{i}(x_{0})}^{\alpha _{i}(x)} \int _{\beta _{i}(y_{0})}^{\beta _{i}(y)}f_{i}(x,y,s,t) \omega _{i}\bigl(u(s,t)\bigr)\,dt\,ds \\ &\phantom{\varphi \bigl(u(x,y)\bigr)\leq}{}+\sum^{n+m}_{j=n+1} \int _{\alpha _{j}(x_{0})}^{\alpha _{j}(x)} \int _{ \beta _{j}(y_{0})}^{\beta _{j}(y)}f_{j}(x,y,s,t)\omega _{i} \Bigl( \max_{\xi \in [s-h,s]} g\bigl(u(\xi ,t)\bigr) \Bigr)\,dt\,ds,\\ &\quad (x,y)\in [x_{0},x_{1})\times [y_{0},y_{1}), \end{aligned} \\ &u(x,y)\leq \psi (x,y),\quad (x,y)\in \bigl[\alpha _{\ast}(x_{0})-h,x_{0} \bigr] \times [y_{0},y_{1}). \end{aligned}$$

(1.4)

In 2014, Thiramanus et al. [28] investigated the Henry–Gronwall integral inequalities with maxima

$$\begin{aligned} \begin{aligned} &u(t)\leq r(t)+ \int _{t_{0}}^{t}(t-s)^{\alpha -1} \Bigl[p(s)u(s)+q(s) \max_{\xi \in [\beta s,s]}u(\xi ) \Bigr]\,ds,\quad t\in [t_{0},T), \\ &u(t)\leq \phi (t),\quad t\in [\beta t_{0},t_{0}]. \end{aligned} \end{aligned}$$

(1.5)

In 2015, Yan [23] investigated some new weakly singular Volterra integral inequalities with maxima

$$\begin{aligned} \begin{aligned} & \varphi \bigl(u(t)\bigr)\leq a(t)+\sum ^{m}_{i=1} \int _{b_{i}(t_{0})}^{b_{i}(t)} \bigl(t^{\alpha _{i}}-s^{\alpha _{i}} \bigr)^{k_{i}(\beta _{i}-1)}s^{q_{i}( \gamma _{i}-1)}g_{i}(t,s)\omega _{i} \bigl(u(s)\bigr)\,ds \\ &\phantom{ \varphi \bigl(u(t)\bigr)\leq}{}+\sum^{m+n}_{j=m+1} \int _{b_{j}(t_{0})}^{b_{j}(t)} \bigl(t^{\alpha _{j}}-s^{ \alpha _{j}} \bigr)^{k_{j}(\beta _{j}-1)}s^{q_{j}(\gamma _{j}-1)}g_{j}(t,s) \\ &\omega _{j} \Bigl(\max_{\xi \in [c_{j}(s)-h,c_{j}(s)]}f\bigl(u(\xi )\bigr) \Bigr)\,ds, \quad t\in [t_{0},t_{1}), \\ &u(t)\leq \psi (t),\quad t\in \bigl[b^{\ast}(t_{0})-h,t_{0} \bigr].\end{aligned} \end{aligned}$$

(1.6)

In 2017, Xu and Ma [29] investigated some new retarded nonlinear Volterra–Fredholm type integral inequalities with maxima in two variables

$$\begin{aligned} & \varphi \bigl(u(x,y)\bigr)\leq k(x,y)+ \int _{\alpha (x)}^{\infty}a(s,y) \psi \bigl(u(s,y)\bigr)\,ds \\ &\phantom{\varphi \bigl(u(x,y)\bigr)\leq }{}+ \sum_{i=1}^{l_{1}} \int _{\alpha _{i}(x)}^{\infty} \int _{\beta _{i}(y)}^{\infty} \biggl[b_{i}(s,t,x,y) \varphi _{1}\bigl(u(s,t)\bigr) \\ &\phantom{\varphi \bigl(u(x,y)\bigr)\leq }{}+ \int _{s}^{\infty} \int _{t}^{\infty}c_{i}(\xi ,\eta ,x,y) \varphi _{2} \Bigl(\max_{\sigma \in [\xi ,h\xi ]}u(\sigma ,\eta ) \Bigr) \,d\xi \,d\eta \biggr]\,ds\,dt \end{aligned}$$

(1.7)

$$\begin{aligned} &\phantom{\varphi \bigl(u(x,y)\bigr)\leq }{}+\sum_{j=1}^{l_{2}} \int _{\alpha _{j}(M)}^{\infty} \int _{\beta _{j}(N)}^{ \infty} \biggl[b_{j}(s,t,x,y) \psi \bigl(u(s,t)\bigr) \\ &\phantom{\varphi \bigl(u(x,y)\bigr)\leq }{}+ \int _{s}^{\infty} \int _{t}^{\infty}e_{j}(\xi ,\eta ,x,y) \psi \Bigl(\max_{\sigma \in [\xi ,h\xi ]}u(\sigma ,\eta ) \Bigr)\,d \xi \,d\eta \biggr] \,ds\,dt,\quad (x,y)\in \Delta , \\ &\begin{aligned} u(x,y)^{p}\leq {}&k(x,y)+ \int _{\alpha (x)}^{\infty}a(s,y)u^{p}(s,y)\,ds+ \sum _{i=1}^{l_{1}} \int _{\alpha _{i}(x)}^{\infty} \int _{\beta _{i}(y)}^{ \infty} \biggl[b_{i}(s,t,x,y)u^{p}(s,t) \\ &{}+ \int _{s}^{\infty} \int _{t}^{\infty}c_{i}(\xi ,\eta ,x,y) \max _{\sigma \in [\xi ,h\xi ]}u^{r_{i}}(\sigma ,\eta )\,d\xi \,d\eta \biggr]\,ds \,dt \\ &{}+\sum_{j=1}^{l_{2}} \int _{\alpha _{j}(M)}^{\infty} \int _{\beta _{j}(N)}^{ \infty} \biggl[b_{j}(s,t,x,y)u^{\varepsilon _{j}}(s,t) \\ &{}+ \int _{s}^{\infty} \int _{t}^{\infty}e_{j}(\xi ,\eta ,x,y) \max _{\sigma \in [\xi ,h\xi ]}u^{\delta _{j}}(\sigma ,\eta )\,d\xi \,d \eta \biggr] \,ds\,dt,\quad (x,y)\in \Delta . \end{aligned} \end{aligned}$$

(1.8)

In this paper, we are concerned with the following weakly singular Volterra integral inequalities with maxima in two variables:

$$\begin{aligned} \begin{aligned} &\varphi \bigl(u(x,y)\bigr)\leq a(x,y)+ \sum_{i=1}^{m} \int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)}\bigl(x^{\alpha _{i}}-t^{\alpha _{i}} \bigr)^{k_{i}( \beta _{i}-1)}\\ &\phantom{\varphi \bigl(u(x,y)\bigr)\leq}{}\times t^{v_{i}(\gamma _{i}-1)}\bigl(y^{\alpha _{i}}-s^{\alpha _{i}} \bigr)^{k_{i}( \beta _{i}-1)}s^{v_{i}(\gamma _{i}-1)} \\ &\phantom{\varphi \bigl(u(x,y)\bigr)\leq}{}\times g_{i}(x,y,t,s)\omega _{i}\bigl(u(t,s)\bigr)\,dt\,ds \\ &\phantom{\varphi \bigl(u(x,y)\bigr)\leq}{}+\sum_{j=m+1}^{m+n} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)}\bigl(x^{ \alpha _{j}}-t^{\alpha _{j}} \bigr)^{k_{j}(\beta _{j}-1)} t^{v_{j}(\gamma _{j}-1)}\bigl(y^{ \alpha _{j}}-s^{\alpha _{j}} \bigr)^{k_{j}(\beta _{j}-1)}s^{v_{j}(\gamma _{j}-1)} \\ &\phantom{\varphi \bigl(u(x,y)\bigr)\leq}{}\times g_{j}(x,y,t,s)\omega _{j} \Bigl(\max _{(\xi ,\eta )\in [t-h,t] \times [s-k,s]}f\bigl(u(\xi ,\eta )\bigr) \Bigr)\,dt\,ds,\quad (x,y)\in \Omega , \\ &u(x,y)\leq \psi (x,y),\quad (x,y)\in \Omega _{0}, \end{aligned} \end{aligned}$$

(1.9)

where $\Omega =[x_{0},x_{1})\times [y_{0},y_{1}), \Omega _{0}=[\alpha _{\ast}(x_{0})-h,x_{0})\times [\beta _{\ast}(y_{0})-k,y_{1}) \cup [x_{0},x_{1})\times [\beta _{\ast}(y_{0})-k,y_{0})$.

