Some new integral inequalities of Wendorff type for discontinuous functions with integral jump conditions

Abstract

In this paper, we investigate some new integral inequalities of Wendorff type for discontinuous functions with two independent variables and integral jump conditions. These integral inequalities with discontinuities are of non-Lipschitz type. New lower bounds are obtained, integral inequalities with retardation are also involved.

1 Introduction

The differential equations with impulse perturbations lie in a special important position in the theory of differential equations. Among these theories, integral inequality method is an important tool to investigate the qualitative characteristics of solutions of different kinds of equations such as difference equations, differential equations, impulsive differential equations, and partial differential equations (see [1–11] for details). For some summary papers, the readers are refereed to [11–18]. In papers [19–24], the authors give qualitative analysis of some integro-differential equations using certain integral inequalities; in papers [2, 25–28], the authors give some integral inequalities with more than two independent variables; papers [24, 29–33] give integral inequalities with weak singular kernels and some qualitative properties of fractional differential equations; the dynamic integral inequalities on time scales are given in papers [34, 35], and the inequalities for essentially bounded functions of one or three variables are investigated by [36, 37].

Phoilakrit Thiramanus and Jessade Tariboon [1] investigated impulsive integral inequality of one independent variable

$$\begin{aligned} \varphi (t)\leq C+ \int _{t_{0}}^{t}b(s)\varphi (s)\,\mathrm{d}s+\sum _{t_{0}< t_{i}< t} \gamma _{i}\varphi (t_{i}) +\sum_{t_{0}< t_{i}< t}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}}\varphi (s)\,\mathrm{d}s, \end{aligned}$$

(1)

where $0\leq t_{0}< t_{1}<\cdots$ , $\gamma _{i}, \beta _{i}\geq 0$, $0\leq \sigma _{i}\leq \tau _{i}\leq t_{i}-t_{i-1}$, $C\geq 0$ is a constant, and the points $t_{i}$ are of the first discontinuities.

In 1989, Borysenko [7] investigated impulsive integral inequality of two independent variables of the form

$$\begin{aligned} \varphi (t, x)\leq a(t, x)+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )\varphi (\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d}\eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i}\varphi \bigl(t_{i}^{-}, x_{i}^{-} \bigr), \end{aligned}$$

(2)

where $\varphi (t, x)$ is continuous in Ω, with the exception of the points ${t_{i}, x_{i}}$ where there are finite jumps: $\varphi (t_{i}^{-}, x_{i}^{-})\neq \varphi (t_{i}^{+}, x_{i}^{+})$, $\forall i=1, 2, \ldots$ .

In 2007, Borysenko and Iovane [10] investigated some integral inequalities of Wendorff type

$$\begin{aligned}& \varphi (t, x)\leq a(t, x)+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )\varphi (\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d}\eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i}\varphi ^{m}\bigl(t_{i}^{-}, x_{i}^{-}\bigr), \end{aligned}$$

(3)

$$\begin{aligned}& \varphi (t, x)\leq a(t, x)+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )\varphi ^{m}( \xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)} \gamma _{i}\varphi ^{m}\bigl(t_{i}^{-}, x_{i}^{-}\bigr), \end{aligned}$$

(4)

$$\begin{aligned}& \varphi (t, x)\leq a(t, x)+g(t, x) \int _{t_{0}}^{t} \int _{x_{0}}^{x}b( \xi , \eta )\varphi ^{m}( \xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta + \sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)} \gamma _{i}\varphi ^{m}\bigl(t_{i}^{-}, x_{i}^{-}\bigr), \end{aligned}$$

(5)

$$\begin{aligned}& \begin{aligned}[b] \varphi (t, x)&\leq a(t, x)+g(t, x) \int _{t_{0}}^{t} \int _{x_{0}}^{x}b( \xi , \eta )\varphi \bigl(\sigma ( \xi ), \sigma (\eta )\bigr)\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad {}+\sum _{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i} \varphi ^{m} \bigl(t_{i}^{-}, x_{i}^{-}\bigr), \end{aligned} \end{aligned}$$

(6)

$$\begin{aligned}& \begin{aligned}[b] \varphi (t, x)&\leq a(t, x)+g(t, x) \int _{t_{0}}^{t} \int _{x_{0}}^{x}b( \xi , \eta )\varphi ^{m} \bigl(\sigma (\xi ), \sigma (\eta )\bigr)\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad {}+\sum _{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i} \varphi ^{m} \bigl(t_{i}^{-}, x_{i}^{-}\bigr), \end{aligned} \end{aligned}$$

(7)

where $a(t, x)>0$ is nondecreasing with respect to $(t, x)$, and $g(t, x)\geq 1$, $b(t, x)\geq 0$, $\gamma _{i}\geq 0$ are constants. The delay term $\sigma (t)$ is continuous and nondecreasing in $[t_{0}, +\infty )$, $\lim_{t\rightarrow \infty }\sigma (t)\leq \infty$ for all $t\geq t_{0}$ and $\sigma (t)\leq t$.

In this paper, in a similar way to [8–12] for the inequalities of the functions with one independent variable, we investigate a new Wendorff type inequality for discontinuous functions with two independent variables and give some integro-sum functional inequalities with delay.

