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Generalized nonlinear weakly singular retarded integral inequalities with maxima and their applications

Journal of Inequalities and Applications20182018:294

https://doi.org/10.1186/s13660-018-1885-6

  • Received: 19 March 2018
  • Accepted: 18 October 2018
  • Published:

Abstract

This paper deals with a generalized nonlinear weakly singular retarded Wendroff-type integral inequality with maxima of an unknown function of two variables. The key is that a technique of monotonization without separability and monotonicity of given functions is used for estimating the boundedness of unknown functions. Then our outcomes can be helpful to weaken conditions for some known results. By applying our results, the uniqueness of solutions for some singular integral equation with maxima may be proven.

Keywords

  • Integral inequalities
  • Maxima
  • Monotonicity
  • Uniqueness

MSC

  • 26D15
  • 34A08
  • 34A34

1 Introduction

The Gronwall inequality [1] holds a vital place in studying qualitative properties of the solutions of integral equations and differential equations. Some linear and nonlinear generalizations (e.g. [211]) of the Gronwall inequality have been extensively discussed. With further study of fractional differential equations, integral inequalities with weakly singular kernels have attracted more and more attention (see [1220]). In [14], a new method was presented to analyze the nonlinear singular integral inequalities of Henry type:
$$ u(t)\le a(t)+b(t) \int_{t_{0}}^{t}(t-s)^{\beta-1}s^{\gamma -1}F(s)u(s) \,ds,\quad t\ge0. $$
(1.1)
In 2008, Cheung et al. [20] solved the nonlinear weakly singular inequality
$$\begin{aligned} u^{p}(x,y) \le& a(x,y)+b(x,y) \int_{0}^{x} \int _{0}^{y}\bigl(x^{\alpha}-s^{\alpha} \bigr)^{\beta-1}s^{\gamma-1} \bigl(y^{\alpha}-t^{\alpha} \bigr)^{\beta-1}t^{\gamma-1} \\ &{} \cdot f(s,t)u^{q}(s,t)\,dt\,ds. \end{aligned}$$
(1.2)
On the other hand, since differential equations with maxima of the unknown function [2126] can be applied in control theory, some significant results for integral inequalities containing the maxima of the unknown function [22, 2730] have been obtained. The integral inequality with maxima
$$\begin{aligned}& u(x,y)\leq a(x,y)+ \int_{x_{0}}^{x} \int_{y_{0}}^{y} f(s,t) u^{p}(s,t)\,dt\,ds \\& \hphantom{u(x,y)\leq{}}{} + \int_{\alpha(x_{0})}^{\alpha(x)} \int_{y_{0}}^{y} g(s,t) \Bigl(\max _{\tilde{\eta}\in[s-h,s]}u^{p}(\tilde{\eta },t) \Bigr)\,dt\,ds,\quad x \ge x_{0}, y\ge y_{0}, \\& u(x,y)\leq \psi(x,y), \quad x\in \bigl[\alpha(x_{0})-h, x_{0}\bigr], y\ge y_{0}, \end{aligned}$$
(1.3)
where f, g, and ψ are nonnegative continuous functions and \(a(x,y)>0\) is a nondecreasing continuous function, was discussed in [22].
Combining (1.2) with (1.3), we will consider the integral inequality with maxima
$$ \begin{aligned} &\varphi\bigl(u(x,y)\bigr) \leq a(x,y)+ \sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{\beta_{j}-1}s^{\gamma_{j}-1}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{\bar{\beta}_{j}-1} t^{\bar{\gamma}_{j}-1} \\ &\hphantom{\varphi(u(x,y)) \leq{}}{}\cdot f_{j}(x,y,s,t) \omega_{j} \bigl(u(s,t)\bigr)\mu_{j} \Bigl(\max_{\tilde{\eta}\in[s-h, s]}g\bigl(u( \tilde{\eta},t)\bigr) \Bigr)\,dt\,ds, \\ &\hphantom{\varphi(u(x,y)) \leq{}}{}(x,y)\in [x_{0},x_{1}) \times[y_{0}, y_{1}), \\ &u(x,y) \leq \psi (x,y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times [y_{0}, y_{1}), \end{aligned} $$
(1.4)
where a, g, \(\omega_{j}\), \(f_{j}\), \(b_{j}\), and \(c_{j}\) are nonnegative continuous functions, \(b_{j}\) and \(c_{j}\) are increasing functions and belong to \(C^{1}\), \(b_{*}(x_{0}):=\min_{1\le j\le{m}}b_{j}(x_{0})\), \(h>0\) is a constant. Specially, the monotonicity of a, \(\omega_{j}\), \(\mu_{j}\), \(f_{j}\), and g is not required. Further, \(\omega_{j}\)’s are used to construct a sequence of stronger monotonized functions. Then the obtained result is applied for considering the uniqueness of solutions to a boundary value problem of an integral equation with maxima.

2 Main result

Let \(\mathbb {R}:=(-\infty, +\infty)\), \(\mathbb {R}_{+}:=[0,\infty)\), \(\Delta:=[x_{0},x_{1})\times[y_{0}, y_{1})\) and \(\Xi:= [b_{*}(x_{0})-h,x_{0}]\times[y_{0}, y_{1})\). Define \(\Phi_{1}, \Phi_{2}: B\subset\mathbb {R} \rightarrow \mathbb {R}\setminus\{0\}\). As in [4], if \(\Phi_{1}/\Phi _{2}\) is nondecreasing on B, then \(\Phi_{1}\varpropto\Phi_{2}\). Considering inequality (1.4), we make the following assumptions for all \(j=1,\ldots,m\):
(A1): 

\(b_{j}\in C^{1}([x_{0},x_{1}),\mathbb {R}_{+})\) and \(c_{j}\in C^{1}([y_{0},y_{1}), [y_{0},y_{1}))\) are nondecreasing such that \(b_{j}(x)\leq x\) and \(c_{j}(y)\le y\), and \(c_{j}(y_{0})=y_{0}\);

(A2): 

\(a\in C(\Delta,\mathbb {R}_{+}) \), \(f_{j}\in C(\Delta\times [b_{*}(x_{0}),x_{1})\times[y_{0},y_{1}), \mathbb{R}_{+})\), \(\omega_{j},\mu_{j}\in C(\mathbb{R}_{+},\mathbb {R}_{+}) \) with \(\omega_{j}(t)>0\), \(\mu _{j}(t)>0\) for \(t>0\);

(A3): 

\(g, \varphi\in C(\mathbb {R}_{+},\mathbb{R}_{+})\) and \(\psi\in C(\Xi, \mathbb {R}_{+})\), and φ is strictly increasing such that \(\lim_{t\rightarrow\infty}\varphi(t)=\infty\);

(A4): 

\(\alpha_{j}, \bar{\alpha}_{j}\in(0,1]\), \(\beta_{j},\bar{\beta}_{j}\in(0,1)\), \(\gamma_{j}>1-\frac{1}{p}\), \(\bar{\gamma}_{j}>1-\frac{1}{p}\) such that \(\frac{1}{p}+\alpha_{j}(\beta_{j}-1)+\gamma_{j}-1\ge0\), \(\frac {1}{p}+\bar{\alpha}_{j}(\bar{\beta}_{j}-1)+\bar{\gamma}_{j}-1\ge0\), \(p(\beta_{j}-1)+1>0\), \(p(\bar{\beta}_{j}-1)+1>0\), \(p>1\).

For those \(\omega_{j}\)’s, \(\mu_{j}\)’s given in (A4), define \(\tilde {\omega}_{j}(t)\) inductively by
$$ \tilde{\omega}_{j}(t):= \textstyle\begin{cases} \hat{\omega}_{1}(t)\max_{\tau\in[0, t] }\{\hat{\mu }_{1}(\tilde{g}(\tau))\}, & t\ge0, j=1, \\ \max_{\tau\in[0, t] }\{\frac{\hat{\omega}_{j}(\tau)\hat {\mu}_{j+1}(\tilde{g}((\tau))}{\tilde{\omega}_{i-1}(\tau)}\}\tilde{\omega}_{i-1}(t),& t\ge0, j=2,\ldots,m, \end{cases} $$
(2.1)
where \(\hat{\omega}_{j}(t):=\max_{\tau\in[0, t] }\{\bar {\omega}_{j}(\tau)\}\), \(\hat{\mu}_{j}(t):=\max_{\tau\in[0, t] }\{\bar{\mu}_{j}(\tau)\}\), \(\tilde{g}(t):=\max_{\tau\in[0, t] }\{g(\tau)\}\), \(\bar{\omega}_{j}(t):=\omega_{j}(t)+\varepsilon_{j}\), \(\bar{\mu }_{j}(t):=\mu_{j}(t)+\varepsilon_{j}\) for \(t\ge0\), \(\epsilon_{j}:= \varepsilon\) if \(\omega_{j}(0)=0\) or \(:=0\) if \(\omega_{j}(0)\neq0\) for all \(j=1,2,\ldots,m\), where \(\varepsilon>0\) is an arbitrarily given constant.

Lemma 1

([16])

Let α, β, γ, and p be positive constants. Then
$$ \int^{t}_{0}\bigl(t^{\alpha}-s^{\alpha} \bigr)^{p(\beta-1)}s^{p(\gamma -1)}\,ds=\frac{t^{\theta}}{\alpha}B\biggl( \frac{p(\gamma-1)+1}{\alpha}, p(\beta-1)+1\biggr),\quad t\in{\mathbb{R}_{+}}, $$
where \(\theta:=p[\alpha(\beta-1)+\gamma-1]+1\), \(B(\xi,\eta)=\int ^{1}_{0}s^{\xi-1}(1-s)^{\eta-1}\,ds\) (\(\operatorname{Re} \xi>0\), \(\operatorname{Re} \eta>0\)) is the beta function.

Lemma 2

Suppose that
  1. (C1)

    \(b_{j}\in C^{1}([x_{0},x_{1}),\mathbb {R}_{+})\) and \(c_{j}\in C^{1}([y_{0},y_{1}), [y_{0},y_{1}))\) are nondecreasing with \(b_{j}(x)\leq x\) on \([x_{0},x_{1})\), \(c_{j}(y)\le y\) on \([y_{0},y_{1})\) and \(c_{j}(y_{0})=y_{0}\) for all \(j=1,\ldots,m\);

     
  2. (C2)

    \(\psi\in C(\Xi,\mathbb {R}_{+})\), \(g_{j}\in C(\Delta\times\mathbb {R}^{2}_{+},\mathbb {R}_{+})\) are nondecreasing functions in x and y for all \(j=1,\ldots,m\);

     
  3. (C3)

    \(h_{j}, \bar{h}_{j}\in C(\mathbb {R}_{+},\mathbb{R}_{+})\) (\(j=1,\ldots,m\)) are all nondecreasing with \(h_{j}(t)>0\), \(\bar{h}_{j}(t)>0\) for \(t>0\), and \(h_{j}\bar{h}_{j}\propto h_{j+1}\bar{h}_{j+1}\) (\(j=1,\ldots,m-1\));

     
  4. (C4)

    \(b\in C(\Delta, \mathbb{R}_{+})\), \(b_{x}, b_{y}\in(\Delta, \mathbb{R})\), and \(\max_{s\in[b_{*}(x_{0})-h,x_{0}]}\psi(s,t)\le b(x_{0},t)\) for all \(t\in [y_{0},y_{1})\).

     
If \(u\in C([b_{*}(x_{0})-h,x_{1})\times[y_{0},y_{1}),\mathbb {R}_{+})\) satisfies the integral inequality
$$\begin{aligned}& u(x,y)\leq b(x,y)+\sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} g_{j}(x,y,s,t) \\& \hphantom{u(x,y)\leq{}}{}\times h_{j}\bigl(u(s,t)\bigr)\tilde{{h}}_{j} \Bigl(\max_{\tilde{\eta }\in[s-h, s]}u(\tilde{\eta},t) \Bigr)\,dt\,ds, \quad (x,y) \in\Delta, \\& u(x,y)\leq \psi (x,y), \quad (x,y)\in\Xi, \end{aligned}$$
(2.2)
then
$$ u(x,y)\leq H_{m}^{-1} \biggl(H_{m} \bigl(\eta_{m}(x,y)\bigr)+ \int _{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} g_{m}(x,y,s,t)\,dt\,ds \biggr) $$
(2.3)
for all \((x,y)\in[x_{0}, X_{1}^{*}]\times[y_{0},Y_{1}^{*}]\), where \(H_{j}^{-1}\) is the inverse of the function
$$ H_{j}(t):= \int_{t_{j}}^{t}\frac{ds}{h_{j}(s)\bar{h}_{j}(s)}, \quad t\ge t_{j}>0, j=1,\ldots,m, $$
(2.4)
\(t_{j}\) is a given constant, and \(\eta_{j}\) is defined by
$$ \begin{aligned} &\eta_{1}(x,y):=b(x_{0},y_{0})+ \int_{x_{0}}^{x} \bigl\vert b_{x}(s,y_{0}) \bigr\vert \,ds+ \int_{y_{0}}^{y} \bigl\vert b_{x}(x,t) \bigr\vert \, dt, \\ &\eta_{j+1}(x,y):=H_{j}^{-1} \biggl(H_{j}\bigl(\eta_{j}(x,y)\bigr)+ \int_{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}g_{j}(x,y,s,t)dt \,ds \biggr) \end{aligned} $$
(2.5)
for \(j=1,\ldots,m-1\), and \(x_{0}\le X_{1}^{*}< x_{1}\), \(y_{0}\le Y_{1}^{*}< y_{1}\) are chosen such that
$$ H_{j}\bigl(\eta_{j}\bigl(X^{*}_{1},Y^{*}_{1} \bigr)\bigr)+ \int_{a_{j}(x_{0})}^{a_{j}(X^{*}_{1})} \int_{b_{j}(y_{0})}^{b_{j}(Y^{*}_{1})}g_{j}\bigl(X^{*}_{1},Y^{*}_{1},s,t \bigr)\, dt \,ds\le \int _{u_{j}}^{\infty}\frac{ds}{{h_{j}(s)}\tilde{h}(s)} $$
(2.6)
for \(j=1,\ldots,m\).