2 Main results

In this section, we consider the integral inequality (1.9) with $x_{0}< x_{1}$ and $y_{0}< y_{1}$. First, we give the following conditions:

$(A_{1})$ $b_{i}(x):[x_{0},x_{1})\rightarrow [0,\infty )\ (i=1,2,\ldots ,m+n)$ and $c_{i}(y):[y_{0},y_{1})\rightarrow [y_{0},y_{1})\ (i=1,2,\ldots ,m+n)$ are differentiable continuously and nondecreasing such that $b_{i}(x)\leq x$ on $[x_{0},x_{1})$, $c_{i}(y)\leq y$ on $[y_{0},y_{1})$;

$(A_{2})$ All $g_{i}\ (i=1,2,\ldots ,m+n)$ are continuous nonnegative functions on $\Omega \times \Omega _{0}$;

$(A_{3})$ $f,\varphi :R_{+}\rightarrow R_{+}, \psi :\Omega _{0}\rightarrow R_{+}$ are continuous functions and φ is a strictly increasing function, $\lim_{t\rightarrow \infty}\varphi (t)=+\infty$;

$(A_{4})$ All $\omega _{i}:R_{+}\rightarrow R_{+}\ (i=1,2,\ldots ,m+n)$ are continuous functions;

$(A_{5})$ $a(x,y)$ is a continuous nonnegative function on Ω;

$(A_{6})$ $k_{i},v_{i}\in [0,1], \alpha _{i}\in (0,1], \beta _{i}\in (0,1), pk_{i}(\beta _{i}-1)+1>0, pv_{i}(\gamma _{i}-1)+1>0$ such that $\frac{1}{p}+k_{i}\alpha _{i}(\beta _{i}-1)+v_{i}(\gamma _{i}-1)\geq 0 \ (p>1, i=1,2,\ldots ,m+n), h,k$ are positive constants;

$(A_{7})$ $\alpha _{\ast}(x_{0}):=\min \{\min_{1\leq i\leq m}b_{i}(x_{0}), \min_{m+1\leq j\leq m+n}(b_{j}(x_{0}))\}$, $\beta _{\ast}(y_{0}):=\min \{\min_{1\leq i\leq m}c_{i}(y_{0}), \min_{m+1 \leq j\leq m+n}(c_{j}(y_{0}))\}$;

$(A_{8})$ $\max_{(t,s)\in \Omega _{0}}\psi (t,s)\leq \varphi ^{-1}((1+m+n)^{1- \frac{1}{q}}a(x_{0},y))$ and $u\in C(\Omega _{0},R_{+})$.

For those $\omega _{i}$ given in $(A_{4})$, we can define $\tilde{\omega}_{i}(t)$ $(i=1,2,\ldots ,m+n, t>0)$ by

$$\begin{aligned} \begin{aligned} &\tilde{\omega}_{1}(t)=\max _{\tau \in [0,t]} \bigl\{ \bar{\omega}_{1}(\tau ) \bigr\} , \\ &\tilde{\omega}_{i+1}(t)=\max_{\tau \in [0,t]} \biggl\{ \frac{\bar{\omega}_{i+1}(\tau )}{\tilde{\omega}_{i}(\tau )+\varepsilon _{i}} \biggr\} \tilde{\omega}_{i}(t) \end{aligned} \end{aligned}$$

(2.1)

for $i=1,2,\ldots ,m-1$ and

$$\begin{aligned} \begin{aligned} &\tilde{\omega}_{m+1}(t)=\max _{\tau \in [0,t]} \biggl\{ \frac{\hat{\omega}_{m+1}(\max_{s\in [0,\tau ]}\{f(s)\})}{\tilde{\omega}_{m}(\tau )+\varepsilon _{m}} \biggr\} \tilde{ \omega}_{m}(t), \\ &\tilde{\omega}_{j+1}(t)=\max_{\tau \in [0,t]} \biggl\{ \frac{\hat{\omega}_{j+1}(\max_{s\in [0,\tau ]}\{f(s)\})}{\tilde{\omega}_{j}(\tau )+\varepsilon _{j}} \biggr\} \tilde{\omega}_{j}(t) \end{aligned} \end{aligned}$$

(2.2)

for $j=m+1,\ldots ,m+n-1$, where

$$\begin{aligned} &\hat{\omega}_{j}(t)=\max_{\tau \in [0,t]}\bigl\{ \bar{ \omega}_{j}(\tau )\bigr\} \quad (j=m+1,\ldots ,m+n),\\ &\bar{\omega}_{i}(t):=\omega _{i}(t)+\varepsilon _{i}\quad (i=1,2,\ldots ,m+n). \end{aligned}$$

$\varepsilon _{i}>0$ are very small constants.

Remark 1

If f and $\omega _{i}(u)\ (i=1,2,\ldots ,m)$ given in $(A_{3})$ and $(A_{4})$ are nondecreasing and continuous functions and satisfy

$$\begin{aligned} \omega _{1}\propto \cdots \propto \omega _{m}\propto \omega _{m+1} \circ f\propto \cdots \propto \omega _{m+n}\circ f, \end{aligned}$$

then we define functions $\tilde{\omega}_{i}(u):=\omega _{i}(u)\ (i=1,\ldots ,m)$, $\tilde{\omega}_{j}(u):=\omega _{i}(f(u))\ (j=m+1,\ldots ,m+n)$.

To prove our results, we need the following lemmas.

Lemma 1

([27])

Suppose that $(B_{1})$–$(B_{5})$ hold:

$(B_{1})$ $\alpha _{i}(x):[x_{0},x_{1})\rightarrow [x_{0},x_{1})\ (i=1,2, \ldots ,m+n)$ and $\beta _{i}(y):[y_{0},y_{1})\rightarrow [y_{0},y_{1})\ (i=1,2, \ldots ,m+n)$ are nondecreasing such that $\alpha _{i}(x)\leq x$ on $[x_{0},x_{1})$, $\beta _{i}(y)\leq y$ on $[y_{0},y_{1})$ and $\beta _{i}(y_{0})=y_{0}$;

$(B_{2})$ All $f_{i}\ (i=1,2,\ldots ,m+n)$ are continuous nonnegative functions on $\Lambda \times [\alpha _{\ast}(x_{0}),x_{1})\times [y_{0},y_{1})$;

$(B_{3})$ $g,\varphi :R_{+}\rightarrow R_{+}, \psi :[\alpha _{\ast}(x_{0})-h,x_{1}) \rightarrow R_{+}$ are continuous and φ is strictly increasing such that $\lim_{t\rightarrow \infty}\varphi (t)=+\infty$;

$(B_{4})$ All $\omega _{i}\ (i=1,2,\ldots ,m+n)$ are continuous on $R_{+}$ and positive on $(0,+\infty )$;

$(B_{5})$ $a(x,y)$ is a continuous and nonnegative function on Λ.

Thereinto, $\Lambda :=[x_{0},x_{1}]\times [y_{0},y_{1}], \Omega :=[\alpha _{ \ast}(x_{0}),x_{0})\times [y_{0},y_{1})$, and $x_{0}< x_{1}, y_{0}< y_{1}$ in $R_{+}:=[0,\infty )$. $\max_{s\in [\alpha _{\ast}(x_{0})-h,x_{0}]}\psi (s,y)\leq \varphi ^{-1}(a(x_{0},y))$ for all $y\in [y_{0},y_{1})$ and $u\in C(\Omega ,R_{+})$ satisfies the system of inequalities as follows:

$$\begin{aligned} &\varphi \bigl(u(x,y)\bigr)\leq a(x,y)+\sum _{i=1}^{n} \int _{ \alpha _{i}(x_{0})}^{\alpha _{i}(x)} \int _{\beta _{i}(y_{0})}^{\beta _{i}(y)}f_{i}(x,y,s,t) \omega _{i}\bigl(u(s,t)\bigr)\,dt\,ds \\ &\phantom{\varphi \bigl(u(x,y)\bigr)\leq}{}+\sum_{j=n+1}^{m+n} \int _{\alpha _{j}(x_{0})}^{\alpha _{j}(x)} \int _{ \beta _{j}(y_{0})}^{\beta _{j}(y)}f_{j}(x,y,t,s)\omega _{j} \Bigl( \max_{\xi \in [s-h,s]}g\bigl(u(\xi ,t)\bigr) \Bigr)\,dt\,ds, \\ &\quad (x,y)\in [x_{0},x_{1})\times [y_{0},y_{1}), \\ & u(x,y)\leq \psi (x,y),\quad (x,y)\in \bigl[\alpha _{\ast}(x_{0})-h,x_{0}\bigr] \times [y_{0},y_{1}). \end{aligned}$$

Then

$$\begin{aligned} \begin{aligned} u(x,y)\leq \varphi ^{-1} \bigl(W^{-1}_{m+n} \bigl( \Omega _{m+n}(x,y) \bigr) \bigr) \end{aligned} \end{aligned}$$

for all $(x,y)\in [x_{0},X_{1})\times [y_{0},Y_{1})$, where

$$\begin{aligned} & \Omega _{i}(x,y):=W_{i} \bigl(r_{i}(x,y)\bigr) + \int _{\alpha _{i}(x_{0})}^{ \alpha _{i}(x)} \int _{\beta _{i}(y_{0})}^{\beta _{i}(y)}\max_{( \iota ,\xi )\in [x_{0},x]\times [y_{0},y]}f_{i}( \iota ,\xi ,s,t)\,dt\,ds, \\ & W_{i}(u):= \int ^{u}_{u_{i}} \frac{ds}{\tilde{\omega}_{i}(\varphi ^{-1}(s))},\quad u\geq u_{i}, i=1,2,\ldots ,m+n. \end{aligned}$$

$u_{i}>0$ are given constants, $\tilde{\omega}_{i}$ are defined in (2.1) and (2.2), and $r_{i}(x,y)$ are defined recursively by

$$\begin{aligned} \begin{aligned} &r_{1}(x,y)=\max_{(\iota ,\xi )\in [x_{0},x]\times [y_{0},y]}a( \iota ,\xi ), \\ &r_{i}(x,y)=W^{-1}_{i}\bigl(\Omega _{i}(x,y)\bigr) \end{aligned} \end{aligned}$$

for $i=1,2,\ldots ,m+n$, and $X_{1}\in [x_{0},x_{1}), Y_{1}\in [y_{0},y_{1})$ are chosen such that

$$\begin{aligned} \Omega _{i}(X_{1},Y_{1})\leq \int _{u_{i}}^{\infty} \frac{ds}{\tilde{\omega}_{i}(\varphi ^{-1}(s))} \end{aligned}$$

for $i=1,2,\ldots ,m+n$.