2 Integral inequalities for discontinuous functions with discontinuities of non-Lipschitz type

For a given function a defined in a domain Ω with two variables, we say a is a nondecreasing function if, for all $(p, q), (P, Q)\in \varOmega$ with $p\leq P$, $q\leq Q$, one always has $a(p, q)\leq a(P, Q)$.

Theorem 2.1

Let a nonnegative function$\varphi (t, x)$, determined in the domain

$$\begin{aligned} \varOmega =\bigcup_{k, j\geq 1}\varOmega _{kj}=\bigcup _{k, j\geq 1}\bigl\{ (t, x):t \in [t_{k-1}, t_{k}], x\in [x_{k-1}, x_{k}]\bigr\} , \end{aligned}$$

be continuous inΩ, with the exception of the points$(t_{i}, x_{i})$where there are finite jumps

$$ \varphi \bigl(t_{i}^{+}, x_{i}^{+} \bigr)\neq \varphi \bigl(t_{i}^{-}, x_{i}^{-} \bigr),\quad \forall i=1, 2, \ldots , $$

and satisfy a certain integro-sum inequality inΩ

$$\begin{aligned} \varphi (t, x)&\leq a(t, x)+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )\varphi (\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d}\eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i}\varphi ^{m}\bigl(t_{i}^{-}, x_{i}^{-}\bigr) \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int ^{t_{i}- \tau _{i}}_{t_{i}-\sigma _{i}} \int ^{x_{i}-\delta _{i}}_{x_{i}- \lambda _{i}}\varphi (\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta , \end{aligned}$$

(8)

where$m>0$, $t_{0}\geq 0$, $x_{0}\geq 0$, $\gamma _{i}=\mathrm{const}\geq 0$, $\beta _{i}=\mathrm{const} \geq 0$, and$a(t, x)>0$is nondecreasing, $b(t, x)>0$and satisfies$b(\xi , \eta )=0$. If$(\xi , \eta )\in \varOmega _{ij}$with$i\neq j$, $\lim_{i\rightarrow \infty } t_{i}=\infty$, $\lim_{i\rightarrow \infty } x_{i}=\infty$, then the function$\varphi (t, x)$satisfies the following estimates:

$$\begin{aligned}& \varphi (t, x)\leq a(t, x)\prod_{i=1}^{k-1}A_{i} \cdot \exp \biggl[ \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x}b(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr], \\& \begin{aligned} A_{i}&=\bigl(1+\gamma _{i} a^{m-1}(t_{i}, x_{i})\bigr)\cdot \exp \biggl[ \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}}b(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr] \\ &\quad{}+\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}- \delta _{i}}^{x_{i}-\lambda _{i}}\exp \biggl[ \int _{t_{i-1}}^{\xi } \int _{x_{i-1}}^{\eta }b(s, t)\,\mathrm{d}s\,\mathrm{d}t \biggr] \,\mathrm{d}\xi \,\mathrm{d}\eta \end{aligned} \end{aligned}$$

(9)

if$0< m\leq 1$; and

$$\begin{aligned}& \varphi (t, x)\leq a(t, x)\prod_{i=1}^{k-1} B_{i}^{m^{k-i}}\cdot \exp \biggl[ \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x}b(\xi , \eta ) \,\mathrm{d}\xi \, \mathrm{d}\eta \biggr], \\& \begin{aligned} B_{i}&=\bigl(1+\gamma _{i} a^{m-1}(t_{i}, x_{i})\bigr)\cdot \exp \biggl[m \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}}b(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr] \\ &\quad{}+\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}- \delta _{i}}^{x_{i}-\lambda _{i}}\exp \biggl( \int _{t_{i-1}}^{\xi } \int _{x_{i-1}}^{\eta }b(s, t)\,\mathrm{d}s\,\mathrm{d}t \biggr) \,\mathrm{d}\xi \,\mathrm{d}\eta \end{aligned} \end{aligned}$$

(10)

if$m>1$.

Proof

Due to $a(t, x)>0$, we can obtain that

$$\begin{aligned} \frac{\varphi (t, x)}{a(t, x)}&\leq 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b( \xi , \eta )\frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\, \mathrm{d} \xi \,\mathrm{d}\eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)} \gamma _{i} \frac{\varphi ^{m}(t_{i}^{-}, x_{i}^{-})}{a(t_{i}^{-}, x_{i}^{-})} \\ &\quad {}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}} \frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\,\mathrm{d} \xi \, \mathrm{d}\eta . \end{aligned}$$

(11)

Set

$$\begin{aligned} W(t, x)&= 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta ) \frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\, \mathrm{d}\xi \,\mathrm{d} \eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i} \frac{\varphi ^{m}(t_{i}^{-}, x_{i}^{-})}{a(t_{i}^{-}, x_{i}^{-})} \\ &\quad {}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}} \frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\,\mathrm{d} \xi \, \mathrm{d}\eta , \end{aligned}$$