Proof

Let b be positive on Δ. It means that \(\eta_{1}(x,y)\) is positive on Δ. Under such a circumstance, \(\eta_{1}\) is nondecreasing on Δ and \(\eta_{1}(x,y)>0\),
$$ \eta_{1}(x,y)\ge b(x_{0},y_{0})+ \int_{x_{0}}^{x} b_{x}(s,y_{0}) \,ds+ \int _{y_{0}}^{y}b_{y}(x,t)\, dt=b(x,y). $$
(2.7)
From (2.2) and (2.7), we have
$$ \begin{aligned} &u(x,y)\leq \eta_{1}(x,y)+ \sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} g_{j}(x,y,s,t) \\ &\hphantom{u(x,y)\leq{}}{}\cdot h_{j}\bigl(u(s,t)\bigr)\bar{h}_{j} \Bigl(\max_{\tilde{\eta}\in[s-h, s]}u(\tilde{\eta},t) \Bigr)\,dt\,ds,\quad (x,y) \in\Delta, \\ &u(x,y)\leq \psi (x,y), \quad (x,y)\in\Xi. \end{aligned} $$
(2.8)
Concerning (2.8), we consider the auxiliary inequality
$$ \begin{aligned} &u(x,y) \leq \eta_{1}(x,y)+ \sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} g_{j}(\xi,\eta,s,t) \\ &\hphantom{u(x,y) \leq{}}{} \times h_{j}\bigl(u(s,t)\bigr) \bar{h}_{j} \Bigl(\max_{\tilde{\eta}\in[s-h, s]}u(\tilde{\eta},t) \Bigr)\,dt\,ds, \quad (x,y)\in[x_{0},\xi]\times[y_{0}, \eta], \\ &u(x,y) \leq \psi(x,y), \quad (x,y)\in \bigl[b_{*}(x_{0})-h, x_{0}\bigr]\times[y_{0},\eta], \end{aligned} $$
(2.9)
where \(x_{0}\leq\xi\le X^{*}_{1}\) and \(y_{0}\leq\eta\le Y^{*}_{1}\) are chosen arbitrarily. Having (2.9) we claim
$$ u(x,y)\leq H_{m}^{-1} \biggl(H_{m} \bigl(\eta_{m}(\xi,\eta,x,y)\bigr)+ \int _{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} g_{m}(\xi,\eta,s,t)\,dt \,ds \biggr) $$
(2.10)
for \(x_{0}\le x \le\min\{\xi, X^{*}_{2}\}\), \(y_{0}\le y \le\min\{\eta, Y^{*}_{2}\}\), where \(\tilde{\eta}_{j}(\xi,\eta,x,y)\) is defined inductively by \(\tilde {\eta}_{1}(\xi,\eta,x,y):=\eta_{1}(x,y)\) and
$$ \tilde{\eta}_{j}(\xi,\eta,x,y):= H_{j-1}^{-1} \biggl(H_{j-1}\bigl(\tilde{\eta }_{j-1}(\xi,\eta,x,y)\bigr)+ \int_{b_{j-1}(x_{0})}^{b_{j-1}(x)} \int _{c_{j-1}(y_{0})}^{c_{j-1}(y)} g_{j-1}(\xi,\eta,s,t)\,dt \,ds\biggr) $$
for \(j=2,\ldots, m\), and \(X^{*}_{2}\in[x_{0},x_{1})\), \(Y^{*}_{2}\in[y_{0},y_{1})\) are chosen such that
$$\begin{aligned} &H_{j}\bigl(\tilde{\eta}_{j}\bigl(\xi, \eta,X^{*}_{2},Y^{*}_{2}\bigr)\bigr)+ \int _{b_{j}(x_{0})}^{b_{j}(X^{*}_{2})} \int_{c_{j}(y_{0})}^{c_{j}(Y^{*}_{2})} g_{j}(\xi,\eta,s,t) \\ &\quad \le \int_{t_{j}}^{\infty}\frac{ds}{{h_{j}(s)}\bar{h}_{j}(s)} \end{aligned}$$
(2.11)
for \(j=1,2,\ldots,m\). Note that \(X^{*}_{2}\ge X^{*}_{1}\) and \(Y^{*}_{2}\ge Y^{*}_{1}\). In fact, both \(\tilde{\eta}_{j}(\xi,\eta,x,y)\) and \(g_{j}(\xi,\eta,x,y)\) are nondecreasing in ξ and η. Thus \(X^{*}_{2}\), \(Y^{*}_{2}\) satisfying (2.11) will get smaller as ξ, η are chosen larger.
Since \(\max_{s\in[b^{*}(x_{0})-h,x_{0}]}\psi(s,t)\le b(x_{0},t)\) and \(b(x_{0},t)\le \eta_{1}(x_{0},t)\le\eta_{1}(x,t)\), we obtain
$$ \max_{s\in[b_{*}(x_{0})-h,x_{0}]}\psi(s,t)\leq\eta _{1}(x,t), \quad (x,t)\in[x_{0},x_{1}) \times[y_{0},y_{1}). $$
(2.12)
First, (2.10) holds for \(m=1\). In fact,(2.9) for \(m=1\) is written as
$$ u(x,y)\leq z_{1}(x,y),\quad (x,y)\in \bigl[b_{*}(x_{0})-h, \xi\bigr]\times[y_{0}, \eta], $$
(2.13)
where
$$ z_{1}(x,y)= \textstyle\begin{cases} \eta_{1}(x,y)+ \int_{b_{1}(x_{0})}^{b_{1}(x)}\int_{c_{1}(y_{0})}^{c_{1}(y)} g_{1}(\xi,\eta,s,t) h_{1}(u(s,t)) \\ \quad {}\times\bar{h}_{1} (\max_{\tilde{\eta}\in[s-h, s]}u(\tilde{\eta},t) )\,dt\,ds, \quad (x,y)\in[x_{0},\xi]\times[y_{0},\eta] \\ \eta_{1}(x_{0},y), \quad (x,y)\in[b_{*}(x_{0})-h, x_{0}]\times[y_{0},\eta], \end{cases} $$
(2.14)
\(z_{1}(x,y)\) is a nondecreasing function on \([x_{0}, \xi]\times[y_{0},\eta]\). Then
$$\begin{aligned} \frac{\partial}{\partial x}z_{1}(x,y) =&\frac{\partial }{\partial x} \eta_{1}(x,y)+ \int_{c_{1}(y_{0})}^{c_{1}(y)} g_{1}\bigl(\xi, \eta,b_{1}(x),t\bigr) h_{1}\bigl(u\bigl(b_{1}(x),t \bigr)\bigr) \\ &{}\times\bar{h}_{1} \Bigl(\max_{\tilde{\eta }\in[b_{1}(x)-h, b_{1}(x)]}u(\tilde{ \eta},t) \Bigr)\, dtb'(x) \end{aligned}$$
for all \((x,y)\in[x_{0},\xi]\times[y_{0},\eta] \). We have \(0< h_{1}(u(s,t))\bar{h}_{1}(u(s,t))\le h_{1}(z_{1}(s,t))\bar{h}_{1}(z_{1}(s,t)) \le h_{1}(z_{1}(x,y))\bar{h}_{1}(z_{1}(x,y)) \) by (C3) and (2.13) \(s\le b_{1}(x)\le x\), \(t\le c_{1}(y)\le y\) and both \(z_{1}\) and \(h_{1}\tilde{h}_{1}\) are nondecreasing. Thus
$$\begin{aligned}& \frac{\frac{\partial}{\partial x}z_{1}(x,y)}{h_{1}(z_{1}(x,y))\bar{h}_{1}(z_{1}(x,y))} \\& \quad \le \frac{\frac{\partial}{\partial x}\eta _{1}(x,y)}{h_{1}(\eta_{1}(x,y))\bar{h}_{1}(\eta_{1}(x,y))}+\frac {b'(x)}{h_{1}(z_{1}(x,y))\bar{h}_{1}(z_{1}(x,y))} \\& \qquad {}\times \int_{c_{1}(y_{0})}^{c_{1}(y)} g_{1}\bigl(\xi, \eta,b_{1}(x),t\bigr) h_{1}\bigl(u\bigl(b_{1}(x),t \bigr)\bigr)\bar{h}_{1} \Bigl(\max_{\tilde{\eta}\in [b_{1}(x)-h, b_{1}(x)]}u(\tilde{ \eta},t) \Bigr)\,dt \\& \quad \le \frac{\frac{\partial}{\partial x}\eta_{1}(x,y)}{h_{1}(\eta_{1}(x,y))\bar {h}_{1}(\eta_{1}(x,y))}+b'(x) \int_{c_{1}(y_{0})}^{c_{1}(y)} g_{1}\bigl(\xi, \eta,b_{1}(x),t\bigr)\,dt. \end{aligned}$$
(2.15)
Integrating inequality (2.15) from \(x_{0}\) to x, from (2.4) we get
$$\begin{aligned} H_{1}\bigl(Z_{1}(x,y)\bigr) \le& H_{1}\bigl( \eta_{1}(x,y)\bigr)+ \int _{x_{0}}^{x}b'(s) \int_{c_{1}(y_{0})}^{c_{1}(y)} g_{1}\bigl(\xi, \eta,b_{1}(s),t\bigr)\,dt\,ds \\ =& H_{1}\bigl(\eta_{1}(x,y)\bigr)+ \int_{b_{1}(x_{0})}^{b_{1}(x)} \int _{c_{1}(y_{0})}^{c_{1}(y)} g_{1}(\xi,\eta,s,t)\,dt \,ds \end{aligned}$$
(2.16)
for all \((x,y)\in[x_{0},\xi]\times[y_{0},\eta]\). From (2.14), (2.16), and the monotonicity of \(H^{-1}_{1}\), we have
$$ u(x,y))\le H^{-1}_{1}\biggl( H_{1}\bigl(\eta_{1}(x,y)\bigr)+ \int_{b_{1}(x_{0})}^{b_{1}(x)} \int _{c_{1}(y_{0})}^{c_{1}(y)} g_{1}(\xi,\eta,s,t)\,dt \,ds\biggr) $$
(2.17)
for \(x_{0}\le x\le\xi< X^{*}_{2}\), \(Y_{0}\le y \le\eta< Y^{*}_{2}\), implying that (2.7) is true for \(m=1\).
Assume that (2.10) holds for \(m=k\). Consider
$$\begin{aligned}& u(x,y) \leq \eta_{1}(x,y)+\sum_{j=1}^{k+1} \int_{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} g_{j}(\xi,\eta,s,t) \\& \hphantom{u(x,y) \leq{}}{}\times h_{j}\bigl(u(s,t)\bigr)\bar{h}_{j} \Bigl(\max_{\tilde{\eta}\in[s-h, s]}u(\tilde{\eta},t)\Bigr)\,dt\,ds,\quad (x,y) \in[x_{0},\xi]\times[y_{0},\eta] \\& u(x,y) \leq \psi (x,y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h, x_{0}\bigr]\times[y_{0},\eta] . \end{aligned}$$
(2.18)
Let
$$ z_{2}(x,y)= \textstyle\begin{cases} \eta_{1}(x,y) +\sum_{j=1}^{k+1} \int_{b_{j}(x_{0})}^{b_{j}(x)}\int _{c_{j}(y_{0})}^{c_{j}(y)} g_{j}(\xi,\eta, s,t)h_{j}(u(s,t)) \\ \quad {}\cdot\bar{h}_{j}(\max_{\tilde{\eta}\in[s-h, s]}u(\tilde {\eta},t))\,dt\,ds,\quad (x,y)\in[x_{0},\xi]\times[y_{0},\eta], \\ \eta_{1}(x_{0},y),\quad (x,y)\in [b_{*}(x_{0})-h, x_{0}]\times[y_{0},\eta]. \end{cases} $$
(2.19)
Then \(z_{2}\) is a nondecreasing function on \([x_{0}, x]\times[y_{0},\eta]\). By (2.19) and the definition of \(z_{2}\), it follows that
$$ u(x,y)\leq z_{2}(x,y),\quad (x,y)\in \bigl[b_{*}(x_{0})-h, \xi\bigr]\times [y_{0}, \eta]. $$
(2.20)
Since \(h_{j}\bar{h}_{j}\) is nondecreasing and \(z_{2}(x,y)>0\), \(b'_{j}(x)\ge 0\), and \(b_{j}(x)\le x\), we have
$$\begin{aligned}& \frac{\frac{\partial}{\partial x}z_{2}(x,y)}{h_{1}(z_{2}(x,y))\bar {h}_{1}(z_{2}(x,y))} \\& \quad \le\frac{\frac{\partial}{\partial x}\eta _{1}(x,y)}{h_{1}(z_{2}(x,y))\bar{h}_{1}(z_{2}(x,y))}+\sum_{j=1}^{k+1} \frac{b'_{j}(x)}{ h_{1}(z_{2}(x,y))\bar{h}_{1}(z_{2}(x,y))} \\& \qquad {}\cdot \int_{c_{j}(y_{0})}^{c_{j}(y)}g_{j}\bigl(X,Y,b_{j}(x),t \bigr)h_{j}\bigl(u\bigl(b_{j}(x),t\bigr)\bigr) h_{j}\Bigl(\max_{\xi\in[b_{j}(x)-h,b_{j}(x)]}u(\tilde{\eta},t)\Bigr)\,dt \\& \quad \le\frac{\frac{\partial}{\partial x}\eta_{1}(x,y)}{h_{1}(\eta _{1}(x,y))\bar{h}_{1}(\eta_{1}(x,y))}+\sum_{j=1}^{k+1} \frac{b'_{j}(x)}{ h_{j}(z_{2}(x,y))\bar{h}_{j}(z_{2}(x,y))} \\& \qquad {}\cdot \int_{c_{j}(y_{0})}^{c_{j}(y)}g_{j}\bigl(\xi, \eta,b_{j}(x),t\bigr)h_{j}\bigl(z_{2} \bigl(b_{j}(x),t\bigr)\bigr) \bar{h}_{j}\Bigl(\max _{\tilde{\eta}\in [b_{j}(x)-h,b_{j}(x)]}z_{2}(\tilde{\eta},t)\Bigr)\,dt \\& \quad \le\frac{\frac{\partial}{\partial x}\eta_{1}(x,y)}{h_{1}(\eta _{1}(x,y))\bar{h}_{1}(\eta_{1}(x,y))} +b'_{1}(x) \int_{c_{1}(y_{0})}^{c_{1}(y)}g_{1}\bigl(\xi, \eta,b_{1}(x),t\bigr)\,dt+\sum_{j=1}^{k}b'_{j+1}(x) \\& \qquad {} \cdot \int_{c_{j}(y_{0})}^{c_{j}(y)}g_{j+1}\bigl(\xi,\eta ,b_{j+1}(x),t\bigr)\tilde{h}_{j+1}\bigl(z_{2} \bigl(b_{j+1}(x),t\bigr)\bigr) \hat{h}_{j+1}\Bigl(\max _{\tilde{\eta}\in [b_{j}(x)-h,b_{j}(x)]}z_{2}(\tilde{\eta},t)\Bigr)\,dt \end{aligned}$$