Lemma 2

([14])

$\alpha ,\beta ,\gamma$, and p are positive constants. Then

$$\begin{aligned} \int _{0}^{t}\bigl(t^{\alpha}-s^{\alpha} \bigr)^{p(\beta -1)}s^{p(\gamma -1)}\,ds= \frac{t^{\theta}}{\alpha}B \biggl[ \frac{p(\gamma -1)+1}{\alpha},p( \beta -1)+1 \biggr],\quad t\in R_{+}, \end{aligned}$$

therein $B[\xi ,\eta ]=\int ^{1}_{0}s^{\xi -1}(1-s)^{\eta -1}\,ds\ (\mathrm{Re}\xi >0,\mathrm{Re} \eta >0)$ and $\theta =p[\alpha (\beta -1)+\gamma -1]+1$.

Theorem 2.1

Suppose that $(A_{1})$–$(A_{8})$ hold, $(x,y)\in \Omega \cup \Omega _{0}$, $u(x,y)$ satisfies the integral inequalities (1.9). Then we have

$$\begin{aligned} \begin{aligned} u(x,y)\leq{}& \varphi ^{-1} \biggl[W_{m+n}^{-1} \biggl(W_{m+n} \bigl(r_{m+n}(x,y) \bigr) \\ &{}+ \int _{b_{m+n}(x_{0})}^{b_{m+n}(x)} \int _{c_{m+n}(y_{0})}^{c_{m+n}(y)} \tilde{g}_{m+n}(x,y,t,s)\,dt\,ds \biggr)^{\frac{1}{q}} \biggr] \end{aligned} \end{aligned}$$

(2.3)

for all $(x,y)\in [x_{0},X_{1})\times [y_{0},Y_{1})$, where $W_{i}^{-1}$ are the inverse of the functions

$$\begin{aligned} W_{i}(u):= \int _{u_{i}}^{u} \frac{dx}{\tilde{\omega}_{i}^{q}(\varphi ^{-1}(x^{\frac{1}{q}}))},\quad u \geq u_{i}>0, i=1,\ldots ,m+n, \end{aligned}$$

(2.4)

$u_{i}>0$ are given constants, $r_{i}(t)$ are defined by

$$\begin{aligned} &r_{1}(x,y):=(1+m+n)^{q-1}\tilde{a}^{q}(x,y), \end{aligned}$$

(2.5)

$$\begin{aligned} &\tilde{a}(x,y):=\max_{(\tau ,\xi )\in [x_{0},x]\times [y_{0},y]}\bigl\{ a( \tau ,\xi ) \bigr\} ,\quad (x,y)\in \Omega , \end{aligned}$$

(2.6)

and

$$\begin{aligned} & r_{i+1}(x,y):=W_{i}^{-1} \biggl(W_{i}\bigl(r_{i}(x,y)\bigr)+ \int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)}\tilde{g}_{i}(x,y,t,s)\,dt\,ds \biggr), \\ &\quad i=1, \ldots ,m+n-1, \end{aligned}$$

(2.7)

$$\begin{aligned} &\tilde{g}_{i}(x,y,t,s):=(1+m+n)^{q-1} \bigl((xy)^{\theta _{i}}M_{i}^{2}\bigr)^{ \frac{q}{p}} \hat{g}_{i}^{q}(x,y,t,s),\quad i=1,2,\ldots ,m+n, \end{aligned}$$

(2.8)

$$\begin{aligned} &\hat{g}_{i}(x,y,t,s):=\max_{(\tau ,\xi )\in \Omega} g_{i}(\tau ,\xi ,t,s),\quad i=1,2,\ldots ,m+n, \end{aligned}$$

(2.9)

$$\begin{aligned} &M_{i}=\frac{1}{\alpha _{i}}B \biggl[ \frac{pv_{i}(\gamma _{i}-1)+1}{\alpha _{i}},pk_{i}(\beta _{i}-1)+1 \biggr], \end{aligned}$$

(2.10)

$$\begin{aligned} &\theta _{i}=p\bigl[\alpha _{i}k_{i}( \beta _{i}-1)+v_{i}(\gamma _{i}-1) \bigr]+1, \end{aligned}$$

(2.11)

$\frac{1}{p}+\frac{1}{q}=1, p>1$, $q>0$. $pv_{i}(\gamma _{i}-1)+1>0, pk_{i}(\beta _{i}-1)+1>0$ and $\frac{1}{p}+k_{i}\alpha _{i}(\beta _{i}-1)+v_{i}(\gamma _{i}-1)\geq 0$ for $i=1,2,\ldots,m+n$. $X_{1}\in [x_{0},x_{1}), Y_{1}\in [y_{0},y_{1})$ are the largest numbers such that

$$\begin{aligned} W_{i}\bigl(r_{i}(X_{1},Y_{1}) \bigr)+ \int _{b_{i}(x_{0})}^{b_{i}(X_{1})} \int _{c_{i}(y_{0})}^{c_{i}(Y_{1})} \max_{(\iota ,\xi )\in [x_{0},X_{1})\times [y_{0},Y_{1})} \tilde{g_{i}}(\iota ,\xi ,t,s)\,dt\,ds \leq \int _{u_{i}}^{\infty} \frac{dx}{\tilde{\omega}_{i}^{q}(\varphi ^{-1}(x^{\frac{1}{q}}))}, \end{aligned}$$

$i=1,2,\ldots ,m+n$.

Proof

First of all, for those $f, a(x,y)$ given in $(A_{3})$ and $(A_{5})$, we define $\tilde{a}(x,y)$ by (2.6) and

$$\begin{aligned} \tilde{f}(u):=\max_{\tau \in [0,u]}\bigl\{ f(\tau )\bigr\} ,\quad u\geq 0. \end{aligned}$$

(2.12)

By $(A_{4})$ and Remark 1, the functions $W_{i}$ are strictly increasing. Therefore we know that $W_{i}^{-1}$ are continuous and increasing functions in their domain. The sequence $\{\tilde{\omega}_{i}(t)\}$ defined by $\omega _{i}(t)$ is nondecreasing nonnegative functions on $R_{+}$ and satisfies

$$\begin{aligned} \begin{aligned} &\omega _{i}(t)\leq \tilde{ \omega}_{i}(t),\quad i=1,2,\ldots ,m, \\ &\omega _{i}(t)\leq \hat{\omega}_{i}(t),\quad i=m+1,m+2, \ldots ,m+n, \\ &\hat{\omega}_{i}\bigl(\tilde{f}(t)\bigr)\leq \tilde{ \omega}_{i}(t),\quad i=m+1,m+2,\ldots ,m+n. \end{aligned} \end{aligned}$$

(2.13)

Since the ratios $\frac{\tilde{\omega}_{i+1}(t)}{\tilde{\omega}_{i}(t)}\ (i=1,2, \ldots ,m+n)$ are all nondecreasing, we have $\tilde{\omega}_{i}(t)\propto \tilde{\omega}_{i+1}(t)\ (i=1,2,\ldots ,m+n)$.

Furthermore, $\hat{g}_{i}(x,y,t,s)$ defined by (2.9) are nondecreasing in $x, y$ for each fixed $t, s$ and satisfy $\hat{g}_{i}(x,y,t,s)\geq g_{i}(x,y,t,s)\geq 0$ for all $i=1,2,\ldots ,m+n$. We have $\tilde{a}(x,y)\geq a(x,y)$ and $\hat{g}_{i}(x,y,t,s)\geq g_{i}(x,y,t,s)$, and they are continuous and nondecreasing in $t, s$. From the monotonicity of $\tilde{f}(u)$, we obtain the inequality

$$\begin{aligned} \begin{aligned}\max_{(\xi ,\eta )\in [x-h,x]\times [y-k,y]}f\bigl(u(\xi ,\eta )\bigr)&\leq \max _{(\xi ,\eta )\in [x-h,x]\times [y-k,y]}\tilde{f}\bigl(u(\xi ,\eta )\bigr) \\ &\leq \tilde{f} \Bigl( \max_{(\xi ,\eta )\in [x-h,x]\times [y-k,y]}u( \xi ,\eta ) \Bigr) \end{aligned} \end{aligned}$$

(2.14)

for $(x,y)\in \Omega \cup \Omega _{0}$.

From (1.9), (2.6), (2.9), (2.12), (2.13), and (2.14), we obtain

$$\begin{aligned} \begin{aligned} &\varphi \bigl(u(x,y)\bigr)\\ &\quad\leq \tilde{a}(x,y)+\sum_{i=1}^{m} \int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)}\bigl(x^{ \alpha _{i}}-t^{\alpha _{i}} \bigr)^{k_{i}(\beta _{i}-1)} t^{v_{i}(\gamma _{i}-1)}\bigl(y^{ \alpha _{i}}-s^{\alpha _{i}} \bigr)^{k_{i}(\beta _{i}-1)}\\ &\qquad{}\times s^{v_{i}(\gamma _{i}-1)} \hat{g}_{i}(x,y,t,s)\tilde{\omega}_{i}\bigl(u(t,s) \bigr)\,dt\,ds \\ &\qquad{}+\sum_{j=m+1}^{m+n} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)}\bigl(x^{ \alpha _{j}}-t^{\alpha _{j}} \bigr)^{k_{j}(\beta _{j}-1)} t^{v_{j}(\gamma _{j}-1)}\bigl(y^{ \alpha _{j}}-s^{\alpha _{j}} \bigr)^{k_{j}(\beta _{j}-1)}s^{v_{j}(\gamma _{j}-1)} \\ &\qquad{}\times\hat{g}_{j}(x,y,t,s)\hat{\omega}_{j} \Bigl(\tilde{f} \Bigl(\max_{( \xi ,\eta )\in [t-h,t]\times [s-k,s]}u(\xi ,\eta ) \Bigr) \Bigr)\,dt\,ds \\ &\quad\leq \tilde{a}(x,y)+\sum_{i=1}^{m} \int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)}\bigl(x^{\alpha _{i}}-t^{\alpha _{i}} \bigr)^{k_{i}( \beta _{i}-1)} t^{v_{i}(\gamma _{i}-1)}\bigl(y^{\alpha _{i}}-s^{\alpha _{i}} \bigr)^{k_{i}( \beta _{i}-1)}\\ &\qquad{}\times s^{v_{i}(\gamma _{i}-1)} \hat{g}_{i}(x,y,t,s)\tilde{\omega}_{i}\bigl(u(t,s) \bigr)\,dt\,ds \\ &\qquad{}+\sum_{j=m+1}^{m+n} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)}\bigl(x^{ \alpha _{j}}-t^{\alpha _{j}} \bigr)^{k_{j}(\beta _{j}-1)} t^{v_{j}(\gamma _{j}-1)}\bigl(y^{ \alpha _{j}}-s^{\alpha _{j}} \bigr)^{k_{j}(\beta _{j}-1)}s^{v_{j}(\gamma _{j}-1)}\\ &\qquad{}\times \hat{g}_{j}(x,y,t,s)\tilde{\omega}_{j} \Bigl(\max _{(\xi ,\eta )\in [t-h,t] \times [s-k,s]}u(\xi ,\eta ) \Bigr)\,dt\,ds,\quad (x,y)\in \Omega . \end{aligned} \end{aligned}$$