(12)

with $W(t_{0}, x_{0})=1$, $\varphi (t, x)\leq a(t, x)W(t, x)$, then

$$\begin{aligned} W(t, x)&\leq 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )W( \xi , \eta )\, \mathrm{d}\xi \,\mathrm{d}\eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i}a^{m-1}(t_{i}, x_{i})\bigl[W \bigl(t_{i}^{-}, x_{i}^{-}\bigr) \bigr]^{m} \\ &\quad {}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}}W(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta . \end{aligned}$$

(13)

We give the proof by induction. Firstly, we consider the domain $\varOmega _{11}=\{(t, x):t\in [t_{0}, t_{1}], x\in [x_{0}, x_{1}]\}$, then

$$\begin{aligned} \frac{\varphi (t, x)}{a(t, x)}\leq 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b( \xi , \eta )\frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\, \mathrm{d} \xi \,\mathrm{d}\eta , \end{aligned}$$

(14)

set

$$\begin{aligned} V(t, x)=1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta ) \frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\, \mathrm{d}\xi \,\mathrm{d} \eta , \end{aligned}$$

(15)

thus

$$\begin{aligned}& \varphi (t, x)\leq a(t, x)V(t, x), \\& V(t, x)\leq 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )V(\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d}\eta . \end{aligned}$$

Let

$$\begin{aligned} K(t, x)=1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )V(\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d}\eta . \end{aligned}$$

(16)

Then

$$\begin{aligned} V(t, x)\leq K(t, x),\quad K(t_{0}, x)=K(t, x_{0})=1. \end{aligned}$$

Differentiating $K(t, x)$ with respect to t, the following equation holds:

$$\begin{aligned} K_{t}(t, x)= \int _{x_{0}}^{x}b(t, \eta )V(t, \eta )\,\mathrm{d}\eta , \end{aligned}$$

because $b(t, x)$ and $V(t, x)$ are continuous in $\varOmega _{11}$. Besides, $V(t, x)>0$, it means that $V(t, x)$ maintains the sign in $\varOmega _{11}$. So, on account of generalized mean value theorem of integrals, we can get that

$$\begin{aligned}& K_{t}(t, x)= \int _{x_{0}}^{x}b(t, \eta )V(t, \eta )\,\mathrm{d}\eta \leq \int _{x_{0}}^{x}b(t, \eta )\,\mathrm{d}\eta \cdot K(t, x), \\& \frac{K_{t}(t, x)}{K(t, x)}\leq \int _{x_{0}}^{x}b(t, \eta ) \,\mathrm{d}\eta . \end{aligned}$$

Integrating this inequality from $t_{0}$ to t implies

$$\begin{aligned}& \int _{t_{0}}^{t}\frac{K_{t}(\xi , x)}{K(\xi , x)}\,\mathrm{d}\xi \leq \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta , \\& \ln K(t, x)| _{t_{0}}^{t}=\ln K(t, x)-\ln K(t_{0}, x)=\ln K(t, x), \\& \ln K(t, x)\leq \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta ) \,\mathrm{d}\xi \, \mathrm{d}\eta , \end{aligned}$$

then

$$\begin{aligned} V(t, x)\leq K(t, x)\leq \exp \biggl[ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b( \xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr], \end{aligned}$$

so we can get that

$$\begin{aligned} \varphi (t, x)\leq a(t, x)\exp \biggl[ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b( \xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr]. \end{aligned}$$

(17)

This shows that the estimates are true in $\varOmega _{11}$. Secondly, suppose that (9) and (10) are true in the domain $\varOmega _{kk}$. If $0< m\leq 1$, then for $(t, x)\in \varOmega _{k+1, k+1}$ the following inequality holds:

$$\begin{aligned} W(t, x)&\leq 1+\sum_{i=1}^{k-1}\gamma _{i}a^{m-1}(t_{i}, x_{i})\bigl[W \bigl(t_{i}^{-}, x_{i}^{-}\bigr) \bigr]^{m} +\sum_{i=1}^{k-1}\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}- \sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}-\lambda _{i}} W(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \\ &\quad{}+ \int _{t_{0}}^{t_{k}} \int _{x_{0}}^{x_{k}}b(\xi , \eta )W(\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d}\eta +\gamma _{k}a^{m-1}(t_{k}, x_{k})\bigl[W(t_{k}-0, x_{k}-0) \bigr]^{m} \\ &\quad{}+\beta _{k} \int _{t_{k}-\tau _{k}}^{t_{k}-\sigma _{k}} \int _{x_{k}- \delta _{k}}^{x_{k}-\lambda _{k}} W(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta + \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )W( \xi , \eta )\, \mathrm{d}\xi \,\mathrm{d}\eta \\ &\leq \prod_{i=1}^{k-1}A_{i}\exp \biggl[ \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr] \\ &\quad {}+\gamma _{k}a^{m-1}(t_{k}, x_{k}) \Biggl\{ \prod_{i=1}^{k-1}A_{i} \exp \biggl[ \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr] \Biggr\} ^{m} \\ &\quad{}+\beta _{k} \int _{t_{k}-\tau _{k}}^{t_{k}-\sigma _{k}} \int _{x_{k}- \delta _{k}}^{x_{k}-\lambda _{k}} \prod_{i=1}^{k-1}A_{i} \exp \biggl[ \int _{t_{k-1}}^{ \xi } \int _{x_{k-1}}^{\eta }b(\tau , s)\,\mathrm{d}\tau \, \mathrm{d}s\biggr] \,\mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad {}+ \int _{t_{k}}^{t} \int _{x_{k}}^{x}b( \xi , \eta )W(\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d}\eta \\ &\leq \prod_{i=1}^{k-1}A_{i} \biggl\{ \bigl(1+\gamma _{k}a^{m-1}(t_{k}, x_{k})\bigr) \exp \biggl[ \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr] \\ &\quad{}+\beta _{k} \int _{t_{k}-\tau _{k}}^{t_{k}-\sigma _{k}} \int _{x_{k}- \delta _{k}}^{x_{k}-\lambda _{k}} \exp \biggl[ \int _{t_{k-1}}^{\xi } \int _{x_{k-1}}^{ \eta }b(\tau , s)\,\mathrm{d}\tau \, \mathrm{d}s\biggr]\,\mathrm{d}\xi \,\mathrm{d} \eta \biggr\} \\&\quad{}+ \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )W( \xi , \eta )\, \mathrm{d}\xi \,\mathrm{d}\eta \\ &\leq \prod_{i=1}^{k}A_{i}\exp \biggl[ \int _{t_{k}}^{t} \int _{x_{k}}^{x}b( \xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr], \end{aligned}$$