for all \((x,y)\in[x_{0},X_{1}^{*}]\times[y_{0},Y_{1}^{*}]\), where \(\tilde {h}_{j+1}(u):=h_{j+1}(u)/h_{1}(u)\), \(\hat{h}_{j+1}(u):=\bar{h}_{j+1}(u)/\bar{h}_{1}(u)\), \(j=1,\ldots,k\). Integrating the above inequality from \(x_{0}\) to x, we can obtain
$$\begin{aligned} H_{1}\bigl(z_{2}(x,y)\bigr) \le& H_{1}\bigl( \eta_{1}(x,y)\bigr)+ \int_{b_{1}(x_{0})}^{b_{1}(x)} \int _{c_{1}(y_{0})}^{c_{1}(y)}g_{1}(\xi,\eta,s,t)\,dt \,ds \\ &{} +\sum_{j=1}^{k} \int_{b_{j+1}(x_{0})}^{b_{j+1}(x)} \int _{c_{j+1}(y_{0})}^{c_{j+1}(y)}g_{j+1}(\xi,\eta,s,t) \tilde{h}_{j+1}\bigl(z_{2}(s,t)\bigr) \\ &{} \cdot\hat{h}_{j+1} \Bigl(\max_{\tilde{\eta}\in [s-h,s]}z_{2}( \tilde{\eta},t) \Bigr)\,dt\,ds \end{aligned}$$
(2.21)
for all \((x,y)\in[x_{0},X]\times[y_{0},Y]\). Let
$$ \begin{aligned} &\eta(x,y):=H_{1}\bigl(z_{2}(x,y)\bigr), \\ &\varrho_{1}(x,y):=H_{1}\bigl(\eta_{1}(x,y) \bigr)+ \int_{b_{1}(x_{0})}^{b_{1}(x)} \int _{c_{1}(y_{0})}^{c_{1}(y)}g_{1}(\xi,\eta,s,t)\,dt \,ds. \end{aligned} $$
(2.22)
Then inequality (2.21) can be rewritten as
$$\begin{aligned}& \eta(x,y) \le \varrho_{1}(x,y)+\sum_{j=1}^{k} \int _{b_{j+1}(x_{0})}^{b_{j+1}(x)} \int _{c_{j+1}(y_{0})}^{c_{j+1}(y)}g_{j+1}(\xi,\eta,s,t)\tilde {h}_{j+1}\bigl(H_{1}^{-1}\bigl(z_{2}(s,t) \bigr)\bigr) \\& \hphantom{\eta(x,y) \le{}}{} \cdot\hat{h}_{j+1}\Bigl(\max_{\tilde{\eta}\in [s-h,s]}H_{1}^{-1} \bigl(z_{2}(\tilde{\eta},t)\bigr)\Bigr)\,dt\,ds, \quad (x,y)\in [x_{0},X]\times[y_{0},Y], \\& \eta(x,y) = H_{1}\bigl(\eta_{(}x_{0},y)\bigr)\le \varrho_{1}(x_{0}, y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h, x_{0}\bigr]\times[y_{0},Y], \end{aligned}$$
(2.23)
the same form as (2.9) for \(m=k\). By (C3), each \((\bar {h}_{j+1}\circ H_{1}^{-1})(\tilde{h}_{j+1}\circ H_{1}^{-1})\) (\(j=1,\ldots,k\)) is a nonnegative continuous and increasing function on \(\mathbb{R}_{+}\) and positive on \((0,+\infty)\). Moreover, \((\tilde{h}_{j}\circ H_{1}^{-1})\propto(\hat{h}_{j+1}\circ H_{1}^{-1})\) for all \(j=2,\ldots, k\). By the inductive assumption, we have
$$ \eta(x,y)\le \bar{H}_{k+1}^{-1}\biggl( \bar{H}_{k+1}\bigl(\varrho _{k}(x,y)\bigr)+ \int_{b_{k+1}(x_{0})}^{b_{k+1}(x)} \int_{c_{k+1}(y_{0})}^{c_{k+1}(y)} g_{k+1}(\xi,\eta,s,t)\,dt \,ds\biggr) $$
(2.24)
for \(x_{0}\le x\le\min\{\xi, X_{3}^{*}\}\), \(y_{0}\le y\le\min\{\eta, Y_{3}^{*}\}\), where
$$ \bar{H}_{j+1}(t):= \int_{\tilde{t}_{j+1}}^{t}\frac{ds}{\tilde {h}_{j+1}(H_{1}^{-1}(s))\hat{h}_{j+1}(H_{1}^{-1}(s))},\quad t>0, $$
(2.25)
\(\tilde{t}_{j+1}=H_{1}(t_{j+1})\), \(\bar{H}^{-1}_{j+1}\) is the inverse of \(\bar{H}_{j+1}\), \(j=1,\ldots, k\),
$$ \varrho_{j+1}(x,y):=\bar{H}^{-1}_{j+1} \biggl(\bar{H}_{j+1}\bigl(\varrho _{j}(x,y)\bigr)+ \int_{b_{j+1}(x_{0})}^{b_{j+1}(x)} \int_{c_{j+1}(y_{0})}^{c_{j+1}(y)}g_{j+1}(\xi,\eta,s,t)\,dt \,ds\biggr), $$
(2.26)
\(j=1,\ldots,k-1\), and \(X^{*}_{3}\), \(Y^{*}_{3}\) are chosen such that
$$\begin{aligned}& \bar{H}_{j+1}\bigl(\varrho_{j}\bigl(X^{*}_{3},Y^{*}_{3} \bigr)\bigr)+ \int _{b_{j+1}(x_{0})}^{b_{j+1}(X^{*}_{3})} \int_{c_{j+1}(y_{0})}^{c_{j+1}(Y^{*}_{3})}g_{j+1}(\xi,\eta,t,s)\,dt \,ds \\& \quad \le \int_{\tilde{t}_{j+1}}^{H_{1}(\infty)}\frac{ds}{\tilde {h}_{j+1}(H^{-1}_{1}(s))\hat{h}_{j+1}(H_{1}^{-1}(s))},\quad j=1, \ldots,k. \end{aligned}$$
(2.27)
Note that
$$\begin{aligned} \bar{H}_{j}(t) =& \int_{\tilde{t}_{j}}^{t}\frac {ds}{\tilde{h}_{j}(H_{1}^{-1}(s))\hat{h_{j}}(H_{1}^{-1}(s))} \\ =& \int_{H_{1}(t_{j})}^{t}\frac{h_{1}(H^{-1}_{1}(s))\bar {h}_{1}(H^{-1}_{1}(s))\,ds}{h_{j}(H_{1}^{-1}(s))\bar{h}_{j}(H_{1}^{-1}(s))} \\ =& \int_{H_{1}(t_{j})}^{t}\frac{h_{1}(H^{-1}_{1}(s))\bar {h}_{1}(H^{-1}_{1}(s))\,ds}{h_{j}(H_{1}^{-1}(s))\bar{h}_{j}(H_{1}^{-1}(s))} \\ =& \int_{t_{j}}^{H^{-1}_{1}(t)}\frac{ds}{h_{j}(s)\bar{h}_{j}(s)}=H_{j}\bigl(H^{-1}_{1}(t)\bigr), \quad j=2, \ldots,k+1. \end{aligned}$$
(2.28)
Then, from (2.20), (2.24), and (2.28), we get
$$\begin{aligned} u(x,y) \le& H^{-1}_{1}\bigl(\eta(x,y)\bigr) \\ \le& H_{k+1}^{-1}\biggl(H_{k+1} \bigl(H^{-1}_{1}\bigl(\varrho_{k}(x,y)\bigr) \bigr) + \int_{b_{k+1}(x_{0})}^{b_{k+1}(x)} \int_{c_{k+1}(y_{0})}^{c_{k+1}(y)} g_{k+1}(\xi,\eta,s,t)\,dt \,ds\biggr) \end{aligned}$$
(2.29)
for \(x_{0}\le x\le\min\{X, X_{3}^{*}\}\), \(y_{0}\le y\le\min\{Y, Y_{3}^{*}\} \). Let \(\tilde{\varrho}_{j}(x,y)=H^{-1}_{1}(\varrho_{j}(x,y))\). Then
$$\begin{aligned} \tilde{\varrho}_{1}(x,y) =&H_{1}\bigl( \varrho_{1}(x,y)\bigr) \\ =&H^{-1}_{1}\biggl(H_{1}\bigl( \eta_{1}(x,y)\bigr)+ \int_{b_{1}(x_{0})}^{b_{1}(x)} \int _{c_{1}(y_{0})}^{c_{1}(y)}g_{1}(\xi,\eta,s,t)\,dt \,ds\biggr) \\ =&H^{-1}_{1}\biggl(H_{1}\bigl(\tilde{ \eta}_{1}(\xi,\eta,x,y)\bigr)+ \int _{b_{1}(x_{0})}^{b_{1}(x)} \int_{c_{1}(y_{0})}^{c_{1}(y)}g_{1}(\xi,\eta,s,t)\,dt \,ds\biggr) \\ =&\tilde{\eta}_{2}(X,Y,x,y). \end{aligned}$$
(2.30)
Moreover, with the assumption that \(\tilde{\varrho}_{k}(x,y)=\tilde {\eta}_{k+1}(\xi,\eta,x,y)\), we get
$$\begin{aligned} \tilde{\varrho}_{k+1}(x,y) =&H^{-1}_{1}\biggl( \bar {H}^{-1}_{k+1}\biggl(\bar{H}_{k+1}\bigl( \varrho_{k}(x,y)\bigr)+ \int_{b_{k+1}(x_{0})}^{b_{k+1}(x)} \int_{c_{k+1}(y_{0})}^{c_{k+1}(y)}g_{k+1}(\xi,\eta,t,s)\,dt\,ds \biggr)\biggr) \\ =&H^{-1}_{k+1}\biggl(H_{k+1}\bigl(H^{-1}_{1} \bigl(\varrho_{k}(x,y)\bigr)\bigr)+ \int _{b_{k+1}(x_{0})}^{b_{k+1}(x)} \int_{c_{k+1}(y_{0})}^{c_{k+1}(y)}g_{k+1}(\xi,\eta,t,s)\,dt\,ds \biggr) \\ =&H^{-1}_{k+1}\biggl(H_{k+1}\bigl(\tilde{ \varrho}_{k}(x,y)\bigr)+ \int _{b_{k+1}(x_{0})}^{b_{k+1}(x)} \int_{c_{k+1}(y_{0})}^{c_{k+1}(y)}g_{k+1}(\xi,\eta,t,s)\,dt\,ds \biggr) \\ =&H^{-1}_{k+1}\biggl(H_{k+1}\bigl(\tilde{ \eta}_{k+1}(\xi,\eta ,x,y)\bigr)+ \int_{b_{k+1}(x_{0})}^{b_{k+1}(x)} \int_{c_{k+1}(y_{0})}^{c_{k+1}(y)}g_{k+1}(\xi,\eta,t,s)\,dt\,ds \biggr) \\ =&\tilde{\eta}_{k+2}(\xi,\eta,x,y). \end{aligned}$$
(2.31)
This proves that
$$ \tilde{\varrho}_{j}(x,y)=\tilde{\eta}_{j+1}( \xi,\eta, x,y),\quad j=1,\ldots, k . $$
(2.32)
Therefore, (2.27) becomes
$$\begin{aligned}& H_{j+1}\bigl(\tilde{\eta}_{j+1}\bigl(\xi, \eta,X^{*}_{3},Y^{*}_{3}\bigr) \bigr)+ \int _{b_{j+1}(x_{0})}^{b_{j+1}(X^{*}_{3})} \int_{c_{j+1}(y_{0})}^{c_{j+1}(Y^{*}_{3})}g_{j+1}(\xi,\eta,t,s)\,dt \,ds \\& \quad \le \int_{\tilde{t}_{j+1}}^{H_{1}(\infty)}\frac{ds}{\tilde {h}_{j+1}(H^{-1}_{1}(s))\hat{h}_{j+1}(H_{1}^{-1}(s))} \\& \quad = \int_{t_{j+1}}^{\infty}\frac{ds}{h_{j+1}(s)\bar{h}_{j+1}(s)},\quad j=1, \ldots,k, \end{aligned}$$
(2.33)
which implies that \(X^{*}_{2}=X^{*}_{3}\), \(\xi\le X^{*}_{3}\), \(Y^{*}_{2}=Y^{*}_{3}\), \(\eta\le Y^{*}_{3}\). From (2.29) we obtain
$$ u(x,y)\le H_{k+1}^{-1}\biggl(H_{k+1}\bigl(\tilde{ \eta}_{k+1}(\xi,\eta,x,y)\bigr)+ \int _{b_{k+1}(x_{0})}^{b_{k+1}(x)} \int_{c_{k+1}(y_{0})}^{c_{k+1}(y)} g_{k+1}(\xi,\eta,s,t)\,dt \,ds\biggr) $$
for \(x_{0}\le x\le \min\{X,X_{2}^{*}\}\), \(y_{0}\le y\le \min\{Y,Y_{2}^{*}\}\). This proves (2.10) by induction.
Taking \(x=\xi,\eta\), \(y=\xi,\eta\) in (2.10), we have
$$\begin{aligned} u(\xi,\eta) \leq&H_{m}^{-1} \biggl(H_{m}\bigl( \tilde{\eta}_{m}(\xi,\eta ,\xi,\eta)\bigr)+ \int_{b_{m}(x_{0})}^{b_{m}(X)} \int _{c_{m}(y_{0})}^{c_{m}(\eta)} g_{m}(\xi,\eta,s,t)\,dt \,ds \biggr) \\ = &H_{m}^{-1} \biggl(H_{m}\bigl( \eta_{m}(\xi,\eta)\bigr)+ \int _{b_{m}(x_{0})}^{b_{m}(\xi)} \int_{c_{m}(y_{0})}^{c_{m}(\eta)} g_{m}(\xi,\eta,s,t)\,dt \,ds \biggr) \end{aligned}$$
(2.34)
for \(x_{0}\le\xi\le X^{*}_{1}\), \(y_{0}\le\eta\le Y^{*}_{1}\), since \(x^{*}_{2}\ge X^{*}_{1}\), \(Y^{*}_{2}\ge Y^{*}_{1}\) and \(\tilde{\eta}_{m}(\xi,\eta,\xi,\eta)= \eta_{m}(\xi,\eta)\). Since ξ, η are arbitrary, replacing ξ and η with x and y, respectively, we have
$$ u(x,y) \le H_{m}^{-1} \biggl(H_{m}\bigl( \eta_{m}(x,y)\bigr)+ \int _{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} g_{m}(x,y,s,t)\,dt\,ds \biggr) $$
(2.35)
for all \((x,y)\in[x_{0}, X^{*}_{1}]\times[y_{0},Y^{*}_{1}]\).
Let \(b(x,y)=0\) for some \((x,y)\in\Delta\). Let \(\eta_{1,\epsilon }(x,y):=r_{1}(x,y)+\epsilon\) for any \(\epsilon>0\). Then \(\eta_{1,\epsilon}(x,y)>0\). Using the same arguments as above, where \(\eta_{1}(x,y)\) is replaced with \(\eta_{1,\epsilon}(x,y)\), we get
$$ u(x,y)\leq H_{m}^{-1}\biggl(H_{m}\bigl( \eta_{n,\epsilon}(x,y)\bigr)+ \int _{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} g_{m}(x,y,s,t)\,dt\,ds \biggr) $$
for \(x_{0}\le x\le X^{*}_{1}\), \(y_{0}\le Y^{*}_{1}\). Then consider the continuity of \(\eta_{i,\epsilon}\) in ϵ and the continuity of \(H_{j}\) and \(H_{j}^{-1}\) for \(j=1,\ldots, m\), and let \(\epsilon\rightarrow0^{+}\). Then we obtain (2.7). This completes the proof. □