(2.15)

Let $\frac{1}{p}+\frac{1}{q}=1, p>1$, then $q>0$. Since $pv_{i}(\gamma _{i}-1)+1>0, pk_{i}(\beta _{i}-1)+1>0$ and $\frac{1}{p}+k_{i}\alpha _{i}(\beta _{i}-1)+v_{i}(\gamma _{i}-1)\geq 0$ for $i=1,2,\ldots,m+n$. By Lemma 2 and Holder’s inequality, we get

$$\begin{aligned} &\varphi \bigl(u(x,y)\bigr) \\ &\quad\leq \tilde{a}(x,y)+\sum_{i=1}^{m} \biggl( \int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)} \bigl(x^{ \alpha _{i}}-t^{\alpha _{i}} \bigr)^{pk_{i}(\beta _{i}-1)} \\ &\qquad{}\times t^{pv_{i}( \gamma _{i}-1)}\bigl(y^{\alpha _{i}}-s^{\alpha _{i}} \bigr)^{pk_{i}(\beta _{i}-1)}s^{pv_{i}( \gamma _{i}-1)}\,dt\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)} \hat{g}_{i}^{q}(x,y,t,s) \tilde{\omega}_{i}^{q}\bigl(u(t,s)\bigr)\,dt\,ds \biggr)^{ \frac{1}{q}} \\ &\qquad{}+\sum_{j=m+1}^{m+n} \biggl( \int _{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)}\bigl(x^{ \alpha _{j}}-t^{\alpha _{j}} \bigr)^{pk_{j}(\beta _{j}-1)} \\ &\qquad{}\times t^{pv_{j}( \gamma _{j}-1)}\bigl(y^{\alpha _{j}}-s^{\alpha _{j}} \bigr)^{pk_{j}(\beta _{j}-1)}s^{pv_{j}( \gamma _{j}-1)}\,dt\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \int _{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)} \hat{g}_{j}^{q}(x,y,t,s) \tilde{\omega}_{j}^{q} \Bigl(\max_{(\xi , \eta )\in [t-h,t]\times [s-k,s]}u( \xi ,\eta ) \Bigr)\,dt\,ds \biggr)^{ \frac{1}{q}} \\ &\quad\leq \tilde{a}(x,y)+\sum_{i=1}^{m} \biggl( \int _{0}^{x} \int _{0}^{y}\bigl(x^{ \alpha _{i}}-t^{\alpha _{i}} \bigr)^{pk_{i}(\beta _{i}-1)}\\ &\qquad{}\times t^{pv_{i}( \gamma _{i}-1)}\bigl(y^{\alpha _{i}}-s^{\alpha _{i}} \bigr)^{pk_{i}(\beta _{i}-1)}s^{pv_{i}( \gamma _{i}-1)}\,dt\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{}\times\biggl( \int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)} \hat{g}_{i}^{q}(x,y,t,s) \tilde{\omega}_{i}^{q}\bigl(u(t,s)\bigr)\,dt\,ds \biggr)^{ \frac{1}{q}} \\ &\qquad{}+\sum_{j=m+1}^{m+n} \biggl( \int _{0}^{x} \int _{0}^{y}\bigl(x^{\alpha _{j}}-t^{ \alpha _{j}} \bigr)^{pk_{j}(\beta _{j}-1)} t^{pv_{j}(\gamma _{j}-1)}\bigl(y^{ \alpha _{j}}-s^{\alpha _{j}} \bigr)^{pk_{j}(\beta _{j}-1)}s^{pv_{j}(\gamma _{j}-1)}\,dt\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \int _{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)} \hat{g}_{j}^{q}(x,y,t,s) \tilde{\omega}_{j}^{q} \Bigl(\max_{(\xi , \eta )\in [t-h,t]\times [s-k,s]}u( \xi ,\eta ) \Bigr)\,dt\,ds \biggr)^{ \frac{1}{q}} \\ &\quad=\tilde{a}(x,y)+\sum_{i=1}^{m} \bigl((xy)^{\theta _{i}}M_{i}^{2}\bigr)^{ \frac{1}{p}} \biggl( \int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)} \hat{g}_{i}^{q}(x,y,t,s) \tilde{\omega}_{i}^{q}\bigl(u(t,s)\bigr)\,dt\,ds \biggr)^{ \frac{1}{q}} \\ &\qquad{}+\sum_{j=m+1}^{m+n} \bigl((xy)^{\theta _{i}}M_{i}^{2} \bigr)^{ \frac{1}{p}} \biggl( \int _{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)} \hat{g}_{j}^{q}(x,y,t,s) \tilde{\omega}_{j}^{q} \\ &\qquad{}\times\Bigl(\max_{(\xi , \eta )\in [t-h,t]\times [s-k,s]}u( \xi ,\eta ) \Bigr)\,dt\,ds \biggr)^{ \frac{1}{q}} \end{aligned}$$

(2.16)

for $(x,y)\in \Omega$, where $M_{i}$ and $\theta _{i}$ are defined by (2.10) and (2.11), $i=1,2,\ldots ,m+n$.

By Jensen’s inequality and (2.16), we get for $(x,y)\in \Omega$

$$\begin{aligned} & \varphi ^{q}\bigl(u(x,y)\bigr) \\ &\quad\leq (1+m+n)^{q-1} \Biggl[\tilde{a}^{q}(x,y) \\ &\qquad{}+ \sum _{i=1}^{m} \bigl((xy)^{\theta _{i}}M_{i}^{2} \bigr)^{ \frac{q}{p}} \int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)} \hat{g}_{i}^{q}(x,y,t,s) \tilde{\omega}_{i}^{q}\bigl(u(t,s)\bigr)\,dt\,ds \\ &\qquad{}+\sum_{j=m+1}^{m+n} \bigl((xy)^{\theta _{i}}M_{i}^{2}\bigr)^{ \frac{q}{p}} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)} \hat{g}_{i}^{q}(x,y,t,s) \tilde{\omega}_{j}^{q} \\ &\qquad{}\times \Bigl(\max_{(\xi , \eta )\in [t-h,t]\times [s-k,s]}u( \xi ,\eta ) \Bigr)\,dt\,ds \Biggr],\quad (x,y)\in \Omega . \end{aligned}$$

(2.17)

By (2.5), (2.8), and (2.17), we have

$$\begin{aligned} &\varphi ^{q}\bigl(u(x,y)\bigr) \leq r_{1}(x,y)+\sum_{i=1}^{m} \int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)} \tilde{g}_{i}(x,y,t,s) \tilde{\omega}_{i}^{q}\bigl(u(t,s)\bigr)\,dt\,ds \\ &\phantom{\varphi ^{q}\bigl(u(x,y)\bigr) \leq }{}+\sum_{j=m+1}^{m+n} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)} \tilde{g}_{i}(x,y,t,s) \\ &\phantom{\varphi ^{q}\bigl(u(x,y)\bigr) \leq }{}\times \tilde{\omega}_{j}^{q} \Bigl(\max_{(\xi , \eta )\in [t-h,t]\times [s-k,s]}u( \xi ,\eta ) \Bigr)\,dt\,ds, \\ & \quad(x,y)\in \Omega , \\ & u(x,y)\leq \psi (x,y),\quad (x,y)\in \Omega _{0}. \end{aligned}$$

(2.18)

Concerning (2.18), we consider the auxiliary system of inequalities

$$\begin{aligned} \begin{aligned} \varphi ^{q}\bigl(u(x,y)\bigr) \leq {}&r_{1}(X,Y)+\sum_{i=1}^{m} \int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)} \tilde{g}_{i}(X,Y,t,s) \tilde{\omega}_{i}^{q}\bigl(u(t,s)\bigr)\,dt\,ds \\ &{}+\sum_{j=m+1}^{m+n} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)} \tilde{g}_{j}(X,Y,t,s) \tilde{\omega}_{j}^{q} \Bigl(\max_{(\xi , \eta )\in [t-h,t]\times [s-k,s]}u( \xi ,\eta ) \Bigr)\,dt\,ds \end{aligned} \end{aligned}$$

(2.19)

for all $(x,y)\in [x_{0},X)\times [y_{0},Y)$, where X and Y are chosen arbitrarily such that $x_{0}\leq X\leq X_{1}, y_{0}\leq Y\leq Y_{1}$.

