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. [2‐11]) 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 [12‐20]). In [14], a new method was presented to analyze the nonlinear singular integral inequalities of Henry type:
In 2008, Cheung et al. [20] solved the nonlinear weakly singular inequality
On the other hand, since differential equations with maxima of the unknown function [21‐26] can be applied in control theory, some significant results for integral inequalities containing the maxima of the unknown function [22, 27‐30] have been obtained. The integral inequality with maxima
where f, g, and ψ are nonnegative continuous functions and \(a(x,y)>0\) is a nondecreasing continuous function, was discussed in [22].
$$ u(t)\le a(t)+b(t) \int_{t_{0}}^{t}(t-s)^{\beta-1}s^{\gamma -1}F(s)u(s) \,ds,\quad t\ge0. $$
(1.1)
$$\begin{aligned} u^{p}(x,y) \le& a(x,y)+b(x,y) \int_{0}^{x} \int _{0}^{y}\bigl(x^{\alpha}-s^{\alpha} \bigr)^{\beta-1}s^{\gamma-1} \bigl(y^{\alpha}-t^{\alpha} \bigr)^{\beta-1}t^{\gamma-1} \\ &{} \cdot f(s,t)u^{q}(s,t)\,dt\,ds. \end{aligned}$$
(1.2)
$$\begin{aligned}& u(x,y)\leq a(x,y)+ \int_{x_{0}}^{x} \int_{y_{0}}^{y} f(s,t) u^{p}(s,t)\,dt\,ds \\& \hphantom{u(x,y)\leq{}}{} + \int_{\alpha(x_{0})}^{\alpha(x)} \int_{y_{0}}^{y} g(s,t) \Bigl(\max _{\tilde{\eta}\in[s-h,s]}u^{p}(\tilde{\eta },t) \Bigr)\,dt\,ds,\quad x \ge x_{0}, y\ge y_{0}, \\& u(x,y)\leq \psi(x,y), \quad x\in \bigl[\alpha(x_{0})-h, x_{0}\bigr], y\ge y_{0}, \end{aligned}$$
(1.3)
Combining (1.2) with (1.3), we will consider the integral inequality with maxima
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.
$$ \begin{aligned} &\varphi\bigl(u(x,y)\bigr) \leq a(x,y)+ \sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{\beta_{j}-1}s^{\gamma_{j}-1}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{\bar{\beta}_{j}-1} t^{\bar{\gamma}_{j}-1} \\ &\hphantom{\varphi(u(x,y)) \leq{}}{}\cdot f_{j}(x,y,s,t) \omega_{j} \bigl(u(s,t)\bigr)\mu_{j} \Bigl(\max_{\tilde{\eta}\in[s-h, s]}g\bigl(u( \tilde{\eta},t)\bigr) \Bigr)\,dt\,ds, \\ &\hphantom{\varphi(u(x,y)) \leq{}}{}(x,y)\in [x_{0},x_{1}) \times[y_{0}, y_{1}), \\ &u(x,y) \leq \psi (x,y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times [y_{0}, y_{1}), \end{aligned} $$
(1.4)
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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
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.
$$ \tilde{\omega}_{j}(t):= \textstyle\begin{cases} \hat{\omega}_{1}(t)\max_{\tau\in[0, t] }\{\hat{\mu }_{1}(\tilde{g}(\tau))\}, & t\ge0, j=1, \\ \max_{\tau\in[0, t] }\{\frac{\hat{\omega}_{j}(\tau)\hat {\mu}_{j+1}(\tilde{g}((\tau))}{\tilde{\omega}_{i-1}(\tau)}\}\tilde{\omega}_{i-1}(t),& t\ge0, j=2,\ldots,m, \end{cases} $$
(2.1)
Lemma 1
([16])
Let
α, β, γ, and
p
be positive constants. Then
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.
$$ \int^{t}_{0}\bigl(t^{\alpha}-s^{\alpha} \bigr)^{p(\beta-1)}s^{p(\gamma -1)}\,ds=\frac{t^{\theta}}{\alpha}B\biggl( \frac{p(\gamma-1)+1}{\alpha}, p(\beta-1)+1\biggr),\quad t\in{\mathbb{R}_{+}}, $$
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Lemma 2
Suppose that
If
\(u\in C([b_{*}(x_{0})-h,x_{1})\times[y_{0},y_{1}),\mathbb {R}_{+})\)
satisfies the integral inequality
then
for all
\((x,y)\in[x_{0}, X_{1}^{*}]\times[y_{0},Y_{1}^{*}]\), where
\(H_{j}^{-1}\)
is the inverse of the function
\(t_{j}\)
is a given constant, and
\(\eta_{j}\)
is defined by
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
for
\(j=1,\ldots,m\).
(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
\(j=1,\ldots,m\);
(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
\(j=1,\ldots,m\);
(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
\(h_{j}\bar{h}_{j}\propto h_{j+1}\bar{h}_{j+1}\) (\(j=1,\ldots,m-1\));
(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
\(t\in [y_{0},y_{1})\).
$$\begin{aligned}& u(x,y)\leq b(x,y)+\sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} g_{j}(x,y,s,t) \\& \hphantom{u(x,y)\leq{}}{}\times h_{j}\bigl(u(s,t)\bigr)\tilde{{h}}_{j} \Bigl(\max_{\tilde{\eta }\in[s-h, s]}u(\tilde{\eta},t) \Bigr)\,dt\,ds, \quad (x,y) \in\Delta, \\& u(x,y)\leq \psi (x,y), \quad (x,y)\in\Xi, \end{aligned}$$
(2.2)
$$ u(x,y)\leq H_{m}^{-1} \biggl(H_{m} \bigl(\eta_{m}(x,y)\bigr)+ \int _{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} g_{m}(x,y,s,t)\,dt\,ds \biggr) $$
(2.3)
$$ H_{j}(t):= \int_{t_{j}}^{t}\frac{ds}{h_{j}(s)\bar{h}_{j}(s)}, \quad t\ge t_{j}>0, j=1,\ldots,m, $$
(2.4)
$$ \begin{aligned} &\eta_{1}(x,y):=b(x_{0},y_{0})+ \int_{x_{0}}^{x} \bigl\vert b_{x}(s,y_{0}) \bigr\vert \,ds+ \int_{y_{0}}^{y} \bigl\vert b_{x}(x,t) \bigr\vert \, dt, \\ &\eta_{j+1}(x,y):=H_{j}^{-1} \biggl(H_{j}\bigl(\eta_{j}(x,y)\bigr)+ \int_{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}g_{j}(x,y,s,t)dt \,ds \biggr) \end{aligned} $$
(2.5)
$$ H_{j}\bigl(\eta_{j}\bigl(X^{*}_{1},Y^{*}_{1} \bigr)\bigr)+ \int_{a_{j}(x_{0})}^{a_{j}(X^{*}_{1})} \int_{b_{j}(y_{0})}^{b_{j}(Y^{*}_{1})}g_{j}\bigl(X^{*}_{1},Y^{*}_{1},s,t \bigr)\, dt \,ds\le \int _{u_{j}}^{\infty}\frac{ds}{{h_{j}(s)}\tilde{h}(s)} $$
(2.6)
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 \(\eta_{1}(x,y)>0\),
From (2.2) and (2.7), we have
Concerning (2.8), we consider the auxiliary inequality
where \(x_{0}\leq\xi\le X^{*}_{1}\) and \(y_{0}\leq\eta\le Y^{*}_{1}\) are chosen arbitrarily. Having (2.9) we claim
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
for \(j=2,\ldots, m\), and \(X^{*}_{2}\in[x_{0},x_{1})\), \(Y^{*}_{2}\in[y_{0},y_{1})\) are chosen such that
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.
$$ \eta_{1}(x,y)\ge b(x_{0},y_{0})+ \int_{x_{0}}^{x} b_{x}(s,y_{0}) \,ds+ \int _{y_{0}}^{y}b_{y}(x,t)\, dt=b(x,y). $$
(2.7)
$$ \begin{aligned} &u(x,y)\leq \eta_{1}(x,y)+ \sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} g_{j}(x,y,s,t) \\ &\hphantom{u(x,y)\leq{}}{}\cdot h_{j}\bigl(u(s,t)\bigr)\bar{h}_{j} \Bigl(\max_{\tilde{\eta}\in[s-h, s]}u(\tilde{\eta},t) \Bigr)\,dt\,ds,\quad (x,y) \in\Delta, \\ &u(x,y)\leq \psi (x,y), \quad (x,y)\in\Xi. \end{aligned} $$
(2.8)
$$ \begin{aligned} &u(x,y) \leq \eta_{1}(x,y)+ \sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} g_{j}(\xi,\eta,s,t) \\ &\hphantom{u(x,y) \leq{}}{} \times h_{j}\bigl(u(s,t)\bigr) \bar{h}_{j} \Bigl(\max_{\tilde{\eta}\in[s-h, s]}u(\tilde{\eta},t) \Bigr)\,dt\,ds, \quad (x,y)\in[x_{0},\xi]\times[y_{0}, \eta], \\ &u(x,y) \leq \psi(x,y), \quad (x,y)\in \bigl[b_{*}(x_{0})-h, x_{0}\bigr]\times[y_{0},\eta], \end{aligned} $$
(2.9)
$$ u(x,y)\leq H_{m}^{-1} \biggl(H_{m} \bigl(\eta_{m}(\xi,\eta,x,y)\bigr)+ \int _{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} g_{m}(\xi,\eta,s,t)\,dt \,ds \biggr) $$
(2.10)
$$ \tilde{\eta}_{j}(\xi,\eta,x,y):= H_{j-1}^{-1} \biggl(H_{j-1}\bigl(\tilde{\eta }_{j-1}(\xi,\eta,x,y)\bigr)+ \int_{b_{j-1}(x_{0})}^{b_{j-1}(x)} \int _{c_{j-1}(y_{0})}^{c_{j-1}(y)} g_{j-1}(\xi,\eta,s,t)\,dt \,ds\biggr) $$
$$\begin{aligned} &H_{j}\bigl(\tilde{\eta}_{j}\bigl(\xi, \eta,X^{*}_{2},Y^{*}_{2}\bigr)\bigr)+ \int _{b_{j}(x_{0})}^{b_{j}(X^{*}_{2})} \int_{c_{j}(y_{0})}^{c_{j}(Y^{*}_{2})} g_{j}(\xi,\eta,s,t) \\ &\quad \le \int_{t_{j}}^{\infty}\frac{ds}{{h_{j}(s)}\bar{h}_{j}(s)} \end{aligned}$$
(2.11)
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
where
\(z_{1}(x,y)\) is a nondecreasing function on \([x_{0}, \xi]\times[y_{0},\eta]\). Then
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
Integrating inequality (2.15) from \(x_{0}\) to x, from (2.4) we get
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
for \(x_{0}\le x\le\xi< X^{*}_{2}\), \(Y_{0}\le y \le\eta< Y^{*}_{2}\), implying that (2.7) is true for \(m=1\).