Theorem 2.1

Suppose that (A1)(A4) hold. \(\max_{s\in [b_{*}(x_{0})-h,x_{0}]}\psi(s,y)\leq\varphi^{-1}( (1+m)^{1-1/q}a(x_{0}, y))\) for \(y\in[y_{0},y_{1})\) and \(u\in C([b_{*}(x_{0})-h,x_{1})\times[y_{0},y_{1}),\mathbb {R}_{+})\) are satisfied (1.4). Then, for all \((x,y)\in[x_{0}, X_{1})\times[y_{0},Y_{1})\), we have
$$ u(x,y)\leq\varphi^{-1}\biggl(\biggl(W_{m}^{-1} \bigl(W_{m}\bigl(r_{m}(x,y)\bigr)\bigr)+ \int_{\alpha _{m}(x_{0})}^{\alpha_{m}(x)} \int_{\beta_{m}(y_{0})}^{\beta_{m}(y)} \tilde{f}_{m}(x,y,s,t)\,dt \,ds\biggr)^{1/q}\biggr), $$
(2.36)
where \(W_{j}^{-1}\)is the inverse of the function
$$ W_{j}(t):= \int_{t_{j}}^{t}\frac{ds}{\tilde{\omega}^{q}_{j}(\varphi ^{-1}(s^{1/q}))}, \quad t\ge t_{j}>0, j=1,\ldots,m. $$
(2.37)
In (2.36) and (2.37), \(t_{j}\) is a given constant, \(\frac {1}{p}+\frac{1}{q}=1\), \(\tilde{\omega}_{j}\) (\(j=1,2,\ldots,m\)) are defined by (2.1),
$$\begin{aligned}& r_{1}(x,y) := (1+m)^{q-1}\Bigl(\max_{(\tau,\xi)\in [x_{0}, x]\times[y_{0},y] } \bigl\{ a(\tau,\xi)\bigr\} \Bigr)^{q}, \\& r_{j}(x,y): = W_{j-1}^{-1} \biggl[W_{j-1} \bigl(r_{j-1}(x,y)\bigr)+ \int_{b_{i-1}(x_{0})}^{b_{i-1}(x)} \int _{c_{i-1}(y_{0})}^{c_{i-1}(y)} \tilde{f}_{i-1}(x,y,s,t) \,dt\,ds \biggr], \\& \quad j=2,\ldots, m, \end{aligned}$$
(2.38)
$$\begin{aligned}& \begin{aligned}[b] &\tilde{f}_{j}(x,y,s,t):=(1+m)^{q-1} \bigl({M_{j}} x^{\theta_{j}}{\bar{M}_{j}} y^{\bar{\theta}_{j}}\bigr)^{q/p}\Bigl(\max_{(\iota,\xi)\in[x_{0}, x ]\times[y_{0},y]}f_{j}( \iota,\xi,s,t)\Bigr)^{q}, \\ &\quad (x,y)\in [x_{0},x_{1})\times[y_{0},y_{1}), \end{aligned} \end{aligned}$$
(2.39)
$$\begin{aligned}& \begin{aligned} &M_{j}=\alpha_{j}^{-1}B \biggl(\frac{p(\gamma_{j}-1)+1}{\alpha_{j}}, p(\beta_{j}-1)+1\biggr), \\ &\bar{M}_{j}=\bar{\alpha}_{j}^{-1}B\biggl( \frac{p(\bar{\gamma }_{j}-1)+1}{\bar{\alpha}_{j}}, p(\beta_{j}-1)+1\biggr), \\ &\theta_{j}=p\bigl(\alpha_{j}(\beta_{j}-1)+ \gamma_{j}-1\bigr)+1, \\ &\bar{\theta}_{j}=p\bigl(\bar{\alpha}_{j}(\bar{ \beta}_{j}-1)+\bar{\gamma }_{j}-1\bigr)+1, \quad j=1, \ldots,m, \end{aligned} \end{aligned}$$
(2.40)
\(X_{1}\in[x_{0}, x_{1})\), \(Y_{1}\in[y_{0}, y_{1})\) are chosen such that
$$ W_{j}\bigl(r_{j}(X_{1},Y_{1}) \bigr)+ \int_{b_{j}(x_{0})}^{b_{j}(X_{1})} \int _{c_{j}(y_{0})}^{c_{j}(Y_{1})} \tilde{f}_{j}(x,y,s,t) \,dt\,ds\le \int_{t_{j}}^{\infty}\frac{ds}{\tilde {\omega}^{q}_{j}(\varphi^{-1}(s^{1/q}))} $$
(2.41)
for \(j=1,\ldots,m\).

Proof

Above all, we monotonize functions \(f_{j}\), \(\omega_{j}\), \(\mu_{j}\), g, and a in (1.4). Let
$$ \hat{a}(x,y): = \max_{(\tau,\xi)\in[x_{0}, x]\times[y_{0},y] }\bigl\{ a(\tau,\xi) \bigr\} ,\quad (x,y)\in[x_{0},x_{1})\times[y_{0},y_{1}), $$
which is increasing in x and y. The sequence \(\{\tilde{\omega}_{j}\}\), defined by \(\omega_{j}(s)\) and \(\mu_{j}(s)\) in (2.1), consists of nonnegative and nondecreasing functions on \(\mathbb{R}_{+} \) and satisfies
$$ \omega_{j}(t)\le\hat{ {\omega}}_{j}(t),\qquad \mu_{j}(t)\le\hat{\mu}_{j}(t),\qquad \hat{\omega}_{j}(t) \hat{{\mu}}_{j}\bigl(\tilde{g}(t)\bigr)\le\tilde{\omega }_{j}(t), \quad j=1,\ldots,m. $$
(2.42)
Moreover, because the ratios \({\tilde{\omega}_{j+1}}/{\tilde{\omega }_{j}}\) (\(j=1,\ldots,m-1\)) are all nondecreasing, we have
$$ \tilde{\omega}_{j}\varpropto\tilde{\omega}_{j+1}, \quad j=1,2,\ldots,m-1. $$
(2.43)
Let
$$ \hat{f}_{j}(x,y,s,t) :=\max_{(\iota,\xi)\in[x_{0}, x ]\times[y_{0},y]}f_{j}( \iota ,\xi,s,t), $$
(2.44)
which are increasing in x and y and satisfy \(\tilde {f}_{j}(x,y,s,t)\geq f_{j}(x,y,s,t)\geq0\) for \(j=1,2,\ldots,m\). Since is nondecreasing, we obtain
$$ \max_{\tilde{\eta}\in[s-h,s]} g\bigl(u(\xi,y)\bigr)\le\max _{\tilde{\eta}\in[s-h,s]} \tilde{g}\bigl(u(\xi,y)\bigr) \le\tilde{g}\Bigl(\max _{\tilde{\eta}\in[s-h,s]} u(\xi,y)\Bigr) $$
(2.45)
for all \((s,y)\in[b_{*}(x_{0}), x_{1})\times[y_{0},y_{1})\). From (1.4), (2.42), (2.45), and the definition of \(\hat{f}_{j}\), we can obtain
$$ \begin{aligned} &\varphi\bigl(u(x,y)\bigr)\leq \hat{a}(x,y)+\sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{\beta_{j}-1}s^{\gamma_{j}-1}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{\bar{\beta}_{j}-1} t^{\bar{\gamma}_{j}-1} \\ &\hphantom{\varphi(u(x,y))\leq{}}{}\times\hat{f}_{j}(x,y,s,t) \hat{ \omega}_{j}\bigl(u(s,t)\bigr)\hat{\mu}_{j}\Bigl(\tilde{g} \Bigl(\max_{\tilde {\eta}\in[s-h,s]}u(\tilde{\eta},t)\Bigr)\Bigr)\,dt\,ds, \quad \\ &\hphantom{\varphi(u(x,y))\leq{}}{}(x,y)\in[x_{0},x_{1}) \times[y_{0}, y_{1}), \\ &u(x,y) \leq \psi (x,y),\quad (x,y)\in\bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times [y_{0}, y_{1}). \end{aligned} $$
(2.46)
Let \(\frac{1}{p}+\frac{1}{q}=1\), \(p>1\), then \(q>0\). By Lemma 1, Hölder’s inequality, (A4) and (2.46), we obtain, for all \((x,y)\in[x_{0},x_{1})\times[y_{0}, y_{1})\),
$$\begin{aligned}& \varphi\bigl(u(x,y)\bigr) \\& \quad \leq \hat{a}(x,y)+\sum _{j=1}^{m} \biggl( \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{p(\beta_{j}-1)}s^{p(\gamma_{j}-1)}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{p(\bar{\beta}_{j}-1)} t^{(\bar{\gamma}_{j}-1)}\,dt\,ds\biggr)^{1/p} \\& \qquad {}\cdot\biggl( \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\hat{f}^{q}_{j}(x,y,s,t) \hat{\omega}^{q}_{j}\bigl(u(s,t)\bigr) \Bigl(\hat{ \mu}_{j}\Bigl(\tilde{g}\Bigl(\max_{\tilde{\eta}\in[s-h,s]}u(\tilde{ \eta},t)\Bigr)\Bigr)\Bigr)^{q} \,dt\,ds\biggr)^{1/q} \\& \quad \leq \hat{a}(x,y)+\sum_{j=1}^{m} \biggl( \int _{0}^{x} \int_{0}^{y} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{p(\beta_{j}-1)}s^{p(\gamma_{j}-1)}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\beta}_{j}} \bigr)^{p(\bar{\gamma}_{j}-1)} t^{{p(\bar{\gamma}_{j}-1)}}\,dt\,ds\biggr)^{1/p} \\& \qquad {}\cdot\biggl( \int_{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\hat {f}^{q}_{j}(x,y,s,t) \hat{\omega}^{q}_{j}\bigl(u(s,t)\bigr) \Bigl(\hat{ \mu}_{j}\Bigl(\tilde{g}\Bigl(\max_{\tilde{\eta}\in[s-h,s]}u(\tilde{ \eta},t)\Bigr)\Bigr)\Bigr)^{q}dtds\biggr)^{1/q} \\& \quad \leq \hat{a}(x,y)+ \sum_{j=1}^{m} \bigl(M_{j}x^{\theta_{j}}\bar{M}_{j}y^{\bar{\theta}_{j}} \bigr)^{1/p}\biggl( \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\hat{f}^{q}_{j}(x,y,s,t) \\& \qquad {} \cdot\hat{\omega}^{q}_{j}\bigl(u(s,t)\bigr) \Bigl( \hat{\mu}_{j}\Bigl(\tilde{g}\Bigl(\max_{\tilde{\eta}\in[s-h,s]}u( \tilde{\eta},t)\Bigr)\Bigr)\Bigr)^{q} \,dt\,ds\biggr)^{1/q}, \end{aligned}$$
(2.47)
where \(0\le b_{j}(t)\le t \), \(0\le c_{j}(t)\le t\), \(M_{j}\), \(\bar {M}_{j}\), \(\theta_{j}\), and \(\bar{\theta}_{j}\) are given by (2.40) for \(j=1,\ldots,m\).
By Jensen’s inequality and (2.47), we get, for all \((x,y)\in \Delta\),
$$\begin{aligned} \varphi^{q}\bigl(u(x,y)\bigr) \leq& (1+m)^{q-1}\Biggl( \hat{a}^{q}(x,y)+ \sum_{j=1}^{m} \bigl(M_{j}x^{\theta_{j}}\bar{M}_{j}y^{\bar{\theta}_{j}} \bigr)^{q/p} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\hat{f}^{q}_{j}(x,y,s,t) \\ &{} \times\hat{\omega}^{q}_{j}\bigl(u(s,t)\bigr) \Bigl( \hat{\mu}_{j}\Bigl(\tilde{g}\Bigl(\max_{\tilde{\eta}\in[s-h,s]}u( \tilde{\eta},t)\Bigr)\Bigr)\Bigr)\Biggr)^{q} \,dt\,ds. \end{aligned}$$
(2.48)
Then, from (2.38), \(r_{1}\) is increasing on Δ. Then, by the definition of \(r_{1}\) and \(\tilde{f}_{j}\), from (2.48) we have
$$\begin{aligned} \varphi^{q}\bigl(u(x,y)\bigr) \leq& r_{1}(x,y)+ \sum _{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\tilde{f}_{j}(x,y,s,t) \hat{\omega}^{q}_{j}\bigl(u(s,t)\bigr) \\ &{}\cdot\Bigl(\hat{\mu}_{j}\Bigl(\tilde{g}\Bigl(\max _{\tilde{\eta}\in [s-h,s]}u(\tilde{\eta},t)\Bigr)\Bigr)\Bigr)^{q}\,dt \,ds, \quad (x,y)\in\Delta. \end{aligned}$$
(2.49)
According to (2.49), we consider the inequalities
$$ \begin{aligned} &\varphi^{q}\bigl(u(x,y)\bigr) \leq r_{1}(X,Y)+\sum_{j=1}^{m} \int_{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \tilde{f}_{j}(X,Y,s,t) \\ &\hphantom{\varphi^{q}(u(x,y)) \leq{}}{}\cdot\hat{\omega}^{q}_{j}\bigl(u(s,t) \bigr) \Bigl(\hat{\mu}_{j}\Bigl(\hat{g}\Bigl(\max_{\tilde{\eta}\in[\tilde{\eta}-h,s]}u( \tilde{\eta },t)\Bigr)\Bigr)\Bigr)^{q} \,dt\,ds, \\ &\hphantom{\varphi^{q}(u(x,y)) \leq{}}{} (x,y) \in[x_{0},X]\times[y_{0},Y], \\ &u(x,y) \leq \psi(x,y), \quad (x,y)\in \bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times[y_{0},Y], \end{aligned} $$
(2.50)
where \(x_{0}\leq X\le X_{1}\) and \(y_{0}\leq Y\le Y_{1}\) are chosen arbitrarily.
Since \(\max_{s\in[b_{*}(x_{0})-h,x_{0}]}\psi(s,y)\leq\varphi ^{-1}((1+m)^{1-1/q}a(x_{0},y)) \) for \(y\in[y_{0},y_{1})\), \(a(x_{0},y)\le\hat{ a}(x_{0},y)\), we have \(\max_{s\in[b_{*}(x_{0})-h,x_{0}]}\psi(s,y)\leq\varphi ^{-1}(r^{1/q}_{1}(X,Y))\), \(y\in[y_{0},Y]\). Define a function \(z(x,y): [b_{*}(x_{0})-h, X)\times[y_{0},Y)\rightarrow \mathbb {R}_{+}\) by
$$ z(x,y)= \textstyle\begin{cases} r_{1}(X,Y)+\sum_{j=1}^{m} \int_{b_{j}(x_{0})}^{b_{j}(x)}\int _{c_{j}(y_{0})}^{c_{j}(y)}\tilde{f}_{j}(X,Y,s,t) \hat{\omega}^{q}_{j}(u(s,t)) \\ \quad {}\times (\hat{\mu}_{j}(\hat{g}(\max_{\tilde{\eta}\in [s-h,s]}u(\tilde{\eta},t))))^{q}\,dt\,ds, \quad (x,y)\in[x_{0},X]\times [y_{0},Y], \\ r_{1}(X,Y), \quad (x,y)\in[b_{*}(x_{0})-h,x_{0}]\times [y_{0}, Y]. \end{cases} $$
Clearly, \(z(x,y)\) is increasing in x. By the definition of \(z(x,y)\) and (2.50), we have
$$ u(x,y)\leq\varphi^{-1}\bigl(z^{1/q}(x,y) \bigr), \quad (x,y)\in\bigl[b_{*}(x_{0})-h, X\bigr]\times[y_{0},Y]. $$
(2.51)
Since \(\varphi(t)\) is strictly increasing and \(z(x,y)\) is nondecreasing, from (2.51) we get, for \((s,y)\in[b_{*}(x_{0}), X]\times[y_{0},Y]\),
$$ \max_{\xi\in[s-h, s]} u(\xi,y) \leq \max _{\xi\in[s-h, s]} \varphi^{-1}\bigl(z^{1/q}(\xi,y) \bigr) \leq \varphi^{-1}\bigl(z^{1/q}(s,y)\bigr). $$
(2.52)
From the definition of \(z(x,y)\), (2.42), (2.51), and (2.52), it follows that
$$\begin{aligned}& z(x,y) \leq r_{1}(X,Y)+\sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{b_{j}(y_{0})}^{b_{j}(y)} \tilde{f}_{j}(X,Y,s,t) \hat{\omega}^{q}_{j}\bigl(\varphi^{-1} \bigl(z^{1/q}(s,t)\bigr)\bigr) \\& \hphantom{z(x,y) \leq{}} {}\cdot\Bigl(\hat{\mu}_{j}\Bigl(\hat{g}\Bigl(\max _{\tilde{\eta}\in[s-h, s]} \varphi^{-1}\bigl(z^{1/q}(\tilde{ \eta},t)\bigr)\Bigr)\Bigr)\Bigr)^{q}\,dt\,ds \\& \hphantom{z(x,y) } \leq r_{1}(X,Y)+\sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{b_{j}(y_{0})}^{b_{j}(y)} \tilde{f}_{j}(X,Y,s,t) \hat{\omega}^{q}_{j}\bigl(\varphi^{-1} \bigl(z^{1/q}(s,t)\bigr)\bigr) \\& \hphantom{z(x,y) \leq{}} {}\cdot\bigl(\hat{\mu}_{j}\bigl(\tilde{g}\bigl( \varphi^{-1}\bigl(z^{1/q}(s,t)\bigr)\bigr)\bigr) \bigr)^{q}\,dt\,ds \\& \hphantom{z(x,y) } \leq r_{1}(X,Y)+\sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{b_{j}(y_{0})}^{b_{j}(y)} \tilde{f}_{j}(X,Y,s,t) \tilde{\omega}^{q}_{j}\bigl(\varphi^{-1} \bigl(z^{1/q}(s,t)\bigr)\bigr) \\& \hphantom{z(x,y) \leq{}} {} \cdot\vartheta_{j}\Bigl(\max_{\tilde{\eta}\in[s-h, s]} u(\tilde {\eta},t)\Bigr)\,dt\,ds, \quad (x,y)\in[x_{0}, X] \times[y_{0},Y], \\& z(x,y) \leq r_{1}(X,Y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times[y_{0}, Y], \end{aligned}$$
(2.53)
where \(\vartheta_{j}(t)\equiv1\), \(t\ge0\).