Since

$$\begin{aligned} \max_{(x,y)\in \Omega _{0}}\psi (x,y)\leq \varphi ^{-1} \bigl((1+m+n)^{ \frac{q-1}{q}}a(x_{0},y_{0}) \bigr)\leq \varphi ^{-1} \bigl(r_{1}^{ \frac{1}{q}}(X,Y) \bigr), \end{aligned}$$

we get

$$\begin{aligned} \max_{(x,y)\in \Omega _{0}}\psi (x,y)\leq \varphi ^{-1} \bigl(r_{1}^{ \frac{1}{q}}(X,Y) \bigr). \end{aligned}$$

Now we can define the function

$$\begin{aligned} z(x,y)=\textstyle\begin{cases} r_{1}(X,Y)+\sum_{i=1}^{m}\int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)}\tilde{g}_{i}(X,Y,t,s)\tilde{\omega}_{i}^{q}(u(t,s))\,dt\,ds \\ \quad{}+\sum_{j=m+1}^{m+n}\int _{b_{j}(x_{0})}^{b_{j}(x)}\int _{c_{j}(y_{0})}^{c_{j}(y)} \tilde{g}_{j}(X,Y,t,s) \tilde{\omega}_{j}^{q}\\ \quad{}\times (\max_{(\xi , \eta )\in [t-h,t]\times [s-k,s]}u(\xi ,\eta ) )\,dt\,ds, \\ \qquad (x,y)\in [x_{0},X) \times [y_{0},Y), \\ r_{1}(X,Y), \quad (x,y)\in \Omega _{0}. \end{cases}\displaystyle \end{aligned}$$

(2.20)

Obviously, $z(x,y)$ is nondecreasing.

By (2.19) and (2.20), we have

$$\begin{aligned} & u(x,y)\leq \varphi ^{-1} \bigl(z^{\frac{1}{q}}(x,y) \bigr),\quad (x,y)\in \bigl[\alpha _{\ast}(x_{0})-h,X\bigr)\times \bigl[\beta _{\ast}(y_{0})-k,Y\bigr), \end{aligned}$$

(2.21)

$$\begin{aligned} \begin{aligned}&\max_{(\xi ,\eta )\in [x-h,x]\times [y-k,y]}u(\xi ,\eta )\leq \max _{( \xi ,\eta )\in [x-h,x]\times [y-k,y]}\varphi ^{-1} \bigl(z^{ \frac{1}{q}}(\xi ,\eta ) \bigr)\\ &\phantom{\max_{(\xi ,\eta )\in [x-h,x]\times [y-k,y]}u(\xi ,\eta )}\leq \varphi ^{-1} \Bigl(\max_{(\xi , \eta )\in [x-h,x]\times [y-k,y]} \bigl(z^{\frac{1}{q}}(\xi ,\eta ) \bigr) \Bigr).\end{aligned} \end{aligned}$$

(2.22)

From (2.21) and (2.22) we have

$$\begin{aligned} & z(x,y) \\ &\quad\leq r_{1}(X,Y)+\sum _{i=1}^{m} \int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)}\tilde{g}_{i}(X,Y,t,s) \tilde{\omega}_{i}^{q}\bigl( \varphi ^{-1} \bigl(z^{\frac{1}{q}}(t,s)\bigr)\bigr)\,dt\,ds \\ &\qquad{}+\sum_{j=m+1}^{m+n} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)} \tilde{g}_{j}(X,Y,t,s) \tilde {\omega}_{j}^{q} \Bigl(\varphi ^{-1} \Bigl( \max_{(\xi ,\eta )\in [t-h,t]\times [s-k,s]}z^{\frac{1}{q}}( \xi ,\eta ) \Bigr) \Bigr)\,dt\,ds, \\ &\quad (x,y) \in [x_{0},X)\times [y_{0},Y), \\ &z(x,y)\leq r_{1}(x,y),\quad \bigl(x,y\in \bigl[\alpha _{\ast}(x_{0})-h,X \bigr)\times \bigl[\beta _{ \ast}(y_{0})-k,Y\bigr). \end{aligned}$$

(2.23)

Let $e(z):=\varphi ^{-1}(z^{\frac{1}{q}})$ and $e(z)$ is a continuous and nondecreasing function on $R_{+}$. Thus, $\tilde{w}_{i}(e(z))$ is continuous and nondecreasing on $R_{+}\ (i=1,2,\ldots ,m+n)$, $\tilde{w}_{i}(e(z))>0$ for $z>0$.

Since $\tilde{w}_{i}(z)\propto \tilde{w}_{i+1}(z)$, we get that $\frac{\tilde{w}_{i+1}(e(z))}{\tilde{w}_{i}(e(z))}$ is also continuous and nondecreasing on $R_{+}$. So we obtain $\tilde{w}_{i}^{q}(e(z))\propto \tilde{w}_{i+1}^{q}(e(z)), i=1,2,3, \ldots ,m+n-1$. By (2.23), we let $\varphi (u(x,y))=z(x,y), a(x,y)=r_{1}(X,Y), f_{i}(x,y,s,t)= \tilde{g}_{i}(X,Y,t,s), \omega _{i}(u(s,t))=\tilde{\omega}_{i}(e(z))$, applying Lemma 1, we have

$$\begin{aligned} \begin{aligned} z(x,y)\leq{}& W^{-1}_{m+n} \biggl(W_{m+n} \biggl(r_{m+n}(X,Y,x,y)\\ &{}+ \int ^{b_{m+n}(x)}_{b_{m+n}(x_{0})} \int ^{c_{m+n}(y)}_{c_{m+n}(y_{0})} \tilde{g}_{m+n}(X,Y,s,t)\,dt\,ds \biggr) \biggr) \end{aligned} \end{aligned}$$

(2.24)

for all $x_{0}\leq x\leq \min \{X,X_{2}\}$ and $y_{0}\leq y\leq \min \{Y,Y_{2}\}$, where $W_{m+n}$ is defined in (2.4),

$$\begin{aligned} &r_{1}(X,Y,x,y):=r_{1}(X,Y),\\ &r_{i+1}(X,Y,x,y):=W_{i}^{-1} \biggl(W_{i}\bigl(r_{i}(X,Y,x,y)\bigr)+ \int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)}\tilde{g}_{i}(x,y,t,s)\,dt\,ds \biggr),\\ &\quad i=1,2, \ldots ,m+n, \end{aligned}$$

$X_{2}\leq x_{1}, Y_{2}\leq y_{1}$ are the largest numbers such that

$$\begin{aligned} &W_{i}\bigl(r_{i}(X,Y,X_{2},Y_{2}) \bigr)\\ &\qquad{}+ \int _{b_{i}(x_{0})}^{b_{i}(X_{1})} \int _{c_{i}(y_{0})}^{c_{i}(Y_{1})}\max_{(\iota ,\xi )\in [x_{0},X_{1}) \times [y_{0},Y_{1})}f(\iota ,\xi ,t,s)\,dt\,ds \\ &\quad\leq \int _{u_{i}}^{ \infty} \frac{dx}{\tilde{\omega}_{i}^{q}(\varphi ^{-1}(x^{\frac{1}{q}}))}, \end{aligned}$$

$i=1,2,\ldots ,m+n$.

It follows from (2.21) and (2.24) that we have

$$\begin{aligned} \begin{aligned} u(x,y)\leq{}& \varphi ^{-1} \biggl(W^{-1}_{m+n} \biggl(W_{m+n} \biggl(r_{m+n}(X,Y,x,y)\\ &{} + \int ^{b_{m+n}(x)}_{b_{m+n}(x_{0})} \int ^{c_{m+n}(y)}_{c_{m+n}(y_{0})} \tilde{g}_{m+n}(X,Y,s,t)\,dt\,ds \biggr) \biggr)^{\frac{1}{q}} \biggr) \end{aligned} \end{aligned}$$

(2.25)

for all $x_{0}\leq x\leq \min \{X,X_{2}\}$ and $y_{0}\leq y\leq \min \{Y,Y_{2}\}$.

Let $x=X, y=Y, X_{2}=X_{1}, Y_{2}=Y_{1}$, we have

$$\begin{aligned} \begin{aligned} u(X,Y)\leq{}& \varphi ^{-1} \biggl(W^{-1}_{m+n} \biggl(W_{m+n} \biggl(r_{m+n}(X,Y,X,Y) \\ &{}+ \int ^{b_{m+n}(x)}_{b_{m+n}(x_{0})} \int ^{c_{m+n}(y)}_{c_{m+n}(y_{0})} \tilde{g}_{m+n}(X,Y,s,t)\,dt\,ds \biggr) \biggr)^{\frac{1}{q}} \biggr) \end{aligned} \end{aligned}$$

(2.26)

for all $x_{0}\leq X\leq X_{1}$ and $y_{0}\leq Y\leq Y_{1}$.

It is easy to obtain $r_{m+n}(X,Y,X,Y)=r_{m+n}(X,Y)$. So (2.26) can be restated as

$$\begin{aligned} \begin{aligned} u(X,Y)\leq{}& \varphi ^{-1} \biggl(W^{-1}_{m+n} \biggl(W_{m+n} \biggl(r_{m+n}(X,Y)\\ &{} + \int ^{b_{m+n}(x)}_{b_{m+n}(x_{0})} \int ^{c_{m+n}(y)}_{c_{m+n}(y_{0})} \tilde{g}_{m+n}(X,Y,s,t)\,dt\,ds \biggr) \biggr)^{\frac{1}{q}} \biggr). \end{aligned} \end{aligned}$$

(2.27)

Because $X,Y$ are arbitrary, we can replace X and Y with x and y. Thus we get

$$\begin{aligned} \begin{aligned} u(x,y)\leq{}& \varphi ^{-1} \biggl(W^{-1}_{m+n} \biggl(W_{m+n} \biggl(r_{m+n}(x,y)\\ &{} + \int ^{b_{m+n}(x)}_{b_{m+n}(x_{0})} \int ^{c_{m+n}(y)}_{c_{m+n}(y_{0})} \tilde{g}_{m+n}(x,y,s,t)\,dt\,ds \biggr) \biggr)^{\frac{1}{q}} \biggr) \end{aligned} \end{aligned}$$

(2.28)

for all $(x,y)\in [x_{0},X_{1})\times [y_{0},Y_{1})$.