$$ u(x,y)\leq z_{1}(x,y),\quad (x,y)\in \bigl[b_{*}(x_{0})-h, \xi\bigr]\times[y_{0}, \eta], $$
(2.13)
$$ z_{1}(x,y)= \textstyle\begin{cases} \eta_{1}(x,y)+ \int_{b_{1}(x_{0})}^{b_{1}(x)}\int_{c_{1}(y_{0})}^{c_{1}(y)} g_{1}(\xi,\eta,s,t) h_{1}(u(s,t)) \\ \quad {}\times\bar{h}_{1} (\max_{\tilde{\eta}\in[s-h, s]}u(\tilde{\eta},t) )\,dt\,ds, \quad (x,y)\in[x_{0},\xi]\times[y_{0},\eta] \\ \eta_{1}(x_{0},y), \quad (x,y)\in[b_{*}(x_{0})-h, x_{0}]\times[y_{0},\eta], \end{cases} $$
(2.14)
$$\begin{aligned} \frac{\partial}{\partial x}z_{1}(x,y) =&\frac{\partial }{\partial x} \eta_{1}(x,y)+ \int_{c_{1}(y_{0})}^{c_{1}(y)} g_{1}\bigl(\xi, \eta,b_{1}(x),t\bigr) h_{1}\bigl(u\bigl(b_{1}(x),t \bigr)\bigr) \\ &{}\times\bar{h}_{1} \Bigl(\max_{\tilde{\eta }\in[b_{1}(x)-h, b_{1}(x)]}u(\tilde{ \eta},t) \Bigr)\, dtb'(x) \end{aligned}$$
$$\begin{aligned}& \frac{\frac{\partial}{\partial x}z_{1}(x,y)}{h_{1}(z_{1}(x,y))\bar{h}_{1}(z_{1}(x,y))} \\& \quad \le \frac{\frac{\partial}{\partial x}\eta _{1}(x,y)}{h_{1}(\eta_{1}(x,y))\bar{h}_{1}(\eta_{1}(x,y))}+\frac {b'(x)}{h_{1}(z_{1}(x,y))\bar{h}_{1}(z_{1}(x,y))} \\& \qquad {}\times \int_{c_{1}(y_{0})}^{c_{1}(y)} g_{1}\bigl(\xi, \eta,b_{1}(x),t\bigr) h_{1}\bigl(u\bigl(b_{1}(x),t \bigr)\bigr)\bar{h}_{1} \Bigl(\max_{\tilde{\eta}\in [b_{1}(x)-h, b_{1}(x)]}u(\tilde{ \eta},t) \Bigr)\,dt \\& \quad \le \frac{\frac{\partial}{\partial x}\eta_{1}(x,y)}{h_{1}(\eta_{1}(x,y))\bar {h}_{1}(\eta_{1}(x,y))}+b'(x) \int_{c_{1}(y_{0})}^{c_{1}(y)} g_{1}\bigl(\xi, \eta,b_{1}(x),t\bigr)\,dt. \end{aligned}$$
(2.15)
$$\begin{aligned} H_{1}\bigl(Z_{1}(x,y)\bigr) \le& H_{1}\bigl( \eta_{1}(x,y)\bigr)+ \int _{x_{0}}^{x}b'(s) \int_{c_{1}(y_{0})}^{c_{1}(y)} g_{1}\bigl(\xi, \eta,b_{1}(s),t\bigr)\,dt\,ds \\ =& H_{1}\bigl(\eta_{1}(x,y)\bigr)+ \int_{b_{1}(x_{0})}^{b_{1}(x)} \int _{c_{1}(y_{0})}^{c_{1}(y)} g_{1}(\xi,\eta,s,t)\,dt \,ds \end{aligned}$$
(2.16)
$$ u(x,y))\le H^{-1}_{1}\biggl( H_{1}\bigl(\eta_{1}(x,y)\bigr)+ \int_{b_{1}(x_{0})}^{b_{1}(x)} \int _{c_{1}(y_{0})}^{c_{1}(y)} g_{1}(\xi,\eta,s,t)\,dt \,ds\biggr) $$
(2.17)
Assume that (2.10) holds for \(m=k\). Consider
Let
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
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
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
for all \((x,y)\in[x_{0},X]\times[y_{0},Y]\). Let
Then inequality (2.21) can be rewritten as
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
for \(x_{0}\le x\le\min\{\xi, X_{3}^{*}\}\), \(y_{0}\le y\le\min\{\eta, Y_{3}^{*}\}\), where
\(\tilde{t}_{j+1}=H_{1}(t_{j+1})\), \(\bar{H}^{-1}_{j+1}\) is the inverse of \(\bar{H}_{j+1}\), \(j=1,\ldots, k\),
\(j=1,\ldots,k-1\), and \(X^{*}_{3}\), \(Y^{*}_{3}\) are chosen such that
Note that
Then, from (2.20), (2.24), and (2.28), we get
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
Moreover, with the assumption that \(\tilde{\varrho}_{k}(x,y)=\tilde {\eta}_{k+1}(\xi,\eta,x,y)\), we get
This proves that
Therefore, (2.27) becomes
which implies that \(X^{*}_{2}=X^{*}_{3}\), \(\xi\le X^{*}_{3}\), \(Y^{*}_{2}=Y^{*}_{3}\), \(\eta\le Y^{*}_{3}\). From (2.29) we obtain
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.
$$\begin{aligned}& u(x,y) \leq \eta_{1}(x,y)+\sum_{j=1}^{k+1} \int_{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} g_{j}(\xi,\eta,s,t) \\& \hphantom{u(x,y) \leq{}}{}\times h_{j}\bigl(u(s,t)\bigr)\bar{h}_{j} \Bigl(\max_{\tilde{\eta}\in[s-h, s]}u(\tilde{\eta},t)\Bigr)\,dt\,ds,\quad (x,y) \in[x_{0},\xi]\times[y_{0},\eta] \\& u(x,y) \leq \psi (x,y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h, x_{0}\bigr]\times[y_{0},\eta] . \end{aligned}$$
(2.18)
$$ z_{2}(x,y)= \textstyle\begin{cases} \eta_{1}(x,y) +\sum_{j=1}^{k+1} \int_{b_{j}(x_{0})}^{b_{j}(x)}\int _{c_{j}(y_{0})}^{c_{j}(y)} g_{j}(\xi,\eta, s,t)h_{j}(u(s,t)) \\ \quad {}\cdot\bar{h}_{j}(\max_{\tilde{\eta}\in[s-h, s]}u(\tilde {\eta},t))\,dt\,ds,\quad (x,y)\in[x_{0},\xi]\times[y_{0},\eta], \\ \eta_{1}(x_{0},y),\quad (x,y)\in [b_{*}(x_{0})-h, x_{0}]\times[y_{0},\eta]. \end{cases} $$
(2.19)
$$ u(x,y)\leq z_{2}(x,y),\quad (x,y)\in \bigl[b_{*}(x_{0})-h, \xi\bigr]\times [y_{0}, \eta]. $$
(2.20)
$$\begin{aligned}& \frac{\frac{\partial}{\partial x}z_{2}(x,y)}{h_{1}(z_{2}(x,y))\bar {h}_{1}(z_{2}(x,y))} \\& \quad \le\frac{\frac{\partial}{\partial x}\eta _{1}(x,y)}{h_{1}(z_{2}(x,y))\bar{h}_{1}(z_{2}(x,y))}+\sum_{j=1}^{k+1} \frac{b'_{j}(x)}{ h_{1}(z_{2}(x,y))\bar{h}_{1}(z_{2}(x,y))} \\& \qquad {}\cdot \int_{c_{j}(y_{0})}^{c_{j}(y)}g_{j}\bigl(X,Y,b_{j}(x),t \bigr)h_{j}\bigl(u\bigl(b_{j}(x),t\bigr)\bigr) h_{j}\Bigl(\max_{\xi\in[b_{j}(x)-h,b_{j}(x)]}u(\tilde{\eta},t)\Bigr)\,dt \\& \quad \le\frac{\frac{\partial}{\partial x}\eta_{1}(x,y)}{h_{1}(\eta _{1}(x,y))\bar{h}_{1}(\eta_{1}(x,y))}+\sum_{j=1}^{k+1} \frac{b'_{j}(x)}{ h_{j}(z_{2}(x,y))\bar{h}_{j}(z_{2}(x,y))} \\& \qquad {}\cdot \int_{c_{j}(y_{0})}^{c_{j}(y)}g_{j}\bigl(\xi, \eta,b_{j}(x),t\bigr)h_{j}\bigl(z_{2} \bigl(b_{j}(x),t\bigr)\bigr) \bar{h}_{j}\Bigl(\max _{\tilde{\eta}\in [b_{j}(x)-h,b_{j}(x)]}z_{2}(\tilde{\eta},t)\Bigr)\,dt \\& \quad \le\frac{\frac{\partial}{\partial x}\eta_{1}(x,y)}{h_{1}(\eta _{1}(x,y))\bar{h}_{1}(\eta_{1}(x,y))} +b'_{1}(x) \int_{c_{1}(y_{0})}^{c_{1}(y)}g_{1}\bigl(\xi, \eta,b_{1}(x),t\bigr)\,dt+\sum_{j=1}^{k}b'_{j+1}(x) \\& \qquad {} \cdot \int_{c_{j}(y_{0})}^{c_{j}(y)}g_{j+1}\bigl(\xi,\eta ,b_{j+1}(x),t\bigr)\tilde{h}_{j+1}\bigl(z_{2} \bigl(b_{j+1}(x),t\bigr)\bigr) \hat{h}_{j+1}\Bigl(\max _{\tilde{\eta}\in [b_{j}(x)-h,b_{j}(x)]}z_{2}(\tilde{\eta},t)\Bigr)\,dt \end{aligned}$$