Let \(v(t):=\varphi^{-1}(t^{1/q})\), which is a continuous and increasing function on \(\mathbb {R}_{+}\). Thus \(\tilde{\omega }^{q}_{j}(h(t))\) (\(j=1,\ldots, m\)) are continuous and increasing on \(\mathbb {R}_{+}\) and satisfy \(\tilde{\omega}_{j}(v(t))>0 \) for \(t>0\). Moreover, since \(\tilde {\omega}_{j}(t)\propto\tilde{\omega}_{j+1}(t)\), \(\tilde{\omega}^{q}_{j+1}(v(t))/\tilde{\omega}^{q}_{j}(v(t))\) are continuous and increasing on \(\mathbb {R}_{+}\) and positive on \((0,\infty)\), then \((\tilde{\omega}_{j}\circ v)\vartheta_{j}\propto (\tilde{\omega}_{j+1}\circ v)\vartheta_{j+1}\) for \(j=1,2,\ldots,m-1\).

Applying Lemma 2 to specified \(g_{j}(x,y,s,t)=\tilde{f}_{j}(X,Y,s,t)\), \(h_{j}(t)=\tilde{\omega}^{q}_{j}(\varphi^{-1}(t^{1/q}))\), \(\bar{h}_{j}(t)=\vartheta_{j}(t)\equiv1\) (\(j=1,2,\ldots,m\)), and (2.53), we obtain
$$\begin{aligned} z(x,y) \le& W_{m}^{-1}\biggl[W_{n}\bigl(\tilde {r}_{m}(X,Y,x,y)\bigr) \\ &{} + \int_{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} \tilde{ f}_{m}(X,Y,s,t) \,dt\,ds\biggr] \end{aligned}$$
(2.54)
for \(x_{0}\le x \le\min\{X, X_{2}\}\), \(y_{0}\le y \le\min\{Y, Y_{2}\}\), where \(\tilde{r}_{j}\) is defined inductively by \(\tilde{r}_{1}(X,Y, x,y):=\gamma _{1}(X,Y)\) and
$$ \tilde{r}_{j}(X,Y,x,y):= W_{i-1}^{-1} \biggl(W_{i-1}\bigl(\tilde {r}_{i-1}(X,Y,x,y)\bigr)+ \int_{b_{i-1}(x_{0})}^{b_{i-1}(x)} \int _{c_{i-1}(y_{0})}^{c_{i-1}(y)} \tilde{f}_{i-1}(X,Y,s,t) \,dt\,ds\biggr) $$
for \(j=2,\ldots, m\), and \(\bar{X}_{1}\), \(\bar{Y}_{1}\) are chosen such that
$$\begin{aligned}& W_{j}\bigl(\tilde{r}_{j}(X,Y,\bar{X}_{1}, \bar{Y}_{1})\bigr)+ \int _{b_{j}(x_{0})}^{b_{j}(X_{2})} \int_{c_{j}(y_{0})}^{c_{j}(\bar{Y}_{1})} \tilde{f}_{j}(X,Y,s,t) \\& \quad \le \int_{t_{j}}^{\infty}\frac{ds}{\tilde{\omega}^{q}_{j}(\varphi ^{-1}(s^{1/q}))} \end{aligned}$$
(2.55)
for \(j=1,\ldots,m\).

Note that \(X_{2}\ge X_{1}\) and \(Y_{2}\ge Y_{1}\). In fact, both \(\tilde {r}_{j}(X,Y,x,y)\) and \(\tilde{f}_{j}(X,Y,x,y)\) are increasing in X and Y. Thus \(X_{2}\), \(Y_{2}\) satisfying (2.55) get smaller as X, Y are chosen larger.

According to (2.51) and (2.54),
$$\begin{aligned} u(x,y) \le& \varphi^{-1}\biggl(W_{m}^{-1} \biggl(W_{n}\bigl(\tilde {r}_{m}(X,Y,x,y)\bigr) \\ &{}+ \int_{\alpha_{m}(x_{0})}^{\alpha_{m}(x)} \int_{\beta _{m}(y_{0})}^{\beta_{m}(y)} \tilde{ f}_{m}(X,Y,s,t) \,dt\,ds\biggr)\biggr) \end{aligned}$$
(2.56)
for \(x_{0}\le x \le\min\{X, X_{2}\}\), \(y_{0}\le y \le\min\{Y, Y_{2}\}\).
Taking \(x=X\), \(y=Y\) in (2.56), we have
$$\begin{aligned} u(X,Y) \le& \varphi^{-1}\biggl(W_{m}^{-1} \biggl(W_{n}\bigl(\tilde {r}_{m}(X,Y,X,Y)\bigr) \\ &{}+ \int_{b_{m}(x_{0})}^{b_{m}(X)} \int_{c_{m}(y_{0})}^{c_{m}(Y)} \tilde{ f}_{m}(X,Y,s,t) \,dt\,ds\biggr)\biggr) \end{aligned}$$
(2.57)
for \(x_{0}\le X\le X_{1}\), \(y_{0}\le Y\le Y_{1}\). It is easy to verify \(\tilde {r}_{m}(X,Y,X,Y)= r_{m}(X,Y)\). Thus, (2.57) can be written as
$$\begin{aligned} u(X,Y) \le& \varphi^{-1}\biggl(W_{m}^{-1} \biggl(W_{n}\bigl(r_{m}(X,Y)\bigr) \\ &{}+ \int_{b_{m}(x_{0})}^{b_{m}(X)} \int_{c_{m}(y_{0})}^{c_{m}(Y)} \tilde{ f}_{m}(X,Y,s,t) \,dt\,ds\biggr)\biggr). \end{aligned}$$
(2.58)
Since X, Y are arbitrary, replacing Y and X with y and x, respectively, we have
$$\begin{aligned} u(x,y) \le& \varphi^{-1}\biggl(W_{m}^{-1} \biggl(W_{n}\bigl(r_{m}(x,y)\bigr) \\ &{}+ \int_{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} \tilde{ f}_{m}(x,y,s,t) \,dt\,ds\biggr)\biggr) \end{aligned}$$
(2.59)
for all \((x,y)\in[x_{0}, X^{*}_{1}]\times[y_{0},Y^{*}_{1}]\).

This completes the proof. □

Theorem 2.2

We make the following assumptions:
(S1): 

\(c(x,y)\in C(\Delta, \mathbb{R}_{+}) \) and \(b_{j}\in C^{1}([x_{0},x_{1}),\mathbb {R}_{+})\), and \(c_{j}\in C^{1}([y_{0},y_{1}), [y_{0},y_{1}))\) are nondecreasing with \(b_{j}(x)\leq x\) on \([x_{0},x_{1})\) and \(c_{j}(y)\le y\) on \([y_{0},y_{1})\), and \(c_{j}(y_{0})=y_{0}\) for \(j=1,\ldots,m\);

(S2): 

\(\hat{\psi}\in C(\Xi,\mathbb {R}_{+})\), \(\hat{g}_{j}\in C(\Delta\times[b_{*}(x_{0}),x_{1})\times[y_{0},y_{1}),\mathbb {R}_{+})\) (\(j=1,2,\ldots, m\));

(S3): 

\(\phi_{j}, \hat{\phi}_{j}\in C(\mathbb {R}_{+},\mathbb{R}_{+})\) (\(j=1,\ldots ,m\)) are all nondecreasing with \(\{\phi_{j},\hat{\phi}_{j}\}(t)>0\) for \(t>0\), and \(\phi_{j}\hat{\phi}_{j}\propto\phi_{j+1}\hat{\phi}_{j+1}\) (\(j=1,\ldots,m-1\));

(S4): 

\(k\ge1\), \(\alpha_{j}, \bar{\alpha}_{j}\in(0,1]\), \(\beta_{j},\bar{\beta }_{j}\in(0,1)\), \(\gamma_{j}>1-\frac{1}{p}\), \(\bar{\gamma}_{j}>1-\frac{1}{p}\) such that \(\frac{1}{p}+\alpha_{j}(\beta_{j}-1)+\gamma_{j}-1\ge0\), \(\frac {1}{p}+\bar{\alpha}_{j}(\bar{\beta}_{j}-1)+\bar{\gamma}_{j}-1\ge0\), \(p(\beta_{j}-1)+1>0\), \(p(\bar{\beta}_{j}-1)+1>0\), \(p>1\) for all \(j=1,\ldots,m\).

If \(u\in C([b_{*}(x_{0})-h,x_{1})\times[y_{0},y_{1}),\mathbb {R}_{+})\) satisfies the integral inequality
$$\begin{aligned}& u^{k}(x,y)\leq c(x,y)+\sum_{j=1}^{M} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{\beta_{j}-1}s^{\gamma_{j}-1}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{\bar{\beta}_{j}-1} \\& \hphantom{u^{k}(x,y)\leq{}}{} \times t^{\bar{\gamma}_{j}-1} \hat{g}_{j}(x,y,s,t) \phi_{j}\bigl(u(s,t)\bigr)\hat{\phi}_{j} \Bigl(\max _{\tilde{\eta}\in [s-h, s]}g\bigl(u(\tilde{\eta},t)\bigr) \Bigr) \\& \hphantom{u^{k}(x,y)\leq{}}{} +\sum_{j=M+1}^{m} \int_{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)}\hat{g}_{j}(x,y,s,t) \phi_{j}\bigl(u(s,t)\bigr)\hat{\phi }_{j} \Bigl(\max _{\tilde{\eta}\in[s-h, s]}u(\tilde{\eta },t) \Bigr), \\& \hphantom{u^{k}(x,y)\leq{}}{}(x,y)\in[x_{0},x_{1})\times [y_{0}, y_{1}), \\& u(x,y) \leq \hat{ \psi }(x,y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times [y_{0}, y_{1}), \end{aligned}$$
(2.60)
where \(\max_{s\in[b_{*}(x_{0})-h,x_{0}]}\hat{\psi}(s,y)\leq( (1+m)^{1-1/q}c(x_{0},y))^{1/k}\) for all \(y\in[y_{0},y_{1})\).
Then
$$ u(x,y) \leq \biggl(G_{m}^{-1} \bigl(G_{m}\bigl(e_{m}(x,y)\bigr)\bigr) + \int_{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} \tilde{g}_{m}(x,y,s,t)\,dt \,ds\biggr)^{1/(kq)} $$
(2.61)
for all \((x,y)\in[x_{0}, X_{2})\times[y_{0},Y_{2})\), where \(G_{j}^{-1}\)is the inverse of the function
$$ G_{j}(u):= \int_{t_{j}}^{t}\frac{ds}{\phi^{q}_{j}(s^{1/(kq)})\hat{\phi }^{q}_{j}(s^{1/(kq)})}, \quad t\ge t_{j}>0, j=1,\ldots,m. $$
(2.62)
In (2.61) and (2.62), \(t_{j}>0\) is a given constant, \(\frac{1}{p}+\frac{1}{q}=1\), \(e_{j}(x,y)\) is defined recursively by
$$\begin{aligned}& \begin{aligned}[b] &e_{1}(x,y)=(1+m)^{q-1}\Bigl( \max_{(\iota,\xi)\in[x_{0}, x ]\times [y_{0},y]}c(\iota,\xi)\Bigr)^{q},\quad \textit{and} \\ &e_{j+1}(x,y):=G_{j}^{-1}\biggl[G_{j} \bigl(e_{j}(x,y)\bigr)+ \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \tilde{g}_{j}(x,y,s,t) \,dt\,ds\biggr], \\ &\quad j=1,\ldots, m-1, \end{aligned} \end{aligned}$$
(2.63)
$$\begin{aligned}& \begin{aligned}[b] &\tilde{g}_{j}(x,y,s,t):=(1+m)^{q-1} \bigl({M_{j}} x^{\theta_{j}}{\bar{M}_{j}} y^{\bar{\theta}_{j}}\bigr)^{q/p}\Bigl(\max_{(\iota,\xi)\in[x_{0}, x ]\times[y_{0},y]} \hat{g}_{j}(\iota,\xi,s,t)\Bigr)^{q}, \\ &\quad (x,y)\in [x_{0},x_{1})\times[y_{0},y_{1}), \end{aligned} \end{aligned}$$
(2.64)
$$\begin{aligned}& \begin{aligned} &M_{j}=\alpha_{j}^{-1}B \biggl(\frac{p(\gamma_{j}-1)+1}{\alpha_{j}}, p(\beta_{j}-1)+1\biggr), \\ &\bar{M}_{j}=\bar{\alpha}_{j}^{-1}B\biggl( \frac{p(\bar{\gamma }_{j}-1)+1}{\bar{\alpha}_{j}}, p(\beta_{j}-1)+1\biggr), \\ &\theta_{j}=p\bigl(\alpha_{j}(\beta_{j}-1)+ \gamma_{j}-1\bigr)+1, \\ &\bar{\theta}_{j}=p\bigl(\bar{\alpha}_{j}(\bar{ \beta}_{j}-1)+\bar{\gamma}_{j}-1\bigr)+1, \quad j=1, \ldots,M \\ &M_{j}=\bar{M}_{j}=1, \qquad \theta_{j}= \bar{\theta}_{j}=1, \quad j=M+1,\ldots,m, \end{aligned} \end{aligned}$$
(2.65)
\(X_{2}\in[x_{0}, x_{1})\), \(Y_{2}\in[y_{0}, y_{1})\) are chosen such that
$$\begin{aligned}& G_{j}\bigl(r_{j}(X_{2},Y_{2})\bigr)+ \int_{b_{j}(x_{0})}^{b_{j}(X_{1})} \int _{c_{j}(y_{0})}^{c_{j}(Y_{2})} \tilde{g}_{j}(X_{2},Y_{2},s,t) \,dt\,ds \\& \quad \le \int_{t_{j}}^{\infty}\frac{ds}{\phi^{q}_{j}(s^{1/q})\hat{\phi }^{q}_{j}(s^{1/q})} \end{aligned}$$
(2.66)
for \(j=1,2,\ldots,m\).