The proof is complete. □

Corollary 2.1

Suppose that $(A_{1})$–$(A_{8})$ hold if $u(x,y)$ satisfy the following inequality:

$$\begin{aligned} &\begin{aligned} &\varphi \bigl(u(x,y)\bigr)\\ &\quad\leq c+\sum _{i=1}^{m} \int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)}\bigl(x^{\alpha _{i}}-t^{\alpha _{i}} \bigr)^{k_{i}( \beta _{i}-1)} t^{v_{i}(\gamma _{i}-1)} \\ &\qquad{}\times\bigl(y^{\alpha _{i}}-s^{\alpha _{i}}\bigr)^{k_{i}(\beta _{i}-1)}s^{v_{i}( \gamma _{i}-1)}g_{i}(x,y,t,s) \omega _{i}\bigl(u(t,s)\bigr)\,dt\,ds \\ &\qquad{}+\sum_{j=m+1}^{m+n} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)}\bigl(x^{ \alpha _{j}}-t^{\alpha _{j}} \bigr)^{k_{j}(\beta _{j}-1)} t^{v_{j}(\gamma _{j}-1)}\bigl(y^{ \alpha _{j}}-s^{\alpha _{j}} \bigr)^{k_{j}(\beta _{j}-1)} \\ &\qquad{}\times s^{v_{j}(\gamma _{j}-1)}g_{j}(x,y,t,s)\omega _{j} \Bigl(f \Bigl( \max_{(\xi ,\eta )\in [t-h,t]\times [s-k,s]}u(\xi ,\eta ) \Bigr) \Bigr)\,dt\,ds, \quad (x,y)\in \Omega , \end{aligned} \\ &u(x,y)\leq \psi (x,y),\quad (x,y)\in \Omega _{0}, \end{aligned}$$

(2.29)

where $u\in C(\Omega \cup \Omega _{0},R_{+})$ and $c\geq 0$ is a constant. Then

$$\begin{aligned} \begin{aligned} u(x,y)\leq{}& \varphi ^{-1} \biggl(W^{-1}_{m+n} \biggl(W_{m+n} \biggl( \bar{r}_{m+n}(x,y) \\ &{}+ \int ^{b_{m+n}(x)}_{b_{m+n}(x_{0})} \int ^{c_{m+n}(y)}_{c_{m+n}(y_{0})} \tilde{g}_{m+n}(x,y,s,t)\,dt\,ds \biggr) \biggr)^{\frac{1}{q}} \biggr) \end{aligned} \end{aligned}$$

(2.30)

for all $(x,y)\in [x_{0},X_{1})\times [y_{0},Y_{1})$, where $\bar{r}_{i}(x,y)$ is defined by $\bar{r}_{1}(x,y):=\varphi ^{q}(M)$ and

$$\begin{aligned} &\begin{aligned}& \bar{r}_{i+1}(x,y):=W^{-1}_{i} \biggl(W_{i}\bigl(\bar{r}_{i}(x,y)\bigr)+ \int ^{b_{i}(x)}_{b_{i}(x_{0})} \int ^{c_{i}(y)}_{c_{i}(y_{0})} \tilde{g}_{m+n}(x,y,s,t)\,dt\,ds \biggr), \\ &\quad i=1,2,\ldots ,m+n-1, \end{aligned} \\ &M:=\max \Bigl[\max_{(s,t)\in \Omega _{0}}\psi (s,t),\varphi ^{-1} \bigl((1+m+n)^{1- \frac{1}{q}}c\bigr) \Bigr], \end{aligned}$$

(2.31)

$\frac{1}{p}+\frac{1}{q}=1, p>1, q>0$. $pv_{i}(\gamma _{i}-1)+1>0, pk_{i}(\beta _{i}-1)+1>0$ and $\frac{1}{p}+k_{i}\alpha _{i}(\beta _{i}-1)+v_{i}(\gamma _{i}-1)\geq 0$ for $i=1,2,\ldots,m+n. X_{1}< x_{1}$, $Y_{1}< y_{1}$ are the largest numbers such that

$$\begin{aligned} \begin{aligned}& W_{i}\bigl( \bar{r}_{i}(X_{1},Y_{1})\bigr)+ \int ^{b_{i}(X_{1})}_{b_{i}(x_{0})} \int ^{c_{i}(Y_{1})}_{c_{i}(y_{0})}\tilde{g}_{i}(X_{1},Y_{1},s,t)\,dt\,ds \\ &\quad\leq \int _{u_{i}}^{\infty} \frac{dz}{\tilde{\omega}^{q}_{i}(\varphi ^{-1}(z^{\frac{1}{q}}))},\quad i=1,2, \ldots ,m+n, \end{aligned} \end{aligned}$$

(2.32)

$W_{i}$ is defined in (2.4) and $W_{i}^{-1}$ is the inverse of $W_{i}$. $\tilde{\omega}_{i}$ is defined in (2.1) and (2.2). $\tilde{g}_{i}$ is defined in (2.7).

Proof

By (2.29) and the definition of M, we have

$$\begin{aligned} \begin{aligned} &\varphi \bigl(u(x,y)\bigr)\leq (1+m+n)^{\frac{1}{q-1}}\varphi (M) \\ &\phantom{\varphi \bigl(u(x,y)\bigr)\leq}{}+\sum_{i=1}^{m} \int _{b_{i}(x_{0})}^{b_{i}(x)} \int _{c_{i}(y_{0})}^{c_{i}(y)}\bigl(x^{ \alpha _{i}}-t^{\alpha _{i}} \bigr)^{k_{i}(\beta _{i}-1)} t^{v_{i}(\gamma _{i}-1)}\bigl(y^{ \alpha _{i}}-s^{\alpha _{i}} \bigr)^{k_{i}(\beta _{i}-1)}s^{v_{i}(\gamma _{i}-1)} \\ &\phantom{\varphi \bigl(u(x,y)\bigr)\leq}{}\times g_{i}(x,y,t,s)\omega _{i}\bigl(u(t,s)\bigr)\,dt\,ds \\ &\phantom{\varphi \bigl(u(x,y)\bigr)\leq}{}+\sum_{j=m+1}^{m+n} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)}\bigl(x^{ \alpha _{j}}-t^{\alpha _{j}} \bigr)^{k_{j}(\beta _{j}-1)} t^{v_{j}(\gamma _{j}-1)}\bigl(y^{ \alpha _{j}}-s^{\alpha _{j}} \bigr)^{k_{j}(\beta _{j}-1)}s^{v_{j}(\gamma _{j}-1)} \\ &\phantom{\varphi \bigl(u(x,y)\bigr)\leq}{}\times g_{i}(x,y,t,s)\omega _{j} \Bigl(f \Bigl(\max _{(\xi ,\eta )\in [t-h,t] \times [s-k,s]}u(\xi ,\eta ) \Bigr) \Bigr)\,dt\,ds,\quad (x,y)\in \Omega , \\ &u(x,y)\leq M,\quad (x,y)\in \Omega _{0}. \end{aligned} \end{aligned}$$

(2.33)

We choose $a(x,y)=(1+m+n)^{\frac{1}{q-1}}\varphi (M)$, then (2.33) can be converted to (2.30) by Theorem 2.1.

The proof is complete. □

3 Applications

In this section, we apply the results to study the boundedness of the solutions for an integral equation with the maxima.

Example 1

Consider the system of integral equations with maxima.

$$\begin{aligned} \begin{aligned} u(x,y)=\textstyle\begin{cases} a(x,y)+\int ^{x}_{x_{0}}\int ^{y}_{y_{0}}(x-t)^{ \beta _{1}-1}t^{\gamma _{1}-1}(y-s)^{\beta _{1}-1}s^{\gamma _{1}-1}\\ \quad{}\times f_{1}(x,y,t,s,u(x,y))\,dt\,ds \\ \quad{}+\int ^{x}_{x_{0}}\int ^{y}_{y_{0}}(x-t)^{\beta _{2}-1}t^{\gamma _{2}-1}(y-s)^{ \beta _{2}-1}s^{\gamma _{2}-1}\\ \quad{}\times f_{2} (x,y,t,s, \max_{(\eta ,\xi ) \in [\bar{\alpha}(t),\hat{\alpha}(t)]\times [\bar{\beta}(s), \hat{\beta}(s)]}u(\eta ,\xi ) )\,dt\,ds, \\ \qquad (x,y)\in [x_{0},x)\times [y_{0},y), \\ \psi (x,y),\quad (x,y)\in [\hat{\alpha}(x_{0})-h,x_{0})\times [ \hat{\beta}(y_{0})-k,y_{0}), \end{cases}\displaystyle \end{aligned} \end{aligned}$$

(3.1)

$\psi \in C([\hat{\alpha}(x_{0})-h,x_{0})\times [\hat{\beta}(y_{0})-k,y_{0}),R)$, $x_{0}\geq 0, y_{0}\geq 0, h>0$.