$$\begin{aligned} H_{1}\bigl(z_{2}(x,y)\bigr) \le& H_{1}\bigl( \eta_{1}(x,y)\bigr)+ \int_{b_{1}(x_{0})}^{b_{1}(x)} \int _{c_{1}(y_{0})}^{c_{1}(y)}g_{1}(\xi,\eta,s,t)\,dt \,ds \\ &{} +\sum_{j=1}^{k} \int_{b_{j+1}(x_{0})}^{b_{j+1}(x)} \int _{c_{j+1}(y_{0})}^{c_{j+1}(y)}g_{j+1}(\xi,\eta,s,t) \tilde{h}_{j+1}\bigl(z_{2}(s,t)\bigr) \\ &{} \cdot\hat{h}_{j+1} \Bigl(\max_{\tilde{\eta}\in [s-h,s]}z_{2}( \tilde{\eta},t) \Bigr)\,dt\,ds \end{aligned}$$
(2.21)
$$ \begin{aligned} &\eta(x,y):=H_{1}\bigl(z_{2}(x,y)\bigr), \\ &\varrho_{1}(x,y):=H_{1}\bigl(\eta_{1}(x,y) \bigr)+ \int_{b_{1}(x_{0})}^{b_{1}(x)} \int _{c_{1}(y_{0})}^{c_{1}(y)}g_{1}(\xi,\eta,s,t)\,dt \,ds. \end{aligned} $$
(2.22)
$$\begin{aligned}& \eta(x,y) \le \varrho_{1}(x,y)+\sum_{j=1}^{k} \int _{b_{j+1}(x_{0})}^{b_{j+1}(x)} \int _{c_{j+1}(y_{0})}^{c_{j+1}(y)}g_{j+1}(\xi,\eta,s,t)\tilde {h}_{j+1}\bigl(H_{1}^{-1}\bigl(z_{2}(s,t) \bigr)\bigr) \\& \hphantom{\eta(x,y) \le{}}{} \cdot\hat{h}_{j+1}\Bigl(\max_{\tilde{\eta}\in [s-h,s]}H_{1}^{-1} \bigl(z_{2}(\tilde{\eta},t)\bigr)\Bigr)\,dt\,ds, \quad (x,y)\in [x_{0},X]\times[y_{0},Y], \\& \eta(x,y) = H_{1}\bigl(\eta_{(}x_{0},y)\bigr)\le \varrho_{1}(x_{0}, y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h, x_{0}\bigr]\times[y_{0},Y], \end{aligned}$$
(2.23)
$$ \eta(x,y)\le \bar{H}_{k+1}^{-1}\biggl( \bar{H}_{k+1}\bigl(\varrho _{k}(x,y)\bigr)+ \int_{b_{k+1}(x_{0})}^{b_{k+1}(x)} \int_{c_{k+1}(y_{0})}^{c_{k+1}(y)} g_{k+1}(\xi,\eta,s,t)\,dt \,ds\biggr) $$
(2.24)
$$ \bar{H}_{j+1}(t):= \int_{\tilde{t}_{j+1}}^{t}\frac{ds}{\tilde {h}_{j+1}(H_{1}^{-1}(s))\hat{h}_{j+1}(H_{1}^{-1}(s))},\quad t>0, $$
(2.25)
$$ \varrho_{j+1}(x,y):=\bar{H}^{-1}_{j+1} \biggl(\bar{H}_{j+1}\bigl(\varrho _{j}(x,y)\bigr)+ \int_{b_{j+1}(x_{0})}^{b_{j+1}(x)} \int_{c_{j+1}(y_{0})}^{c_{j+1}(y)}g_{j+1}(\xi,\eta,s,t)\,dt \,ds\biggr), $$
(2.26)
$$\begin{aligned}& \bar{H}_{j+1}\bigl(\varrho_{j}\bigl(X^{*}_{3},Y^{*}_{3} \bigr)\bigr)+ \int _{b_{j+1}(x_{0})}^{b_{j+1}(X^{*}_{3})} \int_{c_{j+1}(y_{0})}^{c_{j+1}(Y^{*}_{3})}g_{j+1}(\xi,\eta,t,s)\,dt \,ds \\& \quad \le \int_{\tilde{t}_{j+1}}^{H_{1}(\infty)}\frac{ds}{\tilde {h}_{j+1}(H^{-1}_{1}(s))\hat{h}_{j+1}(H_{1}^{-1}(s))},\quad j=1, \ldots,k. \end{aligned}$$
(2.27)
$$\begin{aligned} \bar{H}_{j}(t) =& \int_{\tilde{t}_{j}}^{t}\frac {ds}{\tilde{h}_{j}(H_{1}^{-1}(s))\hat{h_{j}}(H_{1}^{-1}(s))} \\ =& \int_{H_{1}(t_{j})}^{t}\frac{h_{1}(H^{-1}_{1}(s))\bar {h}_{1}(H^{-1}_{1}(s))\,ds}{h_{j}(H_{1}^{-1}(s))\bar{h}_{j}(H_{1}^{-1}(s))} \\ =& \int_{H_{1}(t_{j})}^{t}\frac{h_{1}(H^{-1}_{1}(s))\bar {h}_{1}(H^{-1}_{1}(s))\,ds}{h_{j}(H_{1}^{-1}(s))\bar{h}_{j}(H_{1}^{-1}(s))} \\ =& \int_{t_{j}}^{H^{-1}_{1}(t)}\frac{ds}{h_{j}(s)\bar{h}_{j}(s)}=H_{j}\bigl(H^{-1}_{1}(t)\bigr), \quad j=2, \ldots,k+1. \end{aligned}$$
(2.28)
$$\begin{aligned} u(x,y) \le& H^{-1}_{1}\bigl(\eta(x,y)\bigr) \\ \le& H_{k+1}^{-1}\biggl(H_{k+1} \bigl(H^{-1}_{1}\bigl(\varrho_{k}(x,y)\bigr) \bigr) + \int_{b_{k+1}(x_{0})}^{b_{k+1}(x)} \int_{c_{k+1}(y_{0})}^{c_{k+1}(y)} g_{k+1}(\xi,\eta,s,t)\,dt \,ds\biggr) \end{aligned}$$
(2.29)
$$\begin{aligned} \tilde{\varrho}_{1}(x,y) =&H_{1}\bigl( \varrho_{1}(x,y)\bigr) \\ =&H^{-1}_{1}\biggl(H_{1}\bigl( \eta_{1}(x,y)\bigr)+ \int_{b_{1}(x_{0})}^{b_{1}(x)} \int _{c_{1}(y_{0})}^{c_{1}(y)}g_{1}(\xi,\eta,s,t)\,dt \,ds\biggr) \\ =&H^{-1}_{1}\biggl(H_{1}\bigl(\tilde{ \eta}_{1}(\xi,\eta,x,y)\bigr)+ \int _{b_{1}(x_{0})}^{b_{1}(x)} \int_{c_{1}(y_{0})}^{c_{1}(y)}g_{1}(\xi,\eta,s,t)\,dt \,ds\biggr) \\ =&\tilde{\eta}_{2}(X,Y,x,y). \end{aligned}$$
(2.30)
$$\begin{aligned} \tilde{\varrho}_{k+1}(x,y) =&H^{-1}_{1}\biggl( \bar {H}^{-1}_{k+1}\biggl(\bar{H}_{k+1}\bigl( \varrho_{k}(x,y)\bigr)+ \int_{b_{k+1}(x_{0})}^{b_{k+1}(x)} \int_{c_{k+1}(y_{0})}^{c_{k+1}(y)}g_{k+1}(\xi,\eta,t,s)\,dt\,ds \biggr)\biggr) \\ =&H^{-1}_{k+1}\biggl(H_{k+1}\bigl(H^{-1}_{1} \bigl(\varrho_{k}(x,y)\bigr)\bigr)+ \int _{b_{k+1}(x_{0})}^{b_{k+1}(x)} \int_{c_{k+1}(y_{0})}^{c_{k+1}(y)}g_{k+1}(\xi,\eta,t,s)\,dt\,ds \biggr) \\ =&H^{-1}_{k+1}\biggl(H_{k+1}\bigl(\tilde{ \varrho}_{k}(x,y)\bigr)+ \int _{b_{k+1}(x_{0})}^{b_{k+1}(x)} \int_{c_{k+1}(y_{0})}^{c_{k+1}(y)}g_{k+1}(\xi,\eta,t,s)\,dt\,ds \biggr) \\ =&H^{-1}_{k+1}\biggl(H_{k+1}\bigl(\tilde{ \eta}_{k+1}(\xi,\eta ,x,y)\bigr)+ \int_{b_{k+1}(x_{0})}^{b_{k+1}(x)} \int_{c_{k+1}(y_{0})}^{c_{k+1}(y)}g_{k+1}(\xi,\eta,t,s)\,dt\,ds \biggr) \\ =&\tilde{\eta}_{k+2}(\xi,\eta,x,y). \end{aligned}$$
(2.31)
$$ \tilde{\varrho}_{j}(x,y)=\tilde{\eta}_{j+1}( \xi,\eta, x,y),\quad j=1,\ldots, k . $$
(2.32)
$$\begin{aligned}& H_{j+1}\bigl(\tilde{\eta}_{j+1}\bigl(\xi, \eta,X^{*}_{3},Y^{*}_{3}\bigr) \bigr)+ \int _{b_{j+1}(x_{0})}^{b_{j+1}(X^{*}_{3})} \int_{c_{j+1}(y_{0})}^{c_{j+1}(Y^{*}_{3})}g_{j+1}(\xi,\eta,t,s)\,dt \,ds \\& \quad \le \int_{\tilde{t}_{j+1}}^{H_{1}(\infty)}\frac{ds}{\tilde {h}_{j+1}(H^{-1}_{1}(s))\hat{h}_{j+1}(H_{1}^{-1}(s))} \\& \quad = \int_{t_{j+1}}^{\infty}\frac{ds}{h_{j+1}(s)\bar{h}_{j+1}(s)},\quad j=1, \ldots,k, \end{aligned}$$
(2.33)
$$ u(x,y)\le H_{k+1}^{-1}\biggl(H_{k+1}\bigl(\tilde{ \eta}_{k+1}(\xi,\eta,x,y)\bigr)+ \int _{b_{k+1}(x_{0})}^{b_{k+1}(x)} \int_{c_{k+1}(y_{0})}^{c_{k+1}(y)} g_{k+1}(\xi,\eta,s,t)\,dt \,ds\biggr) $$
Taking \(x=\xi,\eta\), \(y=\xi,\eta\) in (2.10), we have
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
for all \((x,y)\in[x_{0}, X^{*}_{1}]\times[y_{0},Y^{*}_{1}]\).