so when $0< m\leq 1$, (9) stands.

If $m>1$, then for $(t, x)\in \varOmega _{k+1, k+1}$ the following inequality holds:

$$\begin{aligned} W(t, x) &\leq 1+\sum_{i=1}^{k-1}\gamma _{i}a^{m-1}(t_{i}, x_{i})\bigl[W \bigl(t_{i}^{-}, x_{i}^{-}\bigr) \bigr]^{m} +\sum_{i=1}^{k-1}\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}- \sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}-\lambda _{i}} W(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \\ &\quad{}+ \int _{t_{0}}^{t_{k}} \int _{x_{0}}^{x_{k}}b(\xi , \eta )W(\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d}\eta +\gamma _{k}a^{m-1}(t_{k}, x_{k})\bigl[W(t_{k}-0, x_{k}-0) \bigr]^{m} \\ &\quad{}+\beta _{k} \int _{t_{k}-\tau _{k}}^{t_{k}-\sigma _{k}} \int _{x_{k}- \delta _{k}}^{x_{k}-\lambda _{k}} W(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta + \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )W( \xi , \eta )\, \mathrm{d}\xi \,\mathrm{d}\eta \\ &\leq \prod_{i=1}^{k-1}B_{i}^{m^{k-i-1}} \exp \biggl[ \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr]\\&\quad{} +\gamma _{k}a^{m-1}(t_{k}, x_{k}) \Biggl\{ \prod_{i=1}^{k-1}B_{i}^{m^{k-i-1}} \exp \biggl[ \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr] \Biggr\} ^{m} \\ &\quad{}+\beta _{k} \int _{t_{k}-\tau _{k}}^{t_{k}-\sigma _{k}} \int _{x_{k}- \delta _{k}}^{x_{k}-\lambda _{k}} \prod_{i=1}^{k-1}B_{i}^{m^{k-i-1}} \exp \biggl[ \int _{t_{k-1}}^{\xi } \int _{x_{k-1}}^{\eta }b(\tau , s) \,\mathrm{d}\tau \, \mathrm{d}s\biggr]\,\mathrm{d}\xi \,\mathrm{d}\eta \\&\quad{}+ \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )W(\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d} \eta \\ &\leq \prod_{i=1}^{k-1}B_{i}^{m^{k-i}} \exp \biggl[m \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr] \\&\quad{}+\gamma _{k}a^{m-1}(t_{k}, x_{k}) \Biggl\{ \prod_{i=1}^{k-1}B_{i}^{m^{k-i}} \exp \biggl[m \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr] \Biggr\} \\ &\quad{}+\beta _{k} \int _{t_{k}-\tau _{k}}^{t_{k}-\sigma _{k}} \int _{x_{k}- \delta _{k}}^{x_{k}-\lambda _{k}} \prod_{i=1}^{k-1}B_{i}^{m^{k-i}} \exp \biggl[ \int _{t_{k-1}}^{\xi } \int _{x_{k-1}}^{\eta }b(\tau , s) \,\mathrm{d}\tau \, \mathrm{d}s\biggr]\,\mathrm{d}\xi \,\mathrm{d}\eta\\&\quad{} + \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )W(\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d} \eta \\ &\leq \prod_{i=1}^{k-1}B_{i}^{m^{k-i}} \biggl\{ \bigl(1+\gamma _{k}a^{m-1}(t_{k}, x_{k})\bigr)\exp \biggl[m \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b( \xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr] \\ &\quad{}+\beta _{k} \int _{t_{k}-\tau _{k}}^{t_{k}-\sigma _{k}} \int _{x_{k}- \delta _{k}}^{x_{k}-\lambda _{k}} \exp \biggl[ \int _{t_{k-1}}^{\xi } \int _{x_{k-1}}^{ \eta }b(\tau , s)\,\mathrm{d}\tau \, \mathrm{d}s\biggr]\,\mathrm{d}\xi \,\mathrm{d} \eta \biggr\} \\&\quad{} + \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )W(\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d}\eta \\ &\leq \prod_{i=1}^{k}B_{i}^{m^{k-i}} \exp \biggl[ \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr], \end{aligned}$$

hence when $m>1$, (10) stands. Finally, by mathematical induction, we get (9) and (10) hold on Ω. This finishes the proof. □