Suppose that $(C_{1})$ $|f_{i}(x,y,t,s,u)|\leq g_{i}(x,y,t,s)\omega _{i}(|u|)$, $\omega _{i}$ are continuous positive and nondecreasing functions on $R_{+}\ (i=1,2)$, $\omega _{1}\propto \omega _{2}$, $g_{i}(x,y,t,s)$ is nondecreasing in $(x,y)$ for each fixed $(t,s)$, $\beta _{i}\in (0,1), \gamma _{i}>1-\frac{1}{p}, \frac{1}{p}+\beta _{i}+ \gamma _{i}-2\geq 0\ (p>1, i=1,2)$;

$(C_{2})$ $\bar{\alpha}(x),\hat{\alpha}(x)\in C^{1}([x_{0},\infty ),R_{+}), \bar{\beta}(y),\hat{\beta}(y)\in C^{1}([y_{0},\infty ),R_{+}), \bar{\alpha}(x),\hat{\alpha}(x),\bar{\beta}(y),\hat{\beta}(y)$ are nondecreasing, $\hat{\alpha}(x)\leq x, \bar{\alpha}(x)\leq x, 0<\hat{\alpha}(x)- \bar{\alpha}(x)\leq h$ for $x\geq x_{0}$ and $\bar{\beta}(y)\leq y, \hat{\beta}(y)\leq y, 0<\hat{\beta}(y)- \bar{\beta}(y)\leq k$ for $y\geq y_{0}$;

$(C_{3})$ $a(x,y)$ is continuous on $[x_{0},\infty )\times [y_{0},\infty )$;

$(C_{4})$ $\max_{(\eta ,\xi )\in [\bar{\alpha}(t),\hat{\alpha}(t)]\times [ \bar{\beta}(s),\hat{\beta}(s)]}|\psi (\eta ,\xi )|\leq 3^{1- \frac{1}{q}}|a(x,y)|$.

Then we give an estimate for the solutions of (3.1).

Theorem 3.1

Suppose that $(C_{1})$–$(C_{4})$ hold, then from (3.1) we have

$$\begin{aligned} \begin{aligned} u(x,y)\leq \biggl(W_{2}^{-1} \biggl(W_{2} \biggl(r_{2}(x,y)+c_{2}(x,y) \int ^{x}_{x_{0}} \int ^{y}_{y_{0}}g_{2}^{q}(x,y,t,s)\,dt\,ds \biggr) \biggr) \biggr)^{\frac{1}{q}} \end{aligned} \end{aligned}$$

(3.2)

for $(x,y)\in [x_{0},X_{1}]\times [y_{0},Y_{1}]$, where

$$\begin{aligned} &r_{1}(x,y):=3^{q-1}\bigl(\tilde{a}^{q}(x,y) \bigr),\\ &r_{2}(x,y):=W_{1}^{-1} \biggl(W_{1} \biggl(r_{1}(x,y)+c_{1}(x,y) \int ^{x}_{x_{0}} \int ^{y}_{y_{0}}g_{1}^{q}(x,y,t,s)\,dt\,ds \biggr) \biggr),\\ &c_{i}(x,y):=3^{q-1}\bigl(M_{i}^{2}(xy)^{\theta _{i}} \bigr)^{\frac{q}{p}},\\ &M_{i}=B\bigl[p(\gamma _{i}-1)+1,p(\beta _{i}-1)+1\bigr],\qquad \theta _{i}=p[ \beta _{i}+ \gamma _{i}-2]+1,\quad i=1,2, \\ &\tilde{a}(x,y):=\max_{(\tau ,\xi )\in [x_{0},x]\times [y_{0},y]}\bigl\{ \bigl\vert a( \tau ,\xi ) \bigr\vert \bigr\} , \end{aligned}$$

$\tilde{a}(x,y)$ is a continuous and nondecreasing function on $[x_{0},\infty )\times [y_{0},\infty )$.

$W_{i}^{-1}$ are the inverse of the functions

$$\begin{aligned} W_{i}(u):= \int ^{u}_{u_{i}} \frac{dx}{\omega ^{q}_{i}(x^{\frac{1}{q}})},\quad u\geq u_{i}>0, i=1,2. \end{aligned}$$

$X_{1},Y_{1}$ are the largest numbers such that

$$\begin{aligned} &W_{1} \biggl(3^{q-1} \Bigl(\max_{(\tau ,\xi )\in [x_{0},X_{1}]\times [y_{0},Y_{1}]} \bigl\{ \bigl\vert a(\tau ,\xi ) \bigr\vert \bigr\} \Bigr)^{q}+ c_{1}(X_{1},Y_{1}) \int ^{X_{1}}_{x_{0}} \int ^{Y_{1}}_{y_{0}}g_{1}^{q}(X_{1},Y_{1},t,s)\,dt\,ds \biggr)\\ &\quad\leq \int ^{ \infty}_{u_{1}}\frac{dx}{\omega ^{q}_{1}(x^{\frac{1}{q}})},\\ &W_{2} \biggl(r_{2}(X_{1},Y_{1})+c_{2}(X_{1},Y_{1}) \int ^{X_{1}}_{x_{0}} \int ^{Y_{1}}_{y_{0}}g_{2}^{q}(X_{1},Y_{1},t,s)\,dt\,ds \biggr) \leq \int ^{\infty}_{u_{2}}\frac{dx}{\omega ^{q}_{2}(x^{\frac{1}{q}})}. \end{aligned}$$

Proof

From (3.1) and $(C_{1})$, we have

$$\begin{aligned} \begin{aligned} \bigl\vert u(x,y) \bigr\vert \leq{}& \tilde{a}(x,y)+ \int ^{x}_{x_{0}} \int ^{y}_{y_{0}}(x-t)^{ \beta _{1}-1}t^{\gamma _{1}-1}(y-s)^{\beta _{1}-1}s^{\gamma _{1}-1}\\ &{}\times g_{1}(x,y,t,s) \omega _{1}\bigl( \bigl\vert u(x,y) \bigr\vert \bigr)\,dt\,ds \\ &{}+ \int ^{x}_{x_{0}} \int ^{y}_{y_{0}}(x-t)^{\beta _{2}-1}t^{\gamma _{2}-1}(y-s)^{ \beta _{2}-1}s^{\gamma _{2}-1} g_{2}(x,y,t,s)\\ &{}\times \omega _{2} \Bigl( \Bigl\vert \max _{(\eta ,\xi )\in [\bar{\alpha}(t),\hat{\alpha}(t)]\times [ \bar{\beta}(s),\hat{\beta}(s)]}u(\eta ,\xi ) \Bigr\vert \Bigr)\,dt\,ds \\ \leq{}& \tilde{a}(x,y)+ \int ^{x}_{x_{0}} \int ^{y}_{y_{0}}(x-t)^{\beta _{1}-1}t^{ \gamma _{1}-1}(y-s)^{\beta _{1}-1}s^{\gamma _{1}-1}\\ &{}\times g_{1}(x,y,t,s) \omega _{1}\bigl( \bigl\vert u(x,y) \bigr\vert \bigr)\,dt\,ds \\ &{}+ \int ^{x}_{x_{0}} \int ^{y}_{y_{0}}(x-t)^{\beta _{2}-1}t^{\gamma _{2}-1}(y-s)^{ \beta _{2}-1}s^{\gamma _{2}-1} g_{2}(x,y,t,s)\\ &{}\times\omega _{2} \Bigl(\max_{( \eta ,\xi )\in [\bar{\alpha}(t),\hat{\alpha}(t)]\times [\bar{\beta}(s), \hat{\beta}(s)]} \bigl\vert u(\eta ,\xi ) \bigr\vert \Bigr)\,dt\,ds. \end{aligned} \end{aligned}$$

(3.3)

Let $z(x,y)=|u(x,y)|$ for $(x,y)\in [\hat{\alpha}(x_{0})-h,\infty )\times [\hat{\beta}(y_{0})-k, \infty )$, then we have

$$\begin{aligned} \begin{aligned} z(x,y)\leq{}& \tilde{a}(x,y)+ \int ^{x}_{x_{0}} \int ^{y}_{y_{0}}(x-t)^{ \beta _{1}-1}t^{\gamma _{1}-1}(y-s)^{\beta _{1}-1}s^{\gamma _{1}-1}\\ &{}\times g_{1}(x,y,t,s) \omega _{1}\bigl(z(x,y)\bigr)\,dt\,ds\\ &{}+ \int ^{x}_{x_{0}} \int ^{y}_{y_{0}}(x-t)^{\beta _{2}-1}t^{\gamma _{2}-1}(y-s)^{ \beta _{2}-1}s^{\gamma _{2}-1} g_{2}(x,y,t,s)\\ &{}\times\omega _{2} \Bigl(\max_{( \eta ,\xi )\in [\bar{\alpha}(t),\hat{\alpha}(t)]\times [\bar{\beta}(s), \hat{\beta}(s)]}z( \eta ,\xi ) \Bigr)\,dt\,ds, \end{aligned} \end{aligned}$$

(3.4)

$z(x,y)=|\psi (x,y)|, (x,y)\in [\hat{\alpha}(x_{0})-h,x_{0})\times [ \hat{\beta}(y_{0})-k,y_{0})$.