$$\begin{aligned} u(\xi,\eta) \leq&H_{m}^{-1} \biggl(H_{m}\bigl( \tilde{\eta}_{m}(\xi,\eta ,\xi,\eta)\bigr)+ \int_{b_{m}(x_{0})}^{b_{m}(X)} \int _{c_{m}(y_{0})}^{c_{m}(\eta)} g_{m}(\xi,\eta,s,t)\,dt \,ds \biggr) \\ = &H_{m}^{-1} \biggl(H_{m}\bigl( \eta_{m}(\xi,\eta)\bigr)+ \int _{b_{m}(x_{0})}^{b_{m}(\xi)} \int_{c_{m}(y_{0})}^{c_{m}(\eta)} g_{m}(\xi,\eta,s,t)\,dt \,ds \biggr) \end{aligned}$$
(2.34)
$$ u(x,y) \le H_{m}^{-1} \biggl(H_{m}\bigl( \eta_{m}(x,y)\bigr)+ \int _{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} g_{m}(x,y,s,t)\,dt\,ds \biggr) $$
(2.35)
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
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. □
$$ u(x,y)\leq H_{m}^{-1}\biggl(H_{m}\bigl( \eta_{n,\epsilon}(x,y)\bigr)+ \int _{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} g_{m}(x,y,s,t)\,dt\,ds \biggr) $$
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
where
\(W_{j}^{-1}\)is the inverse of the function
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),
\(X_{1}\in[x_{0}, x_{1})\), \(Y_{1}\in[y_{0}, y_{1})\)
are chosen such that
for
\(j=1,\ldots,m\).
$$ u(x,y)\leq\varphi^{-1}\biggl(\biggl(W_{m}^{-1} \bigl(W_{m}\bigl(r_{m}(x,y)\bigr)\bigr)+ \int_{\alpha _{m}(x_{0})}^{\alpha_{m}(x)} \int_{\beta_{m}(y_{0})}^{\beta_{m}(y)} \tilde{f}_{m}(x,y,s,t)\,dt \,ds\biggr)^{1/q}\biggr), $$
(2.36)
$$ W_{j}(t):= \int_{t_{j}}^{t}\frac{ds}{\tilde{\omega}^{q}_{j}(\varphi ^{-1}(s^{1/q}))}, \quad t\ge t_{j}>0, j=1,\ldots,m. $$
(2.37)
$$\begin{aligned}& r_{1}(x,y) := (1+m)^{q-1}\Bigl(\max_{(\tau,\xi)\in [x_{0}, x]\times[y_{0},y] } \bigl\{ a(\tau,\xi)\bigr\} \Bigr)^{q}, \\& r_{j}(x,y): = W_{j-1}^{-1} \biggl[W_{j-1} \bigl(r_{j-1}(x,y)\bigr)+ \int_{b_{i-1}(x_{0})}^{b_{i-1}(x)} \int _{c_{i-1}(y_{0})}^{c_{i-1}(y)} \tilde{f}_{i-1}(x,y,s,t) \,dt\,ds \biggr], \\& \quad j=2,\ldots, m, \end{aligned}$$
(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)
$$ W_{j}\bigl(r_{j}(X_{1},Y_{1}) \bigr)+ \int_{b_{j}(x_{0})}^{b_{j}(X_{1})} \int _{c_{j}(y_{0})}^{c_{j}(Y_{1})} \tilde{f}_{j}(x,y,s,t) \,dt\,ds\le \int_{t_{j}}^{\infty}\frac{ds}{\tilde {\omega}^{q}_{j}(\varphi^{-1}(s^{1/q}))} $$
(2.41)
Proof
Above all, we monotonize functions \(f_{j}\), \(\omega_{j}\), \(\mu_{j}\), g, and a in (1.4). Let
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
Moreover, because the ratios \({\tilde{\omega}_{j+1}}/{\tilde{\omega }_{j}}\) (\(j=1,\ldots,m-1\)) are all nondecreasing, we have
Let
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 g̃ is nondecreasing, we obtain
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
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 \((x,y)\in[x_{0},x_{1})\times[y_{0}, y_{1})\),
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 \(j=1,\ldots,m\).
$$ \hat{a}(x,y): = \max_{(\tau,\xi)\in[x_{0}, x]\times[y_{0},y] }\bigl\{ a(\tau,\xi) \bigr\} ,\quad (x,y)\in[x_{0},x_{1})\times[y_{0},y_{1}), $$
$$ \omega_{j}(t)\le\hat{ {\omega}}_{j}(t),\qquad \mu_{j}(t)\le\hat{\mu}_{j}(t),\qquad \hat{\omega}_{j}(t) \hat{{\mu}}_{j}\bigl(\tilde{g}(t)\bigr)\le\tilde{\omega }_{j}(t), \quad j=1,\ldots,m. $$
(2.42)
$$ \tilde{\omega}_{j}\varpropto\tilde{\omega}_{j+1}, \quad j=1,2,\ldots,m-1. $$
(2.43)
$$ \hat{f}_{j}(x,y,s,t) :=\max_{(\iota,\xi)\in[x_{0}, x ]\times[y_{0},y]}f_{j}( \iota ,\xi,s,t), $$
(2.44)
$$ \max_{\tilde{\eta}\in[s-h,s]} g\bigl(u(\xi,y)\bigr)\le\max _{\tilde{\eta}\in[s-h,s]} \tilde{g}\bigl(u(\xi,y)\bigr) \le\tilde{g}\Bigl(\max _{\tilde{\eta}\in[s-h,s]} u(\xi,y)\Bigr) $$
(2.45)
$$ \begin{aligned} &\varphi\bigl(u(x,y)\bigr)\leq \hat{a}(x,y)+\sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{\beta_{j}-1}s^{\gamma_{j}-1}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{\bar{\beta}_{j}-1} t^{\bar{\gamma}_{j}-1} \\ &\hphantom{\varphi(u(x,y))\leq{}}{}\times\hat{f}_{j}(x,y,s,t) \hat{ \omega}_{j}\bigl(u(s,t)\bigr)\hat{\mu}_{j}\Bigl(\tilde{g} \Bigl(\max_{\tilde {\eta}\in[s-h,s]}u(\tilde{\eta},t)\Bigr)\Bigr)\,dt\,ds, \quad \\ &\hphantom{\varphi(u(x,y))\leq{}}{}(x,y)\in[x_{0},x_{1}) \times[y_{0}, y_{1}), \\ &u(x,y) \leq \psi (x,y),\quad (x,y)\in\bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times [y_{0}, y_{1}). \end{aligned} $$
(2.46)
$$\begin{aligned}& \varphi\bigl(u(x,y)\bigr) \\& \quad \leq \hat{a}(x,y)+\sum _{j=1}^{m} \biggl( \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{p(\beta_{j}-1)}s^{p(\gamma_{j}-1)}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{p(\bar{\beta}_{j}-1)} t^{(\bar{\gamma}_{j}-1)}\,dt\,ds\biggr)^{1/p} \\& \qquad {}\cdot\biggl( \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\hat{f}^{q}_{j}(x,y,s,t) \hat{\omega}^{q}_{j}\bigl(u(s,t)\bigr) \Bigl(\hat{ \mu}_{j}\Bigl(\tilde{g}\Bigl(\max_{\tilde{\eta}\in[s-h,s]}u(\tilde{ \eta},t)\Bigr)\Bigr)\Bigr)^{q} \,dt\,ds\biggr)^{1/q} \\& \quad \leq \hat{a}(x,y)+\sum_{j=1}^{m} \biggl( \int _{0}^{x} \int_{0}^{y} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{p(\beta_{j}-1)}s^{p(\gamma_{j}-1)}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\beta}_{j}} \bigr)^{p(\bar{\gamma}_{j}-1)} t^{{p(\bar{\gamma}_{j}-1)}}\,dt\,ds\biggr)^{1/p} \\& \qquad {}\cdot\biggl( \int_{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\hat {f}^{q}_{j}(x,y,s,t) \hat{\omega}^{q}_{j}\bigl(u(s,t)\bigr) \Bigl(\hat{ \mu}_{j}\Bigl(\tilde{g}\Bigl(\max_{\tilde{\eta}\in[s-h,s]}u(\tilde{ \eta},t)\Bigr)\Bigr)\Bigr)^{q}dtds\biggr)^{1/q} \\& \quad \leq \hat{a}(x,y)+ \sum_{j=1}^{m} \bigl(M_{j}x^{\theta_{j}}\bar{M}_{j}y^{\bar{\theta}_{j}} \bigr)^{1/p}\biggl( \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\hat{f}^{q}_{j}(x,y,s,t) \\& \qquad {} \cdot\hat{\omega}^{q}_{j}\bigl(u(s,t)\bigr) \Bigl( \hat{\mu}_{j}\Bigl(\tilde{g}\Bigl(\max_{\tilde{\eta}\in[s-h,s]}u( \tilde{\eta},t)\Bigr)\Bigr)\Bigr)^{q} \,dt\,ds\biggr)^{1/q}, \end{aligned}$$
(2.47)
By Jensen’s inequality and (2.47), we get, for all \((x,y)\in \Delta\),
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
According to (2.49), we consider the inequalities
where \(x_{0}\leq X\le X_{1}\) and \(y_{0}\leq Y\le Y_{1}\) are chosen arbitrarily.
$$\begin{aligned} \varphi^{q}\bigl(u(x,y)\bigr) \leq& (1+m)^{q-1}\Biggl( \hat{a}^{q}(x,y)+ \sum_{j=1}^{m} \bigl(M_{j}x^{\theta_{j}}\bar{M}_{j}y^{\bar{\theta}_{j}} \bigr)^{q/p} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\hat{f}^{q}_{j}(x,y,s,t) \\ &{} \times\hat{\omega}^{q}_{j}\bigl(u(s,t)\bigr) \Bigl( \hat{\mu}_{j}\Bigl(\tilde{g}\Bigl(\max_{\tilde{\eta}\in[s-h,s]}u( \tilde{\eta},t)\Bigr)\Bigr)\Bigr)\Biggr)^{q} \,dt\,ds. \end{aligned}$$
(2.48)
$$\begin{aligned} \varphi^{q}\bigl(u(x,y)\bigr) \leq& r_{1}(x,y)+ \sum _{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\tilde{f}_{j}(x,y,s,t) \hat{\omega}^{q}_{j}\bigl(u(s,t)\bigr) \\ &{}\cdot\Bigl(\hat{\mu}_{j}\Bigl(\tilde{g}\Bigl(\max _{\tilde{\eta}\in [s-h,s]}u(\tilde{\eta},t)\Bigr)\Bigr)\Bigr)^{q}\,dt \,ds, \quad (x,y)\in\Delta. \end{aligned}$$
(2.49)
$$ \begin{aligned} &\varphi^{q}\bigl(u(x,y)\bigr) \leq r_{1}(X,Y)+\sum_{j=1}^{m} \int_{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \tilde{f}_{j}(X,Y,s,t) \\ &\hphantom{\varphi^{q}(u(x,y)) \leq{}}{}\cdot\hat{\omega}^{q}_{j}\bigl(u(s,t) \bigr) \Bigl(\hat{\mu}_{j}\Bigl(\hat{g}\Bigl(\max_{\tilde{\eta}\in[\tilde{\eta}-h,s]}u( \tilde{\eta },t)\Bigr)\Bigr)\Bigr)^{q} \,dt\,ds, \\ &\hphantom{\varphi^{q}(u(x,y)) \leq{}}{} (x,y) \in[x_{0},X]\times[y_{0},Y], \\ &u(x,y) \leq \psi(x,y), \quad (x,y)\in \bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times[y_{0},Y], \end{aligned} $$
(2.50)
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
Clearly, \(z(x,y)\) is increasing in x. By the definition of \(z(x,y)\) and (2.50), we have
Since \(\varphi(t)\) is strictly increasing and \(z(x,y)\) is nondecreasing, from (2.51) we get, for \((s,y)\in[b_{*}(x_{0}), X]\times[y_{0},Y]\),
From the definition of \(z(x,y)\), (2.42), (2.51), and (2.52), it follows that
where \(\vartheta_{j}(t)\equiv1\), \(t\ge0\).