Theorem 2.2

Suppose that there exists a nonnegative piecewise continuous function$\varphi (t, x)$determined in the domainΩ, with discontinuity of the first kind in the points$(t_{k}, x_{k})$ ($t_{0}< t_{1}< t_{2}<\cdots$ , $x_{0}< x_{1}< x_{2}<\cdots$ , $\lim_{i\rightarrow \infty } t_{i}=\infty$, $\lim_{i\rightarrow \infty } x_{i}=\infty$), and it satisfies the inequality

$$\begin{aligned} \varphi (t, x)&\leq a(t, x)+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )\varphi ^{m}( \xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)} \gamma _{i}\varphi ^{m}\bigl(t_{i}^{-}, x_{i}^{-}\bigr) \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int ^{t_{i}- \tau _{i}}_{t_{i}-\sigma _{i}} \int ^{x_{i}-\delta _{i}}_{x_{i}- \lambda _{i}}\varphi (\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta , \end{aligned}$$

(18)

$m>0$, $m\neq 1$, wherea, b, $\gamma _{i}$, $\beta _{i}$satisfy the conditions of Theorem 2.1. Then, for$(t, x)\in \varOmega$, $k=1, 2, \ldots$ , the following estimates hold:

$$\begin{aligned}& \varphi (t, x)\leq a(t, x)\prod_{i=1}^{k-1}C_{i} \cdot \biggl[1+(1-m) \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \biggr]^{\frac{1}{1-m}}, \\ & \begin{aligned} C_{i}&=\bigl(1+\gamma _{i} a^{m-1}(t_{i}, x_{i})\bigr)\cdot \biggl[1+(1-m) \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}}b(\xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d} \xi \,\mathrm{d}\eta \biggr]^{\frac{1}{1-m}} \\ &\quad{}+\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}- \delta _{i}}^{x_{i}-\lambda _{i}} \biggl[1+(1-m) \int _{t_{i-1}}^{\xi } \int _{x_{i-1}}^{ \eta }b(\tau , s)a^{m-1}(\tau , s) \,\mathrm{d}\tau \,\mathrm{d}s\biggr]^{ \frac{1}{1-m}}\,\mathrm{d}\xi \,\mathrm{d} \eta , \end{aligned} \\ & C_{0}=1, \end{aligned}$$

(19)

if$0< m<1$; and

$$\begin{aligned}& \begin{aligned}[b] \varphi (t, x)&\leq a(t, x)\prod_{i=1}^{k} D_{i}^{m^{k-i}}\\ &\quad {}\cdot \Biggl[1-(m-1) \Biggl(\prod _{i=1}^{k}D_{i-1}^{m^{k-i}} \Biggr)^{m-1} \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \Biggr]^{-\frac{1}{m-1}}, \end{aligned} \\ & \begin{aligned} D_{i}&=\bigl(1+\gamma _{i} a^{m-1}(t_{i}, x_{i})\bigr)\\ &\quad {}\cdot \Biggl[1-(m-1) \Biggl( \prod_{j=1}^{i}D_{j-1}^{m^{i-j}} \Biggr)^{m-1} \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}}b( \xi , \eta )a^{m-1}(\xi , \eta ) \,\mathrm{d}\xi \,\mathrm{d}\eta \Biggr]^{- \frac{m}{m-1}} \\ &\quad{}+\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}- \delta _{i}}^{x_{i}-\lambda _{i}} \Biggl[1-(m-1) \Biggl(\prod _{j=1}^{i}D_{j-1}^{m^{i-j}} \Biggr)^{m-1}\\&\quad{}\times \int _{t_{i-1}}^{\xi } \int _{x_{i-1}}^{\eta }b(\tau , s)a^{m-1}(\tau , s) \,\mathrm{d}\tau \,\mathrm{d}s\Biggr]^{-\frac{1}{m-1}}\,\mathrm{d}\xi \,\mathrm{d} \eta , \quad D_{0}=1, \end{aligned} \end{aligned}$$

(20)

if$m>1$with$\forall (t, x)\in \varOmega$satisfying

$$\begin{aligned} \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \leq \frac{1}{(m-1)(\prod_{i=1}^{k}D_{i-1}^{m^{k-i}})^{m-1}}. \end{aligned}$$

Proof

Because of $a(t, x)>0$, we get

$$\begin{aligned} \frac{\varphi (t, x)}{a(t, x)}&\leq 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b( \xi , \eta )\frac{\varphi ^{m}(\xi , \eta )}{a(\xi , \eta )} \, \mathrm{d}\xi \,\mathrm{d}\eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i} \frac{\varphi ^{m}(t_{i}^{-}, x_{i}^{-})}{a(t_{i}^{-}, x_{i}^{-})} \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}}\frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\,\mathrm{d}\xi \, \mathrm{d}\eta . \end{aligned}$$