Proof

Let
$$ \begin{aligned} &\hat{c}(x,y):=\max_{(\tau,\xi)\in[x_{0}, x]\times[y_{0},y] }\bigl\{ a(\tau,\xi)\bigr\} , \quad (x,y)\in[x_{0},x_{1}) \times[y_{0},y_{1}). \\ &\bar{g}_{j}(x,y,s,t) :=\max_{(\iota,\xi)\in[x_{0}, x ]\times[y_{0},y]}g_{j}( \iota ,\xi,s,t), \end{aligned} $$
(2.67)
which are increasing in x and y and satisfy \(\bar {g}_{j}(x,y,s,t)\geq g_{j}(x,y,s,t)\geq0\) for \(j=1,2,\ldots,m\). From (2.60), (2.67), and the definition of \(\tilde {g}_{j}\), we obtain
$$\begin{aligned}& u^{k}(x,y) \leq \hat{c}(x,y)+\sum_{j=1}^{M} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{\beta_{j}-1}s^{\gamma_{j}-1}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{\bar{\beta}_{j}-1} \\& \hphantom{u^{k}(x,y) \leq{}}{} \times t^{\bar{\gamma}_{j}-1} \bar{g}_{j}(x,y,s,t) \phi_{j}\bigl(u(s,t)\bigr)\hat{\phi}_{j}\Bigl(\max _{\tilde{\eta}\in[s-h, s]}u(\tilde{\eta},t)\Bigr) \\& \hphantom{u^{k}(x,y) \leq{}}{} +\sum_{j=M+1}^{m} \int_{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)}\bar{g}_{j}(x,y,s,t) \phi_{j}\bigl(u(s,t)\bigr)\hat{\phi}_{j}\Bigl(\max _{\tilde{\eta}\in[s-h, s]}u(\tilde{\eta},t)\Bigr), \\& \hphantom{u^{k}(x,y) \leq{}}{} (x,y)\in[x_{0},x_{1})\times [y_{0}, y_{1}), \\& u(x,y) \leq \hat{ \psi }(x,y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times [y_{0}, y_{1}). \end{aligned}$$
(2.68)
Let \(\frac{1}{p}+\frac{1}{q}=1\), \(p>1\), then \(q>0\). By Lemma 1, Hölder’s inequality, (S4), and (2.68), we obtain, for all \((x,y)\in\Delta\),
$$\begin{aligned}& u^{k}(x,y) \\& \quad \leq \hat{c}(x,y)+\sum_{j=1}^{M} \biggl( \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{p(\beta_{j}-1)}s^{p(\gamma_{j}-1)}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{p(\bar{\beta}_{j}-1)}t^{(\bar{\gamma}_{j}-1)}\,dt\,ds\biggr)^{1/p} \\& \qquad {}\times\biggl( \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\bar{g}^{q}_{j}(x,y,s,t) \phi^{q}_{j}\bigl(u(s,t)\bigr)\hat{\phi}^{q}_{j} \Bigl(\max_{\tilde{\eta}\in [s-h,s]}u(\tilde{\eta},t)\Bigr) \,dt\,ds\biggr)^{1/q} \\& \qquad {}+\sum_{j=M+1}^{m} \biggl( \int_{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)} 1^{p}\,dt\,ds \biggr)^{1/p}\biggl( \int_{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)}\bar{g}^{q}_{j}(x,y,s,t) \phi^{q}_{j}\bigl(u(s,t)\bigr) \\& \qquad {}\times\hat{\phi}^{q}_{j}\Bigl(\max_{\tilde{\eta}\in [s-h,s]}u(\tilde{ \eta},t)\Bigr) \,dt\,ds\biggr)^{1/q} \\& \quad \leq \hat{c}(x,y)+\sum_{j=1}^{M} \biggl( \int _{b_{j}(0)}^{x} \int_{0}^{y} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{p(\beta_{j}-1)}s^{p(\gamma_{j}-1)}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{p(\bar{\beta}_{j}-1)}t^{(\bar{\gamma}_{j}-1)}\,dt\,ds\biggr)^{1/p} \\& \qquad {} \times\biggl( \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\bar{g}^{q}_{j}(x,y,s,t) \phi^{q}_{j}\bigl(u(s,t)\bigr)\hat{\phi}^{q}_{j} \Bigl(\max_{\tilde{\eta}\in [s-h,s]}u(\tilde{\eta},t)\Bigr) \,dt\,ds\biggr)^{1/q} \\& \qquad {}+\sum_{j=M+1}^{m} \biggl( \int_{0}^{x} \int_{0)}^{y} 1^{p}\,dt\,ds \biggr)^{1/p}\biggl( \int_{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)}\bar{g}^{q}_{j}(x,y,s,t) \phi^{q}_{j}\bigl(u(s,t)\bigr) \\& \qquad {}\times\hat{\phi}^{q}_{j}\Bigl(\max_{\tilde{\eta}\in [s-h,s]}u(\tilde{ \eta},t)\Bigr) \,dt\,ds\biggr)^{1/q} \\& \quad \leq \hat{c}(x,y)+ \sum_{j=1}^{m} \bigl(M_{j}x^{\theta_{j}}\bar{M}_{j}y^{\bar{\theta}_{j}} \bigr)^{1/p}\biggl( \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\bar{g}^{q}_{j}(x,y,s,t) \\& \qquad {} \times\phi^{q}_{j}\bigl(u(s,t)\bigr) \hat{ \phi}^{q}_{j}\Bigl(\max_{\tilde {\eta}\in[s-h,s]}u(\tilde{ \eta},t)\Bigr) \,dt\,ds\biggr)^{1/q}, \end{aligned}$$
(2.69)
where \(0\le b_{j}(t)\le t \), \(0\le c_{j}(t)\le t\), \(M_{j}\), \(\bar {M}_{j}\), \(\theta_{j}\), and \(\bar{\theta}_{j}\) are given by (2.65) for \(j=1,\ldots,m\).
By Jensen’s inequality and (2.69), we get, for all \((x,y)\in \Delta\),
$$\begin{aligned} u^{kq}(x,y) \leq& (1+m)^{q-1}( \hat{c}^{q}(x,y)+ \sum_{j=1}^{m}\bigl(M_{j}x^{\theta_{j}} \bar{M}_{j}y^{\bar{\theta}_{j}}\bigr)^{q/p} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\bar{g}^{q}_{j}(x,y,s,t) \\ &{} \times\phi^{q}_{j}\bigl(u(s,t)\bigr) \hat{ \phi}^{q}_{j}\bigl(\max_{\tilde{\eta }\in[s-h,s]}u(\tilde{\eta},t)\bigr) \,dt\,ds. \end{aligned}$$
(2.70)
By the definition of \(e_{1}\) and \(\tilde{g}_{j}\), from (2.70) we obtain
$$\begin{aligned} u^{kq}(x,y) \leq& e_{1}(x,y)+ \sum _{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\tilde{g}_{j}(x,y,s,t) \\ &{}\times\phi^{q}_{j}\bigl(u(s,t)\bigr) \hat{ \phi}^{q}_{j}\bigl(\max_{\tilde{\eta }\in[s-h,s]}u(\tilde{\eta},t)\bigr) \,dt\,ds,\quad (x,y)\in\Delta. \end{aligned}$$
(2.71)
Concerning (2.71), we consider the auxiliary inequalities
$$\begin{aligned}& u^{kq}(x,y)\leq e_{1}(X,Y)+ \sum _{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\tilde{g}_{j}(X,Y,s,t) \\& \hphantom{u^{kq}(x,y)\leq{}}{}\times\phi^{q}_{j}\bigl(u(s,t)\bigr) \hat{\phi}^{q}_{j}\bigl(\max_{\tilde{\eta }\in[s-h,s]}u(\tilde{ \eta},t)\bigr) \,dt\,ds, \quad (x,y)\in[x_{0},X] \times[y_{0},Y], \\& u(x,y)\leq \hat{\psi}(x,y), \quad (x,y)\in \bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times[y_{0},Y], \end{aligned}$$
(2.72)
where \(x_{0}\leq X\le X_{2}\) and \(y_{0}\leq Y\le Y_{2}\) are chosen arbitrarily.
Since \(\max_{s\in[b_{*}(x_{0})-h,x_{0}]}\hat{\psi}(s,y)\leq ((1+m)^{1-1/q}a(x_{0},y))^{\frac{1}{k}}\) for \(y\in[y_{0},y_{1})\), \(a(x_{0},y)\le\hat{ c}(x_{0},y)\), we have \(\max_{s\in[b_{*}(x_{0})-h,x_{0}]}\psi(s,y)\leq (e^{1/q}_{1}(X,Y))^{\frac{1}{k}}\), \(y\in[y_{0},Y]\). Define a function \(z(x,y): [b_{*}(x_{0})-h, X)\times[y_{0},Y)\rightarrow \mathbb {R}_{+}\) by
$$ z(x,y)= \textstyle\begin{cases} e_{1}(X,Y)+ \sum_{j=1}^{m}\int_{b_{j}(x_{0})}^{b_{j}(x)}\int _{c_{j}(y_{0})}^{c_{j}(y)}\tilde{g}_{j}(X,Y,s,t) \\ \quad {}\times\phi^{q}_{j}(u(s,t))\hat{\phi}^{q}_{j}\bigl(\max_{\tilde{\eta }\in[s-h,s]}u(\tilde{\eta},t)\bigr) \,dt\,ds, \quad (x,y)\in[x_{0},X]\times [y_{0},Y], \\ e_{1}(X,Y), \quad (x,y)\in[b_{*}(x_{0})-h,x_{0}]\times [y_{0}, Y]. \end{cases} $$
Clearly, \(z(x,y)\) is increasing in x. From (2.72) and the definition of z, we have
$$ u(x,y)\leq z^{1/(kq)}(x,y),\quad (x,y)\in \bigl[b_{*}(x_{0})-h, X\bigr]\times[y_{0},Y]. $$
(2.73)
Then, noting that z is increasing, from (2.51) we get for \((s,y)\in[b_{*}(x_{0}), X]\times[y_{0},Y]\)
$$ \max_{\tilde{\eta}\in[s-h, s]} u(\tilde{\eta},y) \leq\max _{\tilde{\eta}\in[s-h, s]} z^{1/(kq)}(\tilde{\eta},y)\le\bigl( \max_{\tilde{\eta}\in[s-h, s]} z(\tilde{\eta},y)\bigr)^{1/(kq)}. $$
(2.74)
From (2.42), (2.73), (2.74), and the definition of z, we have
$$\begin{aligned}& z(x,y)\leq e_{1}(X,Y)+\sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{b_{j}(y_{0})}^{b_{j}(y)} \tilde{g}_{j}(X,Y,s,t) \phi^{q}_{j}\bigl(z^{1/(kq)}(s,t)\bigr) \\& \hphantom{z(x,y)\leq{}}{}\times\hat{\phi}^{q}_{j} \Bigl(\max _{\tilde{\eta}\in[s-h, s]} \bigl(z^{1/(kq)}(\tilde{\eta},t)\bigr) \Bigr)\,dt\,ds \\& \hphantom{z(x,y)}\leq e_{1}(X,Y)+\sum_{j=1}^{m} \int_{b_{j}(x_{0})}^{b_{j}(x)} \int _{b_{j}(y_{0})}^{b_{j}(y)} \tilde{g}_{j}(X,Y,s,t) \phi^{q}_{j}\bigl(z^{1/(kq)}(s,t)\bigr) \\& \hphantom{z(x,y)\leq{}}{}\times\hat{\phi}^{q}_{j} \Bigl(\bigl(\max _{\tilde{\eta}\in[s-h, s]} z(\tilde{\eta},t)\bigr)^{1/(kq)}\Bigr) \,dt\,ds, \quad (x,y)\in[x_{0}, X] \times[y_{0},Y], \\& z(x,y) \leq e_{1}(X,Y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times[y_{0}, Y]. \end{aligned}$$
(2.75)
Let \(v(t):=t^{1/(kq)}\), which is a continuous and increasing function on \(\mathbb {R}_{+}\). Thus \(\phi^{q}_{j}(v(t)) \) and \(\hat {\phi}^{q}_{j}(v(t))\) (\(j=1,\ldots, m\)) are continuous and increasing on \(\mathbb {R}_{+}\) and positive on \((0,\infty)\). Moreover, since \(\phi_{j}\hat{\phi}_{j}\propto\phi_{j+1}\hat{\phi}_{j+1}\), we have \((\phi_{j+1}\circ v)^{q}(\hat{\phi}_{j+1}\circ v)^{q} \propto(\phi _{j}\circ v)^{q}(\hat{\phi}_{j}\circ v)^{q}\) (\(j=1,\ldots,m-1\)). Taking \(g_{j}(x,y,s,t)=\tilde{g}_{j}(X,Y,s,t)\) and \(h_{j}(t)=\phi^{q}_{j}(v(t))\), \(\bar{h}_{j}(t)=\hat{\phi}^{q}_{j}(v(t))\), \(j=1,2,\ldots,m\), in Lemma 2 and (2.75),we obtain
$$\begin{aligned} z(x,y) \le& G_{m}^{-1}\biggl(G_{m} \bigl(\tilde{e}_{m}(X,Y,x,y)\bigr) \\ &{} + \int_{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} \tilde{ g}_{m}(X,Y,s,t) \,dt\,ds\biggr) \end{aligned}$$
(2.76)
for \(x_{0}\le x \le\min\{X, X^{*}_{2}\}\), \(y_{0}\le y \le\min\{Y, Y^{*}_{2}\}\), where \(\tilde{e}_{j}(X,Y,x,y)\) is defined inductively by \(\tilde {e}_{1}(X,Y,x,y):=e_{1}(X,Y)\) and
$$ \tilde{e}_{j}(X,Y,x,y):= G_{j-1}^{-1} \biggl(G_{j-1}\bigl(\tilde {e}_{j-1}(X,Y,x,y)\bigr)+ \int_{b_{j-1}(x_{0})}^{b_{j-1}(x)} \int _{c_{j-1}(y_{0})}^{c_{j-1}(y)} \tilde{g}_{j-1}(X,Y,s,t) \,dt\,ds\biggr) $$
for \(j=2,\ldots, m\), and \(X^{*}_{2}\), \(Y^{*}_{2}\) are chosen such that
$$\begin{aligned}& G_{j}\bigl(\tilde{e}_{j}(X,Y,\bar{X}_{1}, \bar{Y}_{1})\bigr)+ \int _{b_{j}(x_{0})}^{b_{j}(X_{2})} \int_{c_{j}(y_{0})}^{c_{j}(\bar{Y}_{1})} \tilde{g}_{j}(X,Y,s,t) \\& \quad \le \int_{t_{j}}^{\infty}\frac{ds}{\tilde{\omega}^{q}_{j}(\varphi ^{-1}(s^{1/q}))} \end{aligned}$$
(2.77)
for \(j=1,\ldots,m\).
Note that \(X^{*}_{2}=X_{2}\) and \(Y^{*}_{2}=Y_{2}\). It follows from (2.73) and (2.76) that
$$\begin{aligned} u(x,y) \le& \biggl(G_{m}^{-1}\biggl(G_{n}\bigl( \tilde{g}_{m}(X,Y,x,y)\bigr) \\ &{}+ \int_{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} \tilde{ g}_{m}(X,Y,s,t) \,dt\,ds\biggr)\biggr)^{1/(kq)} \end{aligned}$$
(2.78)
for \(x_{0}\le x \le\min\{X, X^{*}_{2}\}\), \(y_{0}\le y \le\min\{Y, Y^{*}_{2}\}\).
Taking \(x=X\), \(y=Y\) in (2.56), we have
$$\begin{aligned} u(X,Y) \le& \biggl(G_{m}^{-1}\biggl(G_{m}\bigl( \tilde{e}_{m}(X,Y,X,Y)\bigr) \\ &{}+ \int_{b_{m}(x_{0})}^{b_{m}(X)} \int_{c_{m}(y_{0})}^{c_{m}(Y)} \tilde{ g}_{m}(X,Y,s,t) \,dt\,ds\biggr)\biggr)^{1/(kq)} \end{aligned}$$
(2.79)
for \(x_{0}\le X\le X_{2}\), \(y_{0}\le Y\le Y_{2}\). It is easy to verify \(\tilde {e}_{m}(X,Y,X,Y)= e_{m}(X,Y)\). Thus, (2.57) can be written as
$$\begin{aligned} u(X,Y) \le& \biggl(G_{m}^{-1}\biggl(G_{n} \bigl(r_{m}(X,Y)\bigr) \\ &{}+ \int_{b_{m}(x_{0})}^{b_{m}(X)} \int_{c_{m}(y_{0})}^{c_{m}(Y)} \tilde{ g}_{m}(X,Y,s,t) \,dt\,ds\biggr)\biggr)^{1/(kq)}. \end{aligned}$$
(2.80)
Since \(X,Y\) are arbitrary, replacing X and Y with x and y, respectively, we get
$$\begin{aligned} u(x,y) \le& \biggl(G_{m}^{-1}\biggl(G_{n} \bigl(e_{m}(x,y)\bigr) \\ &{}+ \int_{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} \tilde{ g}_{m}(x,y,s,t) \,dt\,ds\biggr)\biggr)^{1/(kq)} \end{aligned}$$
(2.81)
for all \((x,y)\in[x_{0}, X_{2}]\times[y_{0},Y_{2}]\). This completes the proof. □