$$ z(x,y)= \textstyle\begin{cases} r_{1}(X,Y)+\sum_{j=1}^{m} \int_{b_{j}(x_{0})}^{b_{j}(x)}\int _{c_{j}(y_{0})}^{c_{j}(y)}\tilde{f}_{j}(X,Y,s,t) \hat{\omega}^{q}_{j}(u(s,t)) \\ \quad {}\times (\hat{\mu}_{j}(\hat{g}(\max_{\tilde{\eta}\in [s-h,s]}u(\tilde{\eta},t))))^{q}\,dt\,ds, \quad (x,y)\in[x_{0},X]\times [y_{0},Y], \\ r_{1}(X,Y), \quad (x,y)\in[b_{*}(x_{0})-h,x_{0}]\times [y_{0}, Y]. \end{cases} $$
$$ u(x,y)\leq\varphi^{-1}\bigl(z^{1/q}(x,y) \bigr), \quad (x,y)\in\bigl[b_{*}(x_{0})-h, X\bigr]\times[y_{0},Y]. $$
(2.51)
$$ \max_{\xi\in[s-h, s]} u(\xi,y) \leq \max _{\xi\in[s-h, s]} \varphi^{-1}\bigl(z^{1/q}(\xi,y) \bigr) \leq \varphi^{-1}\bigl(z^{1/q}(s,y)\bigr). $$
(2.52)
$$\begin{aligned}& z(x,y) \leq r_{1}(X,Y)+\sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{b_{j}(y_{0})}^{b_{j}(y)} \tilde{f}_{j}(X,Y,s,t) \hat{\omega}^{q}_{j}\bigl(\varphi^{-1} \bigl(z^{1/q}(s,t)\bigr)\bigr) \\& \hphantom{z(x,y) \leq{}} {}\cdot\Bigl(\hat{\mu}_{j}\Bigl(\hat{g}\Bigl(\max _{\tilde{\eta}\in[s-h, s]} \varphi^{-1}\bigl(z^{1/q}(\tilde{ \eta},t)\bigr)\Bigr)\Bigr)\Bigr)^{q}\,dt\,ds \\& \hphantom{z(x,y) } \leq r_{1}(X,Y)+\sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{b_{j}(y_{0})}^{b_{j}(y)} \tilde{f}_{j}(X,Y,s,t) \hat{\omega}^{q}_{j}\bigl(\varphi^{-1} \bigl(z^{1/q}(s,t)\bigr)\bigr) \\& \hphantom{z(x,y) \leq{}} {}\cdot\bigl(\hat{\mu}_{j}\bigl(\tilde{g}\bigl( \varphi^{-1}\bigl(z^{1/q}(s,t)\bigr)\bigr)\bigr) \bigr)^{q}\,dt\,ds \\& \hphantom{z(x,y) } \leq r_{1}(X,Y)+\sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{b_{j}(y_{0})}^{b_{j}(y)} \tilde{f}_{j}(X,Y,s,t) \tilde{\omega}^{q}_{j}\bigl(\varphi^{-1} \bigl(z^{1/q}(s,t)\bigr)\bigr) \\& \hphantom{z(x,y) \leq{}} {} \cdot\vartheta_{j}\Bigl(\max_{\tilde{\eta}\in[s-h, s]} u(\tilde {\eta},t)\Bigr)\,dt\,ds, \quad (x,y)\in[x_{0}, X] \times[y_{0},Y], \\& z(x,y) \leq r_{1}(X,Y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times[y_{0}, Y], \end{aligned}$$
(2.53)
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
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
for \(j=2,\ldots, m\), and \(\bar{X}_{1}\), \(\bar{Y}_{1}\) are chosen such that
for \(j=1,\ldots,m\).
$$\begin{aligned} z(x,y) \le& W_{m}^{-1}\biggl[W_{n}\bigl(\tilde {r}_{m}(X,Y,x,y)\bigr) \\ &{} + \int_{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} \tilde{ f}_{m}(X,Y,s,t) \,dt\,ds\biggr] \end{aligned}$$
(2.54)
$$ \tilde{r}_{j}(X,Y,x,y):= W_{i-1}^{-1} \biggl(W_{i-1}\bigl(\tilde {r}_{i-1}(X,Y,x,y)\bigr)+ \int_{b_{i-1}(x_{0})}^{b_{i-1}(x)} \int _{c_{i-1}(y_{0})}^{c_{i-1}(y)} \tilde{f}_{i-1}(X,Y,s,t) \,dt\,ds\biggr) $$
$$\begin{aligned}& W_{j}\bigl(\tilde{r}_{j}(X,Y,\bar{X}_{1}, \bar{Y}_{1})\bigr)+ \int _{b_{j}(x_{0})}^{b_{j}(X_{2})} \int_{c_{j}(y_{0})}^{c_{j}(\bar{Y}_{1})} \tilde{f}_{j}(X,Y,s,t) \\& \quad \le \int_{t_{j}}^{\infty}\frac{ds}{\tilde{\omega}^{q}_{j}(\varphi ^{-1}(s^{1/q}))} \end{aligned}$$
(2.55)
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),
for \(x_{0}\le x \le\min\{X, X_{2}\}\), \(y_{0}\le y \le\min\{Y, Y_{2}\}\).
$$\begin{aligned} u(x,y) \le& \varphi^{-1}\biggl(W_{m}^{-1} \biggl(W_{n}\bigl(\tilde {r}_{m}(X,Y,x,y)\bigr) \\ &{}+ \int_{\alpha_{m}(x_{0})}^{\alpha_{m}(x)} \int_{\beta _{m}(y_{0})}^{\beta_{m}(y)} \tilde{ f}_{m}(X,Y,s,t) \,dt\,ds\biggr)\biggr) \end{aligned}$$
(2.56)
Taking \(x=X\), \(y=Y\) in (2.56), we have
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
Since X, Y are arbitrary, replacing Y and X with y and x, respectively, we have
for all \((x,y)\in[x_{0}, X^{*}_{1}]\times[y_{0},Y^{*}_{1}]\).
$$\begin{aligned} u(X,Y) \le& \varphi^{-1}\biggl(W_{m}^{-1} \biggl(W_{n}\bigl(\tilde {r}_{m}(X,Y,X,Y)\bigr) \\ &{}+ \int_{b_{m}(x_{0})}^{b_{m}(X)} \int_{c_{m}(y_{0})}^{c_{m}(Y)} \tilde{ f}_{m}(X,Y,s,t) \,dt\,ds\biggr)\biggr) \end{aligned}$$
(2.57)
$$\begin{aligned} u(X,Y) \le& \varphi^{-1}\biggl(W_{m}^{-1} \biggl(W_{n}\bigl(r_{m}(X,Y)\bigr) \\ &{}+ \int_{b_{m}(x_{0})}^{b_{m}(X)} \int_{c_{m}(y_{0})}^{c_{m}(Y)} \tilde{ f}_{m}(X,Y,s,t) \,dt\,ds\biggr)\biggr). \end{aligned}$$
(2.58)
$$\begin{aligned} u(x,y) \le& \varphi^{-1}\biggl(W_{m}^{-1} \biggl(W_{n}\bigl(r_{m}(x,y)\bigr) \\ &{}+ \int_{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} \tilde{ f}_{m}(x,y,s,t) \,dt\,ds\biggr)\biggr) \end{aligned}$$
(2.59)
This completes the proof. □
Theorem 2.2
We make the following assumptions:
If
\(u\in C([b_{*}(x_{0})-h,x_{1})\times[y_{0},y_{1}),\mathbb {R}_{+})\)
satisfies the integral inequality
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
\(y\in[y_{0},y_{1})\).
(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
\(j=1,\ldots,m\).
$$\begin{aligned}& u^{k}(x,y)\leq c(x,y)+\sum_{j=1}^{M} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{\beta_{j}-1}s^{\gamma_{j}-1}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{\bar{\beta}_{j}-1} \\& \hphantom{u^{k}(x,y)\leq{}}{} \times t^{\bar{\gamma}_{j}-1} \hat{g}_{j}(x,y,s,t) \phi_{j}\bigl(u(s,t)\bigr)\hat{\phi}_{j} \Bigl(\max _{\tilde{\eta}\in [s-h, s]}g\bigl(u(\tilde{\eta},t)\bigr) \Bigr) \\& \hphantom{u^{k}(x,y)\leq{}}{} +\sum_{j=M+1}^{m} \int_{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)}\hat{g}_{j}(x,y,s,t) \phi_{j}\bigl(u(s,t)\bigr)\hat{\phi }_{j} \Bigl(\max _{\tilde{\eta}\in[s-h, s]}u(\tilde{\eta },t) \Bigr), \\& \hphantom{u^{k}(x,y)\leq{}}{}(x,y)\in[x_{0},x_{1})\times [y_{0}, y_{1}), \\& u(x,y) \leq \hat{ \psi }(x,y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times [y_{0}, y_{1}), \end{aligned}$$
(2.60)
Then
for all
\((x,y)\in[x_{0}, X_{2})\times[y_{0},Y_{2})\), where
\(G_{j}^{-1}\)is the inverse of the function
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
\(X_{2}\in[x_{0}, x_{1})\), \(Y_{2}\in[y_{0}, y_{1})\)
are chosen such that
for
\(j=1,2,\ldots,m\).