(21)

Set

$$\begin{aligned} W(t, x)&= 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta ) \frac{\varphi ^{m}(\xi , \eta )}{a(\xi , \eta )}\, \mathrm{d}\xi \,\mathrm{d}\eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i} \frac{\varphi ^{m}(t_{i}^{-}, x_{i}^{-})}{a(t_{i}^{-}, x_{i}^{-})} \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}}\frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\,\mathrm{d}\xi \, \mathrm{d}\eta , \end{aligned}$$

(22)

then $W(t_{0}, x)=W(t, x_{0})=1$, $\varphi (t, x)\leq a(t, x)W(t, x)$, and

$$\begin{aligned} W(t, x)&\leq 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )a^{m-1}( \xi , \eta ) \biggl[\frac{\varphi (\xi , \eta )}{a(\xi , \eta )} \biggr]^{m} \,\mathrm{d}\xi \,\mathrm{d} \eta \\&\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i}a^{m-1}(t_{i}, x_{i}) \biggl[ \frac{\varphi (t_{i}^{-}, x_{i}^{-})}{a(t_{i}^{-}, x_{i}^{-})} \biggr]^{m} \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}} \frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\,\mathrm{d} \xi \, \mathrm{d}\eta \\ &\leq 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )W^{m}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \\&\quad{} +\sum _{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i}a^{m-1}(t_{i}, x_{i})\bigl[W\bigl(t_{i}^{-}, x_{i}^{-}\bigr)\bigr]^{m} \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}} W(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta . \end{aligned}$$

(23)

By mathematical induction, we consider the function in the domain $\varOmega _{11}=\{(t, x):t\in [t_{0}, t_{1}], x\in [x_{0}, x_{1}]\}$ firstly. We get

$$\begin{aligned} W(t, x)\leq 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )b(\xi , \eta )a^{m-1}(\xi , \eta )W^{m}(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d} \eta , \end{aligned}$$

(24)

set

$$\begin{aligned} K(t, x)=1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )b(\xi , \eta )a^{m-1}(\xi , \eta )W^{m}(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d} \eta , \end{aligned}$$

(25)

then

$$\begin{aligned} \varphi (t, x)\leq a(t, x)K(t, x),\quad K(t_{0}, x)=1, K(t, x_{0})=1. \end{aligned}$$

Differentiating $K(t, x)$ with respect to t, the following equation holds:

$$\begin{aligned} K_{t}(t, x)= \int _{x_{0}}^{x}b(t, \eta )a^{m-1}(\xi , \eta )W^{m}(t, \eta )\,\mathrm{d}\eta . \end{aligned}$$

Since $b(t, x)$ and $W(t, x)$ are continuous in $\varOmega _{11}$, besides $W(t, x)>0$, it means that $W(t, x)$ maintains the same sign in $\varOmega _{11}$. So, on account of the generalized first mean value theorem of integrals, we can get that

$$\begin{aligned}& K_{t}(t, x)= \int _{x_{0}}^{x}b(t, \eta )a^{m-1}(t, \eta )W^{m}(t, \eta )\,\mathrm{d}\eta \leq \int _{x_{0}}^{x}b(t, \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\eta \cdot K^{m}(t, x), \\& \frac{K_{t}(t, x)}{K^{m}(t, x)}\leq \int _{x_{0}}^{x}b(t, \eta )a^{m-1}(t, \eta )\, \mathrm{d}\eta . \end{aligned}$$

If $0< m<1$,

$$\begin{aligned} (1-m)\frac{K_{t}(t, x)}{K^{m}(t, x)}\leq (1-m) \int _{x_{0}}^{x}b(t, \eta )a^{m-1}(t, \eta )\, \mathrm{d}\eta , \end{aligned}$$

then

$$\begin{aligned} \frac{d}{dt}K^{1-m}(t, x)\leq (1-m) \int _{x_{0}}^{x}b(t, \eta )a^{m-1}(t, \eta )\, \mathrm{d}\eta . \end{aligned}$$

Integrating this inequality from $t_{0}$ to t, we get

$$\begin{aligned} K^{1-m}(t, x)&\leq K^{1-m}(t_{0}, x)+(1-m) \int _{t_{0}}^{t} \int _{x_{0}}^{x}b( \xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &=1+(1-m) \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta , \end{aligned}$$

then

$$\begin{aligned} W(t, x)\leq K(t, x)\leq \biggl[1+(1-m) \int _{t_{0}}^{t} \int _{x_{0}}^{x}b( \xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \biggr]^{ \frac{1}{1-m}}. \end{aligned}$$

For the case of $m>1$,

$$\begin{aligned} (1-m)\frac{K_{t}(t, x)}{K^{m}(t, x)}\geq (1-m) \int _{x_{0}}^{x}b(t, \eta )a^{m-1}(t, \eta )\, \mathrm{d}\eta , \end{aligned}$$

thus

$$\begin{aligned} \frac{d}{dt}K^{1-m}(t, x)\geq (1-m) \int _{x_{0}}^{x}b(t, \eta )a^{m-1}(t, \eta )\, \mathrm{d}\eta . \end{aligned}$$