Corollary 2.3

Let the following conditions be fulfilled:
(B1): 

all \(b_{j}\in C^{1}([x_{0},x_{1}),\mathbb {R}_{+})\) and \(c_{j}\in C^{1}([y_{0},y_{1}), [y_{0},y_{1}))\) are nondecreasing with \(b_{j}(x)\leq x\) on \([x_{0},x_{1})\), \(c_{j}(y)\le y\) on \([y_{0},y_{1})\), and \(c_{j}(y_{0})=y_{0}\) for all \(j=1,\ldots,m\);

(B2): 

\(a\in C(\Delta, \mathbb{R}_{+})\) and \(\hat{\psi}\in C(\Xi,\mathbb {R}_{+})\), \(\varphi_{1} \in C(\mathbb {R}_{+},\mathbb {R}_{+})\), and \(\varphi_{1}\) is strictly increasing such that \(\lim_{t\rightarrow\infty}\varphi(t)=\infty\),and \(f_{j}\in C(\Delta\times[b_{*}(x_{0}),x_{1})\times[y_{0},y_{1}),\mathbb{R}_{+})\) for all \(j=1,\ldots, m\);

(B3): 

all \(\psi_{j}\) (\(j=1,\ldots,m\)) are continuous and increasing functions on \(\mathbb {R}_{+}\) and positive on \((0,+\infty)\) such that \(\psi_{1}\propto\psi_{2}\propto\ldots\propto\psi_{m}\);

(B4): 

\(\alpha_{j}, \bar{\alpha}_{j}\in(0,1]\), \(\beta_{j},\bar{\beta}_{j}\in(0,1)\), \(\gamma_{j}>1-\frac{1}{p}\), \(\bar{\gamma}_{j}>1-\frac{1}{p}\) such that \(\frac{1}{p}+\alpha_{j}(\beta_{j}-1)+\gamma_{j}-1\ge0\), \(\frac {1}{p}+\bar{\alpha}_{j}(\bar{\beta}_{j}-1)+\bar{\gamma}_{j}-1\ge0\), \(p(\beta_{j}-1)+1>0\), \(p(\bar{\beta}_{j}-1)+1>0\), \(p>1\), \(j=1,2,\ldots,m\);

(B5): 
\(u\in C([b_{*}(x_{0})-h,x_{1})\times[y_{0},y_{1}),\mathbb {R}_{+})\) satisfies the integral inequality
$$ \begin{aligned} &\varphi_{1}\bigl(u(x,y) \bigr) \leq a(x,y)+\sum_{j=1}^{M} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{\beta_{j}-1}s^{\gamma_{j}-1}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{\bar{\beta}_{j}-1} \\ &\hphantom{\varphi_{1}(u(x,y)) \leq{}}{} \times t^{\bar{\gamma}_{j}-1} f_{j}(x,y,s,t) \psi_{j}\bigl(u(s,t)\bigr)\,dt\,ds \\ &\hphantom{\varphi_{1}(u(x,y)) \leq{}}{} +\sum_{j=M+1}^{m} \int_{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)}\bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{\beta_{j}-1}s^{\gamma_{j}-1} \bigl(y^{\bar{\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{\bar{\beta}_{j}-1} \\ &\hphantom{\varphi_{1}(u(x,y)) \leq{}}{} \times t^{\bar{\gamma}_{j}-1}f_{j}(x,y,s,t) \psi_{j} \Bigl(\max_{\tilde {\eta}\in[s-h,s]}u(\tilde{\eta},t) \Bigr) \,dt\,ds, \\ &\hphantom{\varphi_{1}(u(x,y)) \leq{}}{} (x,y)\in[x_{0},x_{1})\times [y_{0}, y_{1}), \\ &u(x,y) \leq \hat{\psi }(x,y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times [y_{0}, y_{1}), \end{aligned} $$
(2.82)
where \(\max_{s\in[b_{*}(x_{0})-h,x_{0}]}\hat{\psi}(s,y)\leq \varphi_{1}^{-1}( (1+m)^{1-1/q}a(x_{0},y))\) for all \(y\in[y_{0},y_{1})\).
Then
$$\begin{aligned} u(x,y) \leq& \varphi_{1}^{-1}\biggl(\check{G}_{m}^{-1} \biggl(\check{G}_{m}\bigl(r_{m}(x,y)\bigr) \\ &{} + \int_{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} \tilde{f}_{m}(x,y,s,t)\,dt \,ds\biggr)^{1/q}\biggr) \end{aligned}$$
(2.83)
for all \((x,y)\in[x_{0}, X_{2})\times[y_{0},Y_{2})\), where \(G_{j}^{-1}\) is the inverse of the function
$$ \check{G}_{j}(t):= \int_{t_{j}}^{t}\frac{ds}{\psi^{q}_{j}(\varphi _{1}^{-1}(s^{1/q}))},\quad t\ge t_{j}>0, j=1,2,\ldots,m, $$
(2.84)
\(t_{j}\) is a given constant, \(r_{j}(x,y)\) is defined recursively by
$$\begin{aligned}& r_{1}(x,y)=(1+m)^{q-1}\Bigl(\max_{(\iota,\xi)\in[x_{0}, x ]\times [y_{0},y]}a( \iota,\xi)\Bigr)^{q},\quad \textit{and} \\& r_{j+1}(x,y):= \check{G}_{j}^{-1}\biggl[ \check{G}_{j}\bigl(r_{j}(x,y)\bigr)+ \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \tilde{f}_{j}(x,y,s,t) \,dt\,ds\biggr], \\& \quad j=1,\ldots, m-1, \end{aligned}$$
(2.85)
$$\begin{aligned}& \begin{aligned}[b] &\tilde{f}_{j}(x,y,s,t):=(1+m)^{q-1} \bigl({M_{j}} x^{\theta_{j}}{\bar{M}_{j}} y^{\bar{\theta}_{j}}\bigr)^{q/p}\Bigl(\max_{(\iota,\xi)\in[x_{0}, x ]\times[y_{0},y]} \check{f}_{j}(\iota,\xi,s,t)\Bigr)^{q}, \\ &\quad (x,y)\in [x_{0},x_{1})\times[y_{0},y_{1}), \end{aligned} \end{aligned}$$
(2.86)
\(M_{j}:=\alpha_{j}^{-1}B(\frac{p(\gamma_{j}-1)+1}{\alpha_{j}}, p(\beta_{j}-1)+1)\), \(\bar{M}_{j}:=\bar{\alpha}_{j}^{-1}B(\frac{p(\bar {\gamma}_{j}-1)+1}{\bar{\alpha}_{j}}, p(\beta_{j}-1)+1)\), \(\theta_{j}:=p(\alpha_{j}(\beta_{j}-1)+\gamma _{j}-1)+1\), \(\bar{\theta}_{j}:=p(\bar{\alpha}_{j}(\bar{\beta }_{j}-1)+\bar{\gamma}_{j}-1)+1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(X_{2}\in[x_{0}, x_{1})\), \(Y_{2}\in[y_{0}, y_{1})\) are chosen such that
$$ \check{G}_{j}\bigl(r_{j}(X_{2},Y_{2}) \bigr)+ \int_{b_{j}(x_{0})}^{b_{j}(X_{1})} \int _{c_{j}(y_{0})}^{c_{i-1}(Y_{2})} \tilde{f}_{j}(X_{2},Y_{2},s,t) \,dt\,ds\le \int_{t_{j}}^{\infty}\frac {ds}{\tilde{\omega}^{q}_{j}(\varphi^{-1}(s^{1/q}))} $$
(2.87)
for \(j=1,2,\ldots,m\).

Proof

Applying Theorem 2.1 to specified \(\omega_{j}(u)\equiv\psi _{j}(u)\) (\(j=1,\ldots,M\)), \(\mu_{j}(u)\equiv1\) (\(j=1,\ldots,M\)), \(\omega_{j}(u)\equiv1\) (\(j=M+1,\ldots,m\)), \(\mu_{j}(u)\equiv\psi_{j}(u)\) (\(j=M+1,\ldots,m\)), \(f_{j}(x,y,s,t)=\check{f}_{j}(x,y,s,t)\), \(g(t)=t\), from (2.82) we obtain estimate (2.83). The proof is complete. □

3 Applications

Consider a nonlinear weakly singular integral equation with maxima
$$ \textstyle\begin{cases} z(x,y)=a(x,y)+\int_{x_{0}}^{x}\int_{y_{0}}^{y}(x-s)^{\theta _{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1} \\ \hphantom{z(x,y)={}}{}\times F(x,y, s,t,z(s,t),\max_{ \tilde{\eta}\in [s-h,s]}z( \tilde{\eta},t))\, ds\, dt, \quad (x,y)\in\Delta, \\ z(x,y)=\psi(x,y), \quad (x,y)\in[x_{0}-h,x_{0}]\times[y_{0}, y_{1}), \end{cases} $$
(3.1)
where \(F\in C(\Delta\times\mathbb {R}^{4},\mathbb {R})\), h is a positive constant, \(\psi\in C([x_{0}-h,x_{0}]\times[y_{0},y_{1}),\mathbb{R})\), \(a\in C(\Delta, \mathbb {R})\), \(\theta_{j}\in(0,1)\), and \(p(\gamma_{j}-1)+1>0\) such that \(\frac{1}{p}+\theta_{j}+\gamma_{j}-2\ge0\) and \(p(\theta _{j}-1)+1>0\), \(p>1\), \(j=1,2\).

The following result gives an estimate for its solutions.