$$ u(x,y) \leq \biggl(G_{m}^{-1} \bigl(G_{m}\bigl(e_{m}(x,y)\bigr)\bigr) + \int_{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} \tilde{g}_{m}(x,y,s,t)\,dt \,ds\biggr)^{1/(kq)} $$
(2.61)
$$ G_{j}(u):= \int_{t_{j}}^{t}\frac{ds}{\phi^{q}_{j}(s^{1/(kq)})\hat{\phi }^{q}_{j}(s^{1/(kq)})}, \quad t\ge t_{j}>0, j=1,\ldots,m. $$
(2.62)
$$\begin{aligned}& \begin{aligned}[b] &e_{1}(x,y)=(1+m)^{q-1}\Bigl( \max_{(\iota,\xi)\in[x_{0}, x ]\times [y_{0},y]}c(\iota,\xi)\Bigr)^{q},\quad \textit{and} \\ &e_{j+1}(x,y):=G_{j}^{-1}\biggl[G_{j} \bigl(e_{j}(x,y)\bigr)+ \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \tilde{g}_{j}(x,y,s,t) \,dt\,ds\biggr], \\ &\quad j=1,\ldots, m-1, \end{aligned} \end{aligned}$$
(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)
$$\begin{aligned}& G_{j}\bigl(r_{j}(X_{2},Y_{2})\bigr)+ \int_{b_{j}(x_{0})}^{b_{j}(X_{1})} \int _{c_{j}(y_{0})}^{c_{j}(Y_{2})} \tilde{g}_{j}(X_{2},Y_{2},s,t) \,dt\,ds \\& \quad \le \int_{t_{j}}^{\infty}\frac{ds}{\phi^{q}_{j}(s^{1/q})\hat{\phi }^{q}_{j}(s^{1/q})} \end{aligned}$$
(2.66)
Proof
Let
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
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 \((x,y)\in\Delta\),
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 \(j=1,\ldots,m\).
$$ \begin{aligned} &\hat{c}(x,y):=\max_{(\tau,\xi)\in[x_{0}, x]\times[y_{0},y] }\bigl\{ a(\tau,\xi)\bigr\} , \quad (x,y)\in[x_{0},x_{1}) \times[y_{0},y_{1}). \\ &\bar{g}_{j}(x,y,s,t) :=\max_{(\iota,\xi)\in[x_{0}, x ]\times[y_{0},y]}g_{j}( \iota ,\xi,s,t), \end{aligned} $$
(2.67)
$$\begin{aligned}& u^{k}(x,y) \leq \hat{c}(x,y)+\sum_{j=1}^{M} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{\beta_{j}-1}s^{\gamma_{j}-1}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{\bar{\beta}_{j}-1} \\& \hphantom{u^{k}(x,y) \leq{}}{} \times t^{\bar{\gamma}_{j}-1} \bar{g}_{j}(x,y,s,t) \phi_{j}\bigl(u(s,t)\bigr)\hat{\phi}_{j}\Bigl(\max _{\tilde{\eta}\in[s-h, s]}u(\tilde{\eta},t)\Bigr) \\& \hphantom{u^{k}(x,y) \leq{}}{} +\sum_{j=M+1}^{m} \int_{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)}\bar{g}_{j}(x,y,s,t) \phi_{j}\bigl(u(s,t)\bigr)\hat{\phi}_{j}\Bigl(\max _{\tilde{\eta}\in[s-h, s]}u(\tilde{\eta},t)\Bigr), \\& \hphantom{u^{k}(x,y) \leq{}}{} (x,y)\in[x_{0},x_{1})\times [y_{0}, y_{1}), \\& u(x,y) \leq \hat{ \psi }(x,y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times [y_{0}, y_{1}). \end{aligned}$$
(2.68)
$$\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)
By Jensen’s inequality and (2.69), we get, for all \((x,y)\in \Delta\),
By the definition of \(e_{1}\) and \(\tilde{g}_{j}\), from (2.70) we obtain
Concerning (2.71), we consider the auxiliary inequalities
where \(x_{0}\leq X\le X_{2}\) and \(y_{0}\leq Y\le Y_{2}\) are chosen arbitrarily.
$$\begin{aligned} u^{kq}(x,y) \leq& (1+m)^{q-1}( \hat{c}^{q}(x,y)+ \sum_{j=1}^{m}\bigl(M_{j}x^{\theta_{j}} \bar{M}_{j}y^{\bar{\theta}_{j}}\bigr)^{q/p} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\bar{g}^{q}_{j}(x,y,s,t) \\ &{} \times\phi^{q}_{j}\bigl(u(s,t)\bigr) \hat{ \phi}^{q}_{j}\bigl(\max_{\tilde{\eta }\in[s-h,s]}u(\tilde{\eta},t)\bigr) \,dt\,ds. \end{aligned}$$
(2.70)
$$\begin{aligned} u^{kq}(x,y) \leq& e_{1}(x,y)+ \sum _{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\tilde{g}_{j}(x,y,s,t) \\ &{}\times\phi^{q}_{j}\bigl(u(s,t)\bigr) \hat{ \phi}^{q}_{j}\bigl(\max_{\tilde{\eta }\in[s-h,s]}u(\tilde{\eta},t)\bigr) \,dt\,ds,\quad (x,y)\in\Delta. \end{aligned}$$
(2.71)
$$\begin{aligned}& u^{kq}(x,y)\leq e_{1}(X,Y)+ \sum _{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)}\tilde{g}_{j}(X,Y,s,t) \\& \hphantom{u^{kq}(x,y)\leq{}}{}\times\phi^{q}_{j}\bigl(u(s,t)\bigr) \hat{\phi}^{q}_{j}\bigl(\max_{\tilde{\eta }\in[s-h,s]}u(\tilde{ \eta},t)\bigr) \,dt\,ds, \quad (x,y)\in[x_{0},X] \times[y_{0},Y], \\& u(x,y)\leq \hat{\psi}(x,y), \quad (x,y)\in \bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times[y_{0},Y], \end{aligned}$$
(2.72)
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
Clearly, \(z(x,y)\) is increasing in x. From (2.72) and the definition of z, we have
Then, noting that z is increasing, from (2.51) we get for \((s,y)\in[b_{*}(x_{0}), X]\times[y_{0},Y]\)
From (2.42), (2.73), (2.74), and the definition of z, we have
$$ z(x,y)= \textstyle\begin{cases} e_{1}(X,Y)+ \sum_{j=1}^{m}\int_{b_{j}(x_{0})}^{b_{j}(x)}\int _{c_{j}(y_{0})}^{c_{j}(y)}\tilde{g}_{j}(X,Y,s,t) \\ \quad {}\times\phi^{q}_{j}(u(s,t))\hat{\phi}^{q}_{j}\bigl(\max_{\tilde{\eta }\in[s-h,s]}u(\tilde{\eta},t)\bigr) \,dt\,ds, \quad (x,y)\in[x_{0},X]\times [y_{0},Y], \\ e_{1}(X,Y), \quad (x,y)\in[b_{*}(x_{0})-h,x_{0}]\times [y_{0}, Y]. \end{cases} $$
$$ u(x,y)\leq z^{1/(kq)}(x,y),\quad (x,y)\in \bigl[b_{*}(x_{0})-h, X\bigr]\times[y_{0},Y]. $$
(2.73)
$$ \max_{\tilde{\eta}\in[s-h, s]} u(\tilde{\eta},y) \leq\max _{\tilde{\eta}\in[s-h, s]} z^{1/(kq)}(\tilde{\eta},y)\le\bigl( \max_{\tilde{\eta}\in[s-h, s]} z(\tilde{\eta},y)\bigr)^{1/(kq)}. $$
(2.74)
$$\begin{aligned}& z(x,y)\leq e_{1}(X,Y)+\sum_{j=1}^{m} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{b_{j}(y_{0})}^{b_{j}(y)} \tilde{g}_{j}(X,Y,s,t) \phi^{q}_{j}\bigl(z^{1/(kq)}(s,t)\bigr) \\& \hphantom{z(x,y)\leq{}}{}\times\hat{\phi}^{q}_{j} \Bigl(\max _{\tilde{\eta}\in[s-h, s]} \bigl(z^{1/(kq)}(\tilde{\eta},t)\bigr) \Bigr)\,dt\,ds \\& \hphantom{z(x,y)}\leq e_{1}(X,Y)+\sum_{j=1}^{m} \int_{b_{j}(x_{0})}^{b_{j}(x)} \int _{b_{j}(y_{0})}^{b_{j}(y)} \tilde{g}_{j}(X,Y,s,t) \phi^{q}_{j}\bigl(z^{1/(kq)}(s,t)\bigr) \\& \hphantom{z(x,y)\leq{}}{}\times\hat{\phi}^{q}_{j} \Bigl(\bigl(\max _{\tilde{\eta}\in[s-h, s]} z(\tilde{\eta},t)\bigr)^{1/(kq)}\Bigr) \,dt\,ds, \quad (x,y)\in[x_{0}, X] \times[y_{0},Y], \\& z(x,y) \leq e_{1}(X,Y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times[y_{0}, Y]. \end{aligned}$$
(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
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
for \(j=2,\ldots, m\), and \(X^{*}_{2}\), \(Y^{*}_{2}\) are chosen such that
for \(j=1,\ldots,m\).
$$\begin{aligned} z(x,y) \le& G_{m}^{-1}\biggl(G_{m} \bigl(\tilde{e}_{m}(X,Y,x,y)\bigr) \\ &{} + \int_{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} \tilde{ g}_{m}(X,Y,s,t) \,dt\,ds\biggr) \end{aligned}$$
(2.76)
$$ \tilde{e}_{j}(X,Y,x,y):= G_{j-1}^{-1} \biggl(G_{j-1}\bigl(\tilde {e}_{j-1}(X,Y,x,y)\bigr)+ \int_{b_{j-1}(x_{0})}^{b_{j-1}(x)} \int _{c_{j-1}(y_{0})}^{c_{j-1}(y)} \tilde{g}_{j-1}(X,Y,s,t) \,dt\,ds\biggr) $$
$$\begin{aligned}& G_{j}\bigl(\tilde{e}_{j}(X,Y,\bar{X}_{1}, \bar{Y}_{1})\bigr)+ \int _{b_{j}(x_{0})}^{b_{j}(X_{2})} \int_{c_{j}(y_{0})}^{c_{j}(\bar{Y}_{1})} \tilde{g}_{j}(X,Y,s,t) \\& \quad \le \int_{t_{j}}^{\infty}\frac{ds}{\tilde{\omega}^{q}_{j}(\varphi ^{-1}(s^{1/q}))} \end{aligned}$$
(2.77)
Note that \(X^{*}_{2}=X_{2}\) and \(Y^{*}_{2}=Y_{2}\). It follows from (2.73) and (2.76) that
for \(x_{0}\le x \le\min\{X, X^{*}_{2}\}\), \(y_{0}\le y \le\min\{Y, Y^{*}_{2}\}\).