Integrating this inequality from $t_{0}$ to t, we get

$$\begin{aligned} K^{1-m}(t, x)&\geq K^{1-m}(t_{0}, x)-(m-1) \int _{t_{0}}^{t} \int _{x_{0}}^{x}b( \xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &=1-(m-1) \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta , \end{aligned}$$

then $\forall (t, x)\in \varOmega _{11}$ satisfying

$$\begin{aligned} \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta ) \,\mathrm{d}\xi \,\mathrm{d}\eta < \frac{1}{m-1}, \end{aligned}$$

we get

$$\begin{aligned} W(t, x)\leq K(t, x)\leq \biggl[1-(m-1) \int _{t_{0}}^{t} \int _{x_{0}}^{x}b( \xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \biggr]^{- \frac{1}{m-1}}. \end{aligned}$$

Now, we firstly consider the case of $0< m<1$. Suppose that (19) is justified in the domain $\varOmega _{kk}$, then for $(t, x)\in \varOmega _{k+1, k+1}$ the following inequality holds:

$$\begin{aligned} &W(t, x)\\ &\quad \leq 1+\sum_{i=1}^{k-1}\gamma _{i}a^{m-1}(t_{i}, x_{i})\bigl[W \bigl(t_{i}^{-}, x_{i}^{-}\bigr) \bigr]^{m} +\sum_{i=1}^{k-1}\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}- \sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}-\lambda _{i}} W(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \\ &\qquad{}+ \int _{t_{0}}^{t_{k}} \int _{x_{0}}^{x_{k}}b(\xi , \eta )a^{m-1}( \xi , \eta )W^{m}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta +\gamma _{k}a^{m-1}(t_{k}, x_{k}) \bigl[W(t_{k}-0, x_{k}-0)\bigr]^{m} \\ &\qquad{}+\beta _{k} \int _{t_{k}-\tau _{k}}^{t_{k}-\sigma _{k}} \int _{x_{k}- \delta _{k}}^{x_{k}-\lambda _{k}} W(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta + \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )a^{m-1}( \xi , \eta )W^{m}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad \leq \prod_{i=1}^{k-1}C_{i}\exp \biggl[1+(1-m) \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b(\xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d} \xi \,\mathrm{d}\eta \biggr]^{\frac{1}{1-m}} \\ &\qquad{}+\gamma _{k}a^{m-1}(t_{k}, x_{k}) \Biggl(\prod_{i=1}^{k-1}C_{i} \Biggr)^{m} \biggl[1+(1-m) \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b( \xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \biggr]^{ \frac{m}{1-m}} \\ &\qquad{}+\beta _{k} \int _{t_{k}-\tau _{k}}^{t_{k}-\sigma _{k}} \int _{x_{k}- \delta _{k}}^{x_{k}-\lambda _{k}} \prod_{i=1}^{k-1}C_{i} \biggl[1+(1-m) \int _{t_{k-1}}^{\xi } \int _{x_{k-1}}^{\eta }b(\tau , s)a^{m-1}(\tau , s) \,\mathrm{d}\tau \,\mathrm{d}s \biggr]^{\frac{1}{1-m}}\,\mathrm{d}\xi \,\mathrm{d} \eta \\ &\qquad{}+ \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )W^{m}( \xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad \leq \prod_{i=1}^{k-1}C_{i}\exp \biggl[1+(1-m) \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b(\xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d} \xi \,\mathrm{d}\eta \biggr]^{\frac{1}{1-m}} \\ &\qquad{}+\gamma _{k}a^{m-1}(t_{k}, x_{k})\prod_{i=1}^{k-1}C_{i} \biggl[1+(1-m) \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b(\xi , \eta )a^{m-1}( \xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \biggr]^{\frac{1}{1-m}} \\ &\qquad{}+\beta _{k} \int _{t_{k}-\tau _{k}}^{t_{k}-\sigma _{k}} \int _{x_{k}- \delta _{k}}^{x_{k}-\lambda _{k}} \prod_{i=1}^{k-1}C_{i} \biggl[1+(1-m) \int _{t_{k-1}}^{\xi } \int _{x_{k-1}}^{\eta }b(\tau , s)a^{m-1}(\tau , s) \,\mathrm{d}\tau \,\mathrm{d}s \biggr]^{\frac{1}{1-m}}\,\mathrm{d}\xi \,\mathrm{d} \eta \\ &\qquad{}+ \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )W^{m}( \xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad \leq \prod_{i=1}^{k-1}C_{i} \biggl\{ \bigl(1+\gamma _{k}a^{m-1}(t_{k}, x_{k})\bigr) \biggl[1+(1-m) \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b(\xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \biggr]^{ \frac{1}{1-m}} \\ &\qquad{}+\beta _{k} \biggl[1+(1-m) \int _{t_{k-1}}^{\xi } \int _{x_{k-1}}^{ \eta }b(\tau , s)a^{m-1}(\tau , s) \,\mathrm{d}\tau \,\mathrm{d}s \biggr]^{ \frac{1}{1-m}}\,\mathrm{d}\xi \,\mathrm{d} \eta \biggr\} \\ &\qquad{}+ \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )W^{m}( \xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad \leq \prod_{i=1}^{k}C_{i}+ \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )W^{m}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d} \eta . \end{aligned}$$