Corollary 3.1

Suppose that functions F in (3.1) satisfy
$$ \bigl\vert F(x,y,s,t,u,v) \bigr\vert \le h_{1}(x,y,s,t) \mu_{1}\bigl( \vert u \vert \bigr)+h_{2}(x,y,s,t) \mu_{2}\bigl( \vert v \vert \bigr), $$
(3.2)
where \(h_{j}\in C([x_{0},x_{1})\times[y_{0},y_{1})\times\mathbb {R}^{2},\mathbb {R}_{+})\), and \(h_{j}(x,y,s,t)\) is nondecreasing in x and y for each fixed s and t, and \(\mu_{j}\in C(\mathbb {R}_{+},(0,\infty))\) (\(j=1,2\)) such that \(\mu_{1}\propto\mu_{2}\), \(\max_{s\in[x_{0}-h,x_{0}]}\psi(s,y)\le3^{1-1/q}|a(x_{0}, y)|\) for all \(y\in[y_{0}, y_{1})\).
Then any solution \(z(x,y)\) of (3.1) has the estimate
$$\begin{aligned}& \bigl\vert z(x,y) \bigr\vert \\& \quad \le \biggl[ {Q_{2}}^{-1} \biggl(Q_{2}\bigl(\gamma(x,y)\bigr)+3^{q-1} \bigl(M_{1}x^{\delta _{1}}M_{2}y^{\delta_{2}} \bigr)^{q/p} \int_{x_{0}}^{x} \int_{y_{0}}^{y}h_{2}(x,y,s,t)dt \,ds \biggr) \biggr]^{1/q} \end{aligned}$$
(3.3)
for all \((x,y)\in[x_{0},X_{1})\times[y_{0},Y_{1})\), where
$$\begin{aligned}& \gamma(x,y) := Q_{1}^{-1} \biggl(Q_{1}\bigl( \eta _{1}(x,y)\bigr)+3^{q-1}\bigl(M_{1}x^{\delta_{1}}M_{2}y^{\delta_{2}} \bigr)^{q/p} \int _{x_{0}}^{x} \int_{y_{0}}^{y}h^{q}_{1}(x,y,s,t) \,dt\,ds \biggr), \\ & \eta_{1}(x,y) := 3^{q-1}\Bigl(\max_{(s,t)\in [x_{0},x]\times[y_{0},y]} \bigl\vert a(s,t) \bigr\vert \Bigr)^{q},\qquad Q_{1}(u):= \int_{u_{1}}^{u}\frac{ds}{\mu_{1}^{q}(s^{\frac{1}{q}})}, \quad u\ge u_{1}>0, \\ & Q_{2}(u) := \int_{u_{1}}^{u}\frac{ds}{\mu_{2}^{q}(s^{\frac{1}{q}})},\quad u\ge u_{2}>0, \end{aligned}$$
\(M_{j}:=B(p(\gamma_{j}-1)+1, p(\theta_{j}-1)+1)\) (\(j=1,2\)), \(\delta _{j}:=p(\theta_{j}+\gamma_{j}-2)+1\), \(j=1,2\), \(\frac{1}{p}+\frac{1}{q}=1\), and constants \(u_{1}\), \(u_{2}\) are given arbitrarily, \(X_{1}\in[x_{0}, x_{1})\), \(Y_{1}\in[y_{0}, y_{1})\) are chosen such that
$$\begin{aligned}& Q_{1}\bigl(\gamma_{1}(X_{1},Y_{1}) \bigr)+3^{q-1}\bigl(M_{1}X_{1}^{\delta_{1}}M_{2}Y_{1}^{\delta _{2}} \bigr)^{q/p} \int_{x_{0}}^{X_{1}} \int_{y_{0}}^{Y_{1}}h^{q}_{1}(X_{1},Y_{1},s,t) \,dt\,ds \le \int_{u_{1}}^{\infty}\frac{ds}{\mu_{1}^{q}(s^{\frac{1}{q}})}, \\& Q_{2}\bigl(\gamma_{2}(X_{1},Y_{1}) \bigr)+3^{q-1}\bigl(M_{1}X_{1}^{\delta_{1}}M_{2}Y_{1}^{\delta _{2}} \bigr)^{q/p} \int_{x_{0}}^{x} \int_{y_{0}}^{y}h^{q}_{2}(X_{1},Y_{1},s,t) \,dt\,ds \le \int_{u_{2}}^{\infty}\frac{ds}{\mu_{2}^{q}(s^{\frac{1}{q}})}. \end{aligned}$$

Proof

From (3.1) we obtain
$$ \begin{aligned} & \bigl\vert z(x,y) \bigr\vert \le \bigl\vert a(x,y) \bigr\vert + \int_{x_{0}}^{x} \int _{y_{0}}^{y}(x-s)^{\theta_{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta _{2}-1}t^{\gamma_{2}-1} \\ &\hphantom{ \bigl\vert z(x,y) \bigr\vert \le{}}{}\cdot \Bigl\vert F\Bigl(x,y, s,t,z(s,t),\max _{ \tilde{\eta}\in[s-h,s]}z( \tilde{\eta},t)\Bigr) \Bigr\vert \,dt\,ds \\ &\hphantom{ \bigl\vert z(x,y) \bigr\vert }\le \bigl\vert a(x,y) \bigr\vert + \int_{x_{0}}^{x} \int_{y_{0}}^{y}(x-s)^{\theta _{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1} \\ &\hphantom{ \bigl\vert z(x,y) \bigr\vert \le{}}{}\cdot h_{1}(x,y,s,t) \mu _{1}\bigl( \bigl\vert z(s,t) \bigr\vert \bigr)\,dt\,ds \\ &\hphantom{ \bigl\vert z(x,y) \bigr\vert \le{}}{}+ \int_{x_{0}}^{x} \int_{y_{0}}^{y}(x-s)^{\theta_{1}-1}s^{\gamma _{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1} h_{2}(x,y,s,t) \\ &\hphantom{ \bigl\vert z(x,y) \bigr\vert \le{}}{}\cdot\mu_{2}\Bigl( \Bigl\vert \max _{ \tilde{\eta}\in[s-h,s]}z( \tilde {\eta},t)\Bigr) \Bigr\vert )\,dt\,ds,\quad (x,y)\in\Delta, \\ & \bigl\vert z(x,y) \bigr\vert \le \bigl\vert \psi(x,y) \bigr\vert , \quad (x,y)\in [x_{0}-h,x_{0}]\times[y_{0},y_{1}). \end{aligned} $$
(3.4)
Set \(v(x,y)=|z(x,y)|\) for all \((x,y)\in[x_{0}-h,x_{1})\times[y_{0},y_{1})\). From (3.4) we get
$$\begin{aligned}& v(x,y) \le \bigl\vert a(x,y) \bigr\vert + \int_{x_{0}}^{x} \int _{y_{0}}^{y}(x-s)^{\theta_{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma _{2}-1} \\& \hphantom{v(x,y) \le{}}{} \cdot h_{1}(x,y,s,t) \mu_{1}\bigl(v(s,t)\bigr)\,dt\,ds \\& \hphantom{v(x,y) \le{}}{}+ \int_{x_{0}}^{x} \int_{y_{0}}^{y}(x-s)^{\theta_{1}-1}s^{\gamma _{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1} h_{2}(x,y,s,t) \\& \hphantom{v(x,y) \le{}}{}\cdot\mu_{2}\Bigl(\max_{ \tilde{\eta}\in[s-h,s]} \bigl\vert v( \tilde {\eta},t) \bigr\vert \Bigr)\,dt\,ds,\quad (x,y)\in\Delta, \\& v(x,y) \le \bigl\vert \psi(x,y) \bigr\vert ,\quad (x,y) \in[x_{0}-h,x_{0}]\times[y_{0},y_{1}). \end{aligned}$$
(3.5)
Applying Corollary 2.3 to the specified \(M=1\), \(m=2\), \(\varphi_{1} (u)=u\), \(f_{j}(x,y,s,t)=h_{j}(x,y,s,t)\), \(b_{j}(t)=t\), \(c_{j}(t)=t\), \(\alpha_{j}=\bar{\alpha}_{j}=1\), \(g(t)=t\), we obtain (3.3) from (3.5). □

Corollary 3.2

Suppose that functions F and ψ in (3.1) satisfy
$$ \bigl\vert F(x,y,s_{1},t_{1})-F(x,y,s_{2},t_{2}) \bigr\vert \leq h_{1}(x,y) \vert s_{1}-s_{2} \vert +h_{2}(x,y) \vert t_{1}-t_{2} \vert $$
(3.6)
for all \((x,y)\in\Delta\) and \(s_{j},t_{j}\in\mathbb {R}\) (\(i =1,2\)), where \(h_{j}\in C(\Delta,\mathbb {R}_{+})\). Then system (3.1) has at most one solution on Δ.

Proof

Assume that equation (3.1) has two solutions \(u(x,y)\), \(v(x,y)\). By the equivalent integral equation (3.1), we have
$$\begin{aligned} \bigl\vert u(x,y)-v(x,y) \bigr\vert \le& \int_{x_{0}}^{x} \int _{y_{0}}^{y}(x-s)^{\theta_{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma _{2}-1} h_{1}(s,t) \bigl\vert u(s,t)-v(s,t) \bigr\vert \,dt\,ds \\ &{}+ \int_{x_{0}}^{x} \int_{y_{0}}^{y} (x-s)^{\theta _{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1}h_{2}(s,t) \\ &{} \cdot \Bigl\vert \max_{ \tilde{\eta}\in[s-h,s]}u( \tilde{\eta},t)-\max _{ \tilde{\eta}\in[s-h,s]}v( \tilde{\eta},t) \Bigr\vert \,dt\,ds \end{aligned}$$
(3.7)
for all \((x,y)\in[x_{0},x_{1})\times[y_{0},y_{1})\). Since \(u(x, y)\) is a continuous function, it implies that, for any fixed \(t \in[y_{0}, y]\) and \(s \in[x_{0}, x]\), there exists \(\tau\in [s-h, s]\) such that \(\max_{ \tilde{\eta}\in[s-h,s]}u( \tilde{\eta },t) = u(\tau,t)\) holds. Now we suppose \(\max_{ \tilde{\eta }\in[s-h,s]}u( \tilde{\eta},t)\ge\max_{ \tilde{\eta}\in [s-h,s]}v( \tilde{\eta},t)\) and have
$$\begin{aligned} \Bigl\vert \max_{ \tilde{\eta}\in[s-h,s]}u( \tilde{\eta},t)-\max _{ \tilde{\eta}\in[s-h,s]}v( \tilde{\eta},t) \Bigr\vert =& \Bigl\vert u( \tau,t)-\max_{ \tilde{\eta}\in[s-h,s]}v( \tilde{\eta},t) \Bigr\vert \\ \le& \bigl\vert u(\tau,t)-v(\tau,t) \bigr\vert \le\max_{ \tilde{\eta}\in [s-h,s]} \bigl\vert u( \tilde{\eta},t)-v( \tilde{\eta},t) \bigr\vert . \end{aligned}$$
(3.8)
It follows from (3.7) and (3.8) that
$$\begin{aligned} \bigl\vert u(x,y)-v(x,y) \bigr\vert \le& \int_{x_{0}}^{x} \int_{y_{0}}^{y} (x-s)^{\theta _{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma _{2}-1}h_{1}(s,t) \bigl\vert u(s,t)-v(s,t) \bigr\vert \,dt\,ds \\ &{}+ \int_{x_{0}}^{x} \int_{y_{0}}^{y} (x-s)^{\theta _{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1}h_{2}(s,t) \\ &{} \cdot\max_{ \tilde{\eta}\in[s-h,s]} \bigl\vert u( \tilde{\eta },t)-v( \tilde{\eta},t) \bigr\vert \,dt\,ds. \end{aligned}$$
(3.9)
Let
$$\phi(x,y):= \bigl\vert u(x,y)-v(x,y) \bigr\vert , \quad (x,y)\in\bigl[ \alpha(x_{0})-h, x_{0}\bigr]\times [y_{0}, y_{1}). $$
From (3.7) we obtain
$$\begin{aligned}& \phi(x,y) \le \int_{x_{0}}^{x} \int_{y_{0}}^{y} (x-s)^{\theta _{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1}h_{1}(s,t) \phi(s,t)\,dt\,ds \\& \hphantom{\phi(x,y) \le{}}{} + \int_{x_{0}}^{x} \int_{y_{0}}^{y} (x-s)^{\theta_{1}-1}s^{\gamma _{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1}(x-s)^{\theta_{1}-1}s^{\gamma_{1}-1} \\& \hphantom{\phi(x,y) \le{}}{} \cdot(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1}h_{2}(s,t) \max_{ \tilde{\eta}\in[s-h,s]}\phi( \tilde{\eta},t)\, dt \, d\eta, \\& \hphantom{\phi(x,y) \le{}}{}(x,y)\in[x_{0},x_{1}) \times[y_{0},y_{1}), \\& \phi(x,y) \le 0, \quad (x,y)\in [x_{0}-h,x_{0}] \times[y_{0},y_{1}). \end{aligned}$$
(3.10)
Let \(\varepsilon>0\) be an arbitrary number. Then from (3.10) we have
$$ \begin{aligned} &\phi(x,y)\le \varepsilon+ \int_{x_{0}}^{x} \int_{y_{0}}^{y} (x-s)^{\theta_{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma _{2}-1}h_{1}(s,t) \phi(s,t)\,dt\,ds \\ &\hphantom{\phi(x,y)\le{}}{}+ \int_{x_{0}}^{x} \int_{y_{0}}^{y} (x-s)^{\theta_{1}-1}s^{\gamma _{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1} \\ &\hphantom{\phi(x,y)\le{}}{} \cdot h_{2}(s,t) \max_{ \tilde{\eta}\in[\alpha(s)-h,\alpha(s)]}\phi( \tilde{\eta},t)\, dt\, d\eta, \\ &\hphantom{\phi(x,y)\le{}}{} (x,y)\in[x_{0},x_{1}) \times[y_{0},y_{1}), \\ &\phi(x,y)\le 0, \quad (x,y)\in [x_{0}-h,x_{0}] \times[y_{0},y_{1}). \end{aligned} $$
(3.11)
Applying Corollary 2.3 to specified \(N=1\), \(m=2\), \(\varphi _{1}(u)=u\), \(g(t)=t\), \(b_{j}(t)=c_{j}(t)=t\), \(f_{j}(x,y,s,t)=h_{2}(s,t)\), \(j=12\), \(a(x,y)=\epsilon\), from (3.11) we obtain, for all \((x,y)\in\Delta\),
$$\begin{aligned}& \phi(x,y) \\& \quad \leq 3^{\frac{q-1}{q}}\varepsilon\exp \biggl(q^{-1}\biggl(3^{\frac {q-1}{q}}\bigl(M_{1}x^{\delta_{1}} \bar{M}_{1}y^{\delta_{2}}\bigr)^{\frac{q}{p}} \int_{x_{0}}^{x} \int_{y_{0}}^{y}\bigl(h_{1}^{q}(s,t)+h_{2}^{q}(s,t) \bigr)\,dt\,ds\biggr)\biggr), \end{aligned}$$
(3.12)
where \(\frac{1}{p}+\frac{1}{q}=1\), \(M_{j}\) and \(\delta_{j}\) (\(j=1,2\)) are defined as in Corollary 3.1. Letting \(\varepsilon\rightarrow 0\), we obtain the uniqueness of the solution of equation (3.1). The uniqueness is proved. □

Declarations

Funding

This research was supported by the National Natural Science Foundation of China (No. 11461058).

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Science and Technology, Sichuan Minzu College, Kangding, P.R. China
(2)
Network Information Center, Sichuan Minzu College, Kangding, P.R. China

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