$$\begin{aligned} u(x,y) \le& \biggl(G_{m}^{-1}\biggl(G_{n}\bigl( \tilde{g}_{m}(X,Y,x,y)\bigr) \\ &{}+ \int_{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} \tilde{ g}_{m}(X,Y,s,t) \,dt\,ds\biggr)\biggr)^{1/(kq)} \end{aligned}$$
(2.78)
Taking \(x=X\), \(y=Y\) in (2.56), we have
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
Since \(X,Y\) are arbitrary, replacing X and Y with x and y, respectively, we get
for all \((x,y)\in[x_{0}, X_{2}]\times[y_{0},Y_{2}]\). This completes the proof. □
$$\begin{aligned} u(X,Y) \le& \biggl(G_{m}^{-1}\biggl(G_{m}\bigl( \tilde{e}_{m}(X,Y,X,Y)\bigr) \\ &{}+ \int_{b_{m}(x_{0})}^{b_{m}(X)} \int_{c_{m}(y_{0})}^{c_{m}(Y)} \tilde{ g}_{m}(X,Y,s,t) \,dt\,ds\biggr)\biggr)^{1/(kq)} \end{aligned}$$
(2.79)
$$\begin{aligned} u(X,Y) \le& \biggl(G_{m}^{-1}\biggl(G_{n} \bigl(r_{m}(X,Y)\bigr) \\ &{}+ \int_{b_{m}(x_{0})}^{b_{m}(X)} \int_{c_{m}(y_{0})}^{c_{m}(Y)} \tilde{ g}_{m}(X,Y,s,t) \,dt\,ds\biggr)\biggr)^{1/(kq)}. \end{aligned}$$
(2.80)
$$\begin{aligned} u(x,y) \le& \biggl(G_{m}^{-1}\biggl(G_{n} \bigl(e_{m}(x,y)\bigr) \\ &{}+ \int_{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} \tilde{ g}_{m}(x,y,s,t) \,dt\,ds\biggr)\biggr)^{1/(kq)} \end{aligned}$$
(2.81)
Corollary 2.3
Let the following conditions be fulfilled:
Then
for all
\((x,y)\in[x_{0}, X_{2})\times[y_{0},Y_{2})\), where
\(G_{j}^{-1}\)
is the inverse of the function
\(t_{j}\)
is a given constant, \(r_{j}(x,y)\)
is defined recursively by
\(M_{j}:=\alpha_{j}^{-1}B(\frac{p(\gamma_{j}-1)+1}{\alpha_{j}}, p(\beta_{j}-1)+1)\), \(\bar{M}_{j}:=\bar{\alpha}_{j}^{-1}B(\frac{p(\bar {\gamma}_{j}-1)+1}{\bar{\alpha}_{j}}, p(\beta_{j}-1)+1)\), \(\theta_{j}:=p(\alpha_{j}(\beta_{j}-1)+\gamma _{j}-1)+1\), \(\bar{\theta}_{j}:=p(\bar{\alpha}_{j}(\bar{\beta }_{j}-1)+\bar{\gamma}_{j}-1)+1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(X_{2}\in[x_{0}, x_{1})\), \(Y_{2}\in[y_{0}, y_{1})\)
are chosen such that
for
\(j=1,2,\ldots,m\).
(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
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
\(y\in[y_{0},y_{1})\).
$$ \begin{aligned} &\varphi_{1}\bigl(u(x,y) \bigr) \leq a(x,y)+\sum_{j=1}^{M} \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{\beta_{j}-1}s^{\gamma_{j}-1}\bigl(y^{\bar {\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{\bar{\beta}_{j}-1} \\ &\hphantom{\varphi_{1}(u(x,y)) \leq{}}{} \times t^{\bar{\gamma}_{j}-1} f_{j}(x,y,s,t) \psi_{j}\bigl(u(s,t)\bigr)\,dt\,ds \\ &\hphantom{\varphi_{1}(u(x,y)) \leq{}}{} +\sum_{j=M+1}^{m} \int_{b_{j}(x_{0})}^{b_{j}(x)} \int _{c_{j}(y_{0})}^{c_{j}(y)}\bigl(x^{\alpha_{j}}-s^{\alpha_{j}} \bigr)^{\beta_{j}-1}s^{\gamma_{j}-1} \bigl(y^{\bar{\alpha}_{j}}-t^{\bar{\alpha}_{j}} \bigr)^{\bar{\beta}_{j}-1} \\ &\hphantom{\varphi_{1}(u(x,y)) \leq{}}{} \times t^{\bar{\gamma}_{j}-1}f_{j}(x,y,s,t) \psi_{j} \Bigl(\max_{\tilde {\eta}\in[s-h,s]}u(\tilde{\eta},t) \Bigr) \,dt\,ds, \\ &\hphantom{\varphi_{1}(u(x,y)) \leq{}}{} (x,y)\in[x_{0},x_{1})\times [y_{0}, y_{1}), \\ &u(x,y) \leq \hat{\psi }(x,y), \quad (x,y)\in\bigl[b_{*}(x_{0})-h,x_{0} \bigr]\times [y_{0}, y_{1}), \end{aligned} $$
(2.82)
$$\begin{aligned} u(x,y) \leq& \varphi_{1}^{-1}\biggl(\check{G}_{m}^{-1} \biggl(\check{G}_{m}\bigl(r_{m}(x,y)\bigr) \\ &{} + \int_{b_{m}(x_{0})}^{b_{m}(x)} \int_{c_{m}(y_{0})}^{c_{m}(y)} \tilde{f}_{m}(x,y,s,t)\,dt \,ds\biggr)^{1/q}\biggr) \end{aligned}$$
(2.83)
$$ \check{G}_{j}(t):= \int_{t_{j}}^{t}\frac{ds}{\psi^{q}_{j}(\varphi _{1}^{-1}(s^{1/q}))},\quad t\ge t_{j}>0, j=1,2,\ldots,m, $$
(2.84)
$$\begin{aligned}& r_{1}(x,y)=(1+m)^{q-1}\Bigl(\max_{(\iota,\xi)\in[x_{0}, x ]\times [y_{0},y]}a( \iota,\xi)\Bigr)^{q},\quad \textit{and} \\& r_{j+1}(x,y):= \check{G}_{j}^{-1}\biggl[ \check{G}_{j}\bigl(r_{j}(x,y)\bigr)+ \int _{b_{j}(x_{0})}^{b_{j}(x)} \int_{c_{j}(y_{0})}^{c_{j}(y)} \tilde{f}_{j}(x,y,s,t) \,dt\,ds\biggr], \\& \quad j=1,\ldots, m-1, \end{aligned}$$
(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)
$$ \check{G}_{j}\bigl(r_{j}(X_{2},Y_{2}) \bigr)+ \int_{b_{j}(x_{0})}^{b_{j}(X_{1})} \int _{c_{j}(y_{0})}^{c_{i-1}(Y_{2})} \tilde{f}_{j}(X_{2},Y_{2},s,t) \,dt\,ds\le \int_{t_{j}}^{\infty}\frac {ds}{\tilde{\omega}^{q}_{j}(\varphi^{-1}(s^{1/q}))} $$
(2.87)
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
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 \(p(\theta _{j}-1)+1>0\), \(p>1\), \(j=1,2\).
$$ \textstyle\begin{cases} z(x,y)=a(x,y)+\int_{x_{0}}^{x}\int_{y_{0}}^{y}(x-s)^{\theta _{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1} \\ \hphantom{z(x,y)={}}{}\times F(x,y, s,t,z(s,t),\max_{ \tilde{\eta}\in [s-h,s]}z( \tilde{\eta},t))\, ds\, dt, \quad (x,y)\in\Delta, \\ z(x,y)=\psi(x,y), \quad (x,y)\in[x_{0}-h,x_{0}]\times[y_{0}, y_{1}), \end{cases} $$
(3.1)
The following result gives an estimate for its solutions.
Corollary 3.1
Suppose that functions
F
in (3.1) satisfy
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
\(y\in[y_{0}, y_{1})\).