From condition $(C_{2})$, we have

$$\begin{aligned} \max_{(\eta ,\xi )\in [\bar{\alpha}(t),\hat{\alpha}(t)]\times [ \bar{\beta}(s),\hat{\beta}(s)]}z(\eta ,\xi )\leq \max_{(\eta ,\xi ) \in [\hat{\alpha}(t)-h,\hat{\alpha}(t)]\times [\hat{\beta}(s)-k, \hat{\beta}(s)]}z( \eta ,\xi ). \end{aligned}$$

By (3.4), we have

$$\begin{aligned} z(x,y)\leq{}& \tilde{a}(x,y)+ \int ^{x}_{x_{0}} \int ^{y}_{y_{0}}(x-t)^{ \beta _{1}-1}t^{\gamma _{1}-1}(y-s)^{\beta _{1}-1}s^{\gamma _{1}-1} \\ \begin{aligned} &{}\times g_{1}(x,y,t,s) \omega _{1}\bigl(z(x,y)\bigr)\,dt\,ds \\ &{}+ \int ^{x}_{x_{0}} \int ^{y}_{y_{0}}(x-t)^{\beta _{2}-1}t^{\gamma _{2}-1}(y-s)^{ \beta _{2}-1}s^{\gamma _{2}-1} g_{2}(x,y,t,s) \end{aligned}\\ &{}\times\omega _{2} \Bigl(\max_{( \eta ,\xi )\in [\hat{\alpha}(t)-h,\hat{\alpha}(t)]\times [\hat{\beta}(s)-k, \hat{\beta}(s)]}z( \eta ,\xi ) \Bigr)\,dt\,ds, \end{aligned}$$

(3.5)

$z(x,y)=|\psi (x,y)|, (x,y)\in [\hat{\alpha}(x_{0})-h,x_{0})\times [ \hat{\beta}(y_{0})-k,y_{0})$.

By condition $(C_{4})$, we can obtain

$$\begin{aligned} \max_{(\eta ,\xi )\in [\hat{\alpha}(x_{0})-h,x_{0})\times [ \hat{\beta}(y_{0})-k,y_{0})} \bigl\vert \psi (\eta ,\xi ) \bigr\vert \leq 3^{1-\frac{1}{q}} \tilde{a}(x,y). \end{aligned}$$

Compare with (1.9), we let $\psi (u(x,y))=z(x,y), a(x,y)=\tilde{a}(x,y), m=n=1, b_{i}(x)=x\ (i=1,2), c_{i}(y)=y\ (i=1,2), \alpha _{i}=1\ (i=1,2), k_{i},v_{i}=1\ (i=1,2)$, applying Theorem 2.1, from (3.5) we obtain (3.2).

Then the proof is complete. □

Example 2

Consider the system of integral inequations with maxima:

$$\begin{aligned} \begin{aligned} \textstyle\begin{cases} u(x,y)\\ \quad\leq xy+\int ^{x}_{0}\int ^{y}_{0}(x-t)^{- \frac{1}{2}}(y-s)^{-\frac{1}{2}}u(x,y)\,dt\,ds \\ \qquad{}+\int ^{x}_{0}\int ^{y}_{0}(x-t)^{-\frac{1}{2}}(y-s)^{-\frac{1}{2}} \max_{(\xi ,\eta )\in [t-h,t]\times [s-k,s]}e^{u(\xi ,\eta )}\,dt\,ds, (x,y)\in \Omega , \\ u(x,y)\leq \psi (x,y),\quad (x,y)\in \Omega _{0}, \end{cases}\displaystyle \end{aligned} \end{aligned}$$

(3.6)

where

$$\begin{aligned} &\Omega =[0,1)\times [0,1),\\ &\Omega _{0}=[-h,0)\times [-k,1)\cup [0,1)\times [-k,0), \end{aligned}$$

$h, k$ are constants, $\psi (x,y)\in C(\Omega _{0},R)$.

Theorem 3.2

$u(x,y)$ satisfies the integral inequalities (3.6), $(x,y)\in \Omega \cup \Omega _{0}$, we let $p=\frac{3}{2}, q=3$.

Then we have

$$\begin{aligned} \begin{aligned} u(x,y)\leq \exp \bigl\{ \ln \bigl(9(xy)^{3} \bigr)+(xy)^{ \frac{3}{2}} \bigl(4M^{4}+9M \bigr)\bigr\} ^{\frac{1}{3}} \end{aligned} \end{aligned}$$

(3.7)

for all $(x,y)\in [0,1)\times [0,1)$.

Proof

Compare with (1.9), from (3.6), we let $\varphi (u)=u, m=n=1, b_{i}(x)=x\ (i=1,2), c_{i}(y)=y\ (i=1,2), \alpha _{i}=1\ (i=1,2), v_{i}=0\ (i=1,2), k_{i}=1\ (i=1,2), \beta _{i}=\frac{1}{2}\ (i=1,2), x_{0}=0,\ y_{0}=0, a(x,y)=xy$.

$$\begin{aligned} &\tilde{a}(x,y):=\max_{(\tau ,\xi )\in [0,x]\times [0,y]}\{\tau \xi \}=xy, \quad (x,y)\in \Omega , \\ &g_{1}(x,y,t,s)=g_{2}(x,y,t,s)=1,\quad \hat{g}_{1}= \hat{g}_{2}=1. \end{aligned}$$

Setting $p=\frac{3}{2}, q=3$, we get

$$\begin{aligned} &\tilde{g}(x,y,t,s):=3^{2}\bigl((xy)^{\theta}M^{2} \bigr)^{\frac{q}{p}}=9M^{4}(xy)^{ \frac{1}{2}}, \\ &M=B\biggl[1,1-\frac{p}{2}\biggr]=B\biggl[1,\frac{1}{4}\biggr],\qquad \theta =1-\frac{p}{2}= \frac{1}{4}, \\ &\omega _{1}(u)=u,\qquad \omega _{2}(u)=u,\qquad f(u)=e^{u}. \end{aligned}$$

From the definition

$$\begin{aligned} W_{i}(u):= \int ^{u}_{1}\frac{dx}{{\omega}^{q}_{i}(x^{\frac{1}{q}})},\quad u\geq 1, i=1,2, \end{aligned}$$

we have

$$\begin{aligned} & W_{1}(u)= W_{2}(u)=\ln u, \\ &W^{-1}_{1}(u)= W^{-1}_{2}(u)=e^{u}. \\ &r_{1}(x,y):=3^{q-1}a^{q}(x,y)=9(xy)^{3},\\ &r_{2}(x,y):=W_{1}^{-1} \biggl(W_{1}\bigl(r_{1}(x,y)\bigr)+ \int _{x_{0}}^{x} \int _{y_{0}}^{y}\tilde{g}(x,y,t,s)\,dt\,ds \biggr) \\ &\phantom{r_{2}(x,y)}=\exp \bigl\{ \ln \bigl(9(xy)^{3} \bigr)+4M^{4}(xy)^{\frac{3}{2}} \bigr\} =9(xy)^{3} \exp \bigl\{ 4M^{4}(xy)^{\frac{3}{2}}\bigr\} . \end{aligned}$$

Applying Theorem 2.1, we obtain

$$\begin{aligned} \begin{aligned} u(x,y)&\leq W_{2}^{-1} \bigl(W_{2} \bigl(r_{2}(x,y) \bigr)+3^{q-1}M^{\frac{2p}{q}}(xy)^{\frac{\theta q}{p}}(x-x_{0}) (y-y_{0}) \bigr)^{\frac{1}{q}} \\ &=\exp \bigl\{ \ln \bigl(9(xy)^{3}\exp \bigl\{ 4M^{4}(xy)^{\frac{3}{2}} \bigr\} \bigr)+9M(xy)^{ \frac{3}{2}}\bigr\} ^{\frac{1}{3}} \\ &=\exp \bigl\{ \ln \bigl(9(xy)^{3} \bigr)+4M^{4}(xy)^{\frac{3}{2}}+9M(xy)^{ \frac{3}{2}} \bigr\} ^{\frac{1}{3}} \\ &=\exp \bigl\{ \ln \bigl(9(xy)^{3} \bigr)+(xy)^{\frac{3}{2}} \bigl(4M^{4}+9M \bigr)\bigr\} ^{\frac{1}{3}}. \end{aligned} \end{aligned}$$

(3.8)

We have $\tilde{\omega}_{i}(u)=\omega _{i}(u)=u, i=1,2$. So we obtain

$$\begin{aligned} \begin{aligned} \int ^{\infty}_{1} \frac{dx}{\tilde{\omega}_{i}^{q}(x^{\frac{1}{q}})}= \int ^{\infty}_{1} \frac{dx}{\tilde{\omega}_{i}^{q}(x^{\frac{1}{q}})} = \int ^{\infty}_{1} \frac{dx}{x}=\infty ,\quad i=1,2. \end{aligned} \end{aligned}$$

Then we have (3.8) holds for all $(x,y)\in [0,1)\times [0,1)$.

So the proof is complete. □

More generally, consider the system of differential equations with maxima:

$$\begin{aligned} \begin{aligned} &z(x,y)=a(x,y)+ \int _{x_{0}}^{x} \int _{y_{0}}^{y}F \Bigl(x,y,z(x,y), \max _{(t,s)\in [x-h,x]\times [y-k,y]}z(t,s) \Bigr) \\ &\phantom{z(x,y)=}+ \int _{x_{0}}^{x} \int _{y_{0}}^{y}(x-t)^{-\lambda}(y-s)^{-\lambda}f(t,s)\,dt\,ds,\quad (x,y)\in \Omega , \\ &z(x,y)=\psi (x,y), \quad (x,y)\in \Omega _{0}, \\ &z(x,y_{0})=f(x),\qquad z(x_{0},y)=g(y),\quad x\geq x_{0},y\geq y_{0}, \end{aligned} \end{aligned}$$

(3.9)

where $\lambda (0<\lambda <1), x_{0}\geq 0, y_{0}\geq 0, h,k>0$ are constants, $\psi \in C(\Omega _{0},R), F\in C(\Omega \times R^{2},R), f\in C([x_{0},x_{1}),R), g\in C([y_{0},y_{1}),R), f(x_{0})=g(y_{0})$.

System (3.9) is more generalized than system (3.6). By Theorem 2.1, we can estimate solutions for the nonlinear equation. By analogy with the equation considered in Sect. 3, Corollary 3.2 of [23], we can prove that system (3.9) has at most one solution on Ω.