$$ \bigl\vert F(x,y,s,t,u,v) \bigr\vert \le h_{1}(x,y,s,t) \mu_{1}\bigl( \vert u \vert \bigr)+h_{2}(x,y,s,t) \mu_{2}\bigl( \vert v \vert \bigr), $$
(3.2)
Then any solution
\(z(x,y)\)
of (3.1) has the estimate
for all
\((x,y)\in[x_{0},X_{1})\times[y_{0},Y_{1})\), where
\(M_{j}:=B(p(\gamma_{j}-1)+1, p(\theta_{j}-1)+1)\) (\(j=1,2\)), \(\delta _{j}:=p(\theta_{j}+\gamma_{j}-2)+1\), \(j=1,2\), \(\frac{1}{p}+\frac{1}{q}=1\), 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
$$\begin{aligned}& \bigl\vert z(x,y) \bigr\vert \\& \quad \le \biggl[ {Q_{2}}^{-1} \biggl(Q_{2}\bigl(\gamma(x,y)\bigr)+3^{q-1} \bigl(M_{1}x^{\delta _{1}}M_{2}y^{\delta_{2}} \bigr)^{q/p} \int_{x_{0}}^{x} \int_{y_{0}}^{y}h_{2}(x,y,s,t)dt \,ds \biggr) \biggr]^{1/q} \end{aligned}$$
(3.3)
$$\begin{aligned}& \gamma(x,y) := Q_{1}^{-1} \biggl(Q_{1}\bigl( \eta _{1}(x,y)\bigr)+3^{q-1}\bigl(M_{1}x^{\delta_{1}}M_{2}y^{\delta_{2}} \bigr)^{q/p} \int _{x_{0}}^{x} \int_{y_{0}}^{y}h^{q}_{1}(x,y,s,t) \,dt\,ds \biggr), \\ & \eta_{1}(x,y) := 3^{q-1}\Bigl(\max_{(s,t)\in [x_{0},x]\times[y_{0},y]} \bigl\vert a(s,t) \bigr\vert \Bigr)^{q},\qquad Q_{1}(u):= \int_{u_{1}}^{u}\frac{ds}{\mu_{1}^{q}(s^{\frac{1}{q}})}, \quad u\ge u_{1}>0, \\ & Q_{2}(u) := \int_{u_{1}}^{u}\frac{ds}{\mu_{2}^{q}(s^{\frac{1}{q}})},\quad u\ge u_{2}>0, \end{aligned}$$
$$\begin{aligned}& Q_{1}\bigl(\gamma_{1}(X_{1},Y_{1}) \bigr)+3^{q-1}\bigl(M_{1}X_{1}^{\delta_{1}}M_{2}Y_{1}^{\delta _{2}} \bigr)^{q/p} \int_{x_{0}}^{X_{1}} \int_{y_{0}}^{Y_{1}}h^{q}_{1}(X_{1},Y_{1},s,t) \,dt\,ds \le \int_{u_{1}}^{\infty}\frac{ds}{\mu_{1}^{q}(s^{\frac{1}{q}})}, \\& Q_{2}\bigl(\gamma_{2}(X_{1},Y_{1}) \bigr)+3^{q-1}\bigl(M_{1}X_{1}^{\delta_{1}}M_{2}Y_{1}^{\delta _{2}} \bigr)^{q/p} \int_{x_{0}}^{x} \int_{y_{0}}^{y}h^{q}_{2}(X_{1},Y_{1},s,t) \,dt\,ds \le \int_{u_{2}}^{\infty}\frac{ds}{\mu_{2}^{q}(s^{\frac{1}{q}})}. \end{aligned}$$
Proof
From (3.1) we obtain
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
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). □
$$ \begin{aligned} & \bigl\vert z(x,y) \bigr\vert \le \bigl\vert a(x,y) \bigr\vert + \int_{x_{0}}^{x} \int _{y_{0}}^{y}(x-s)^{\theta_{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta _{2}-1}t^{\gamma_{2}-1} \\ &\hphantom{ \bigl\vert z(x,y) \bigr\vert \le{}}{}\cdot \Bigl\vert F\Bigl(x,y, s,t,z(s,t),\max _{ \tilde{\eta}\in[s-h,s]}z( \tilde{\eta},t)\Bigr) \Bigr\vert \,dt\,ds \\ &\hphantom{ \bigl\vert z(x,y) \bigr\vert }\le \bigl\vert a(x,y) \bigr\vert + \int_{x_{0}}^{x} \int_{y_{0}}^{y}(x-s)^{\theta _{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1} \\ &\hphantom{ \bigl\vert z(x,y) \bigr\vert \le{}}{}\cdot h_{1}(x,y,s,t) \mu _{1}\bigl( \bigl\vert z(s,t) \bigr\vert \bigr)\,dt\,ds \\ &\hphantom{ \bigl\vert z(x,y) \bigr\vert \le{}}{}+ \int_{x_{0}}^{x} \int_{y_{0}}^{y}(x-s)^{\theta_{1}-1}s^{\gamma _{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1} h_{2}(x,y,s,t) \\ &\hphantom{ \bigl\vert z(x,y) \bigr\vert \le{}}{}\cdot\mu_{2}\Bigl( \Bigl\vert \max _{ \tilde{\eta}\in[s-h,s]}z( \tilde {\eta},t)\Bigr) \Bigr\vert )\,dt\,ds,\quad (x,y)\in\Delta, \\ & \bigl\vert z(x,y) \bigr\vert \le \bigl\vert \psi(x,y) \bigr\vert , \quad (x,y)\in [x_{0}-h,x_{0}]\times[y_{0},y_{1}). \end{aligned} $$
(3.4)
$$\begin{aligned}& v(x,y) \le \bigl\vert a(x,y) \bigr\vert + \int_{x_{0}}^{x} \int _{y_{0}}^{y}(x-s)^{\theta_{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma _{2}-1} \\& \hphantom{v(x,y) \le{}}{} \cdot h_{1}(x,y,s,t) \mu_{1}\bigl(v(s,t)\bigr)\,dt\,ds \\& \hphantom{v(x,y) \le{}}{}+ \int_{x_{0}}^{x} \int_{y_{0}}^{y}(x-s)^{\theta_{1}-1}s^{\gamma _{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1} h_{2}(x,y,s,t) \\& \hphantom{v(x,y) \le{}}{}\cdot\mu_{2}\Bigl(\max_{ \tilde{\eta}\in[s-h,s]} \bigl\vert v( \tilde {\eta},t) \bigr\vert \Bigr)\,dt\,ds,\quad (x,y)\in\Delta, \\& v(x,y) \le \bigl\vert \psi(x,y) \bigr\vert ,\quad (x,y) \in[x_{0}-h,x_{0}]\times[y_{0},y_{1}). \end{aligned}$$
(3.5)
Corollary 3.2
Suppose that functions
F
and
ψ
in (3.1) satisfy
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.
$$ \bigl\vert F(x,y,s_{1},t_{1})-F(x,y,s_{2},t_{2}) \bigr\vert \leq h_{1}(x,y) \vert s_{1}-s_{2} \vert +h_{2}(x,y) \vert t_{1}-t_{2} \vert $$
(3.6)
Proof
Assume that equation (3.1) has two solutions \(u(x,y)\), \(v(x,y)\). By the equivalent integral equation (3.1), we have
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
It follows from (3.7) and (3.8) that
Let
From (3.7) we obtain
Let \(\varepsilon>0\) be an arbitrary number. Then from (3.10) we have
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 \((x,y)\in\Delta\),
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. □
$$\begin{aligned} \bigl\vert u(x,y)-v(x,y) \bigr\vert \le& \int_{x_{0}}^{x} \int _{y_{0}}^{y}(x-s)^{\theta_{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma _{2}-1} h_{1}(s,t) \bigl\vert u(s,t)-v(s,t) \bigr\vert \,dt\,ds \\ &{}+ \int_{x_{0}}^{x} \int_{y_{0}}^{y} (x-s)^{\theta _{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1}h_{2}(s,t) \\ &{} \cdot \Bigl\vert \max_{ \tilde{\eta}\in[s-h,s]}u( \tilde{\eta},t)-\max _{ \tilde{\eta}\in[s-h,s]}v( \tilde{\eta},t) \Bigr\vert \,dt\,ds \end{aligned}$$
(3.7)
$$\begin{aligned} \Bigl\vert \max_{ \tilde{\eta}\in[s-h,s]}u( \tilde{\eta},t)-\max _{ \tilde{\eta}\in[s-h,s]}v( \tilde{\eta},t) \Bigr\vert =& \Bigl\vert u( \tau,t)-\max_{ \tilde{\eta}\in[s-h,s]}v( \tilde{\eta},t) \Bigr\vert \\ \le& \bigl\vert u(\tau,t)-v(\tau,t) \bigr\vert \le\max_{ \tilde{\eta}\in [s-h,s]} \bigl\vert u( \tilde{\eta},t)-v( \tilde{\eta},t) \bigr\vert . \end{aligned}$$
(3.8)
$$\begin{aligned} \bigl\vert u(x,y)-v(x,y) \bigr\vert \le& \int_{x_{0}}^{x} \int_{y_{0}}^{y} (x-s)^{\theta _{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma _{2}-1}h_{1}(s,t) \bigl\vert u(s,t)-v(s,t) \bigr\vert \,dt\,ds \\ &{}+ \int_{x_{0}}^{x} \int_{y_{0}}^{y} (x-s)^{\theta _{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1}h_{2}(s,t) \\ &{} \cdot\max_{ \tilde{\eta}\in[s-h,s]} \bigl\vert u( \tilde{\eta },t)-v( \tilde{\eta},t) \bigr\vert \,dt\,ds. \end{aligned}$$
(3.9)
$$\phi(x,y):= \bigl\vert u(x,y)-v(x,y) \bigr\vert , \quad (x,y)\in\bigl[ \alpha(x_{0})-h, x_{0}\bigr]\times [y_{0}, y_{1}). $$
$$\begin{aligned}& \phi(x,y) \le \int_{x_{0}}^{x} \int_{y_{0}}^{y} (x-s)^{\theta _{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1}h_{1}(s,t) \phi(s,t)\,dt\,ds \\& \hphantom{\phi(x,y) \le{}}{} + \int_{x_{0}}^{x} \int_{y_{0}}^{y} (x-s)^{\theta_{1}-1}s^{\gamma _{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1}(x-s)^{\theta_{1}-1}s^{\gamma_{1}-1} \\& \hphantom{\phi(x,y) \le{}}{} \cdot(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1}h_{2}(s,t) \max_{ \tilde{\eta}\in[s-h,s]}\phi( \tilde{\eta},t)\, dt \, d\eta, \\& \hphantom{\phi(x,y) \le{}}{}(x,y)\in[x_{0},x_{1}) \times[y_{0},y_{1}), \\& \phi(x,y) \le 0, \quad (x,y)\in [x_{0}-h,x_{0}] \times[y_{0},y_{1}). \end{aligned}$$
(3.10)
$$ \begin{aligned} &\phi(x,y)\le \varepsilon+ \int_{x_{0}}^{x} \int_{y_{0}}^{y} (x-s)^{\theta_{1}-1}s^{\gamma_{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma _{2}-1}h_{1}(s,t) \phi(s,t)\,dt\,ds \\ &\hphantom{\phi(x,y)\le{}}{}+ \int_{x_{0}}^{x} \int_{y_{0}}^{y} (x-s)^{\theta_{1}-1}s^{\gamma _{1}-1}(y-t)^{\theta_{2}-1}t^{\gamma_{2}-1} \\ &\hphantom{\phi(x,y)\le{}}{} \cdot h_{2}(s,t) \max_{ \tilde{\eta}\in[\alpha(s)-h,\alpha(s)]}\phi( \tilde{\eta},t)\, dt\, d\eta, \\ &\hphantom{\phi(x,y)\le{}}{} (x,y)\in[x_{0},x_{1}) \times[y_{0},y_{1}), \\ &\phi(x,y)\le 0, \quad (x,y)\in [x_{0}-h,x_{0}] \times[y_{0},y_{1}). \end{aligned} $$
(3.11)
$$\begin{aligned}& \phi(x,y) \\& \quad \leq 3^{\frac{q-1}{q}}\varepsilon\exp \biggl(q^{-1}\biggl(3^{\frac {q-1}{q}}\bigl(M_{1}x^{\delta_{1}} \bar{M}_{1}y^{\delta_{2}}\bigr)^{\frac{q}{p}} \int_{x_{0}}^{x} \int_{y_{0}}^{y}\bigl(h_{1}^{q}(s,t)+h_{2}^{q}(s,t) \bigr)\,dt\,ds\biggr)\biggr), \end{aligned}$$
(3.12)
Competing interests
The authors declare that they have no competing interests.
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