The right-hand side of this inequality is defined as $V(t, x)$, then $V(t_{k}, x)=V(t, x_{k})=\prod_{i=1}^{k}C_{i}$ and $W(t, x)\leq V(t, x)$. Differentiating $V(t, x)$ with respect to t, and on account of the generalized first mean value theorem of integrals, we can get that

$$\begin{aligned}& \begin{aligned} V_{t}(t, x)&= \int _{x_{k}}^{x}b(t, \eta )a^{m-1}(t, \eta )W^{m}(t, \eta )\,\mathrm{d}\eta \\ &\leq \int _{x_{k}}^{x}b(t, \eta )a^{m-1}(t, \eta )\, \mathrm{d}\eta V^{m}(t, \eta ), \end{aligned} \\& \frac{V_{t}(t, x)}{V^{m}(t, x)}\leq \int _{x_{k}}^{x}b(t, \eta )a^{m-1}(t, \eta )\, \mathrm{d}\eta . \\& (1-m)\frac{V_{t}(t, x)}{V^{m}(t, x)}\leq (1-m) \int _{x_{k}}^{x}b(t, \eta )a^{m-1}(t, \eta )\, \mathrm{d}\eta , \end{aligned}$$

thus

$$\begin{aligned} \frac{d}{dt}V^{1-m}(t, x)\leq (1-m) \int _{x_{k}}^{x}b(t, \eta )a^{m-1}(t, \eta )\, \mathrm{d}\eta . \end{aligned}$$

Integrating the above inequality from $t_{k}$ to t, we get

$$\begin{aligned} V^{1-m}(t, x)&\leq V^{1-m}(t_{k}, x)+(1-m) \int _{t_{k}}^{t} \int _{x_{1}}^{x}b( \xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &=\Biggl(\prod_{i=1}^{k} \Biggr)C_{i}^{1-m}+(1-m) \int _{t_{k}}^{t} \int _{x_{k}}^{x}b( \xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta , \end{aligned}$$

then we can get that

$$\begin{aligned} V(t, x)&\leq \Biggl[\prod_{i=1}^{k}C_{i}^{1-m}+(1-m) \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta ) \,\mathrm{d}\xi \,\mathrm{d}\eta \Biggr]^{\frac{1}{1-m}} \\ &\leq \prod_{i=1}^{k}C_{i} \Biggl[1+(1-m) \Biggl(\prod_{i=1}^{k} \Biggr)C_{i}^{m-1} \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta ) \,\mathrm{d}\xi \,\mathrm{d}\eta \Biggr]^{\frac{1}{1-m}} \\ &\leq \prod_{i=1}^{k}C_{i} \biggl[1+(1-m) \int _{t_{k}}^{t} \int _{x_{k}}^{x}b( \xi , \eta )a^{m-1}(\xi , \eta ) \,\mathrm{d}\xi \,\mathrm{d}\eta \biggr]^{ \frac{1}{1-m}}, \end{aligned}$$

so when $0< m<1$, (19) stands.

Next, we prove the case of $m>1$. Assume that (20) is fulfilled in the domain $\varOmega _{kk}$, then for $(t, x)\in \varOmega _{k+1, k+1}$ the following inequality holds:

$$\begin{aligned} &W(t, x) \\ &\quad \leq 1+\sum_{i=1}^{k-1}\gamma _{i}a^{m-1}(t_{i}, x_{i})\bigl[W \bigl(t_{i}^{-}, x_{i}^{-}\bigr) \bigr]^{m} +\sum_{i=1}^{k-1}\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}- \sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}-\lambda _{i}} W(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta \\ &\qquad{}+ \int _{t_{0}}^{t_{k}} \int _{x_{0}}^{x_{k}}b(\xi , \eta )a^{m-1}( \xi , \eta )W^{m}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta +\gamma _{k}a^{m-1}(t_{k}, x_{k}) \bigl[W(t_{k}-0, x_{k}-0)\bigr]^{m} \\ &\qquad{}+\beta _{k} \int _{t_{k}-\tau _{k}}^{t_{k}-\sigma _{k}} \int _{x_{k}- \delta _{k}}^{x_{k}-\lambda _{k}} W(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d}\eta + \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )a^{m-1}( \xi , \eta )W^{m}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad \leq \prod_{i=1}^{k}D_{i-1}^{m^{k-i}} \exp \Biggl[1-(m-1) \Biggl( \prod_{i=1}^{k}D_{i-1}^{m^{k-i}} \Biggr)^{m-1} \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b(\xi , \eta ) a^{m-1}(\xi , \eta )\,\mathrm{d} \xi \,\mathrm{d}\eta \Biggr]^{-\frac{1}{m-1}} \\ &\qquad{}+\gamma _{k}a^{m-1}(t_{k}, x_{k}) \Biggl(\prod_{i=1}^{k}D_{i-1}^{m^{k-i}} \Biggr)^{m} \\&\qquad{}\times \Biggl[1-(m-1) \Biggl(\prod_{i=1}^{k}D_{i-1}^{m^{k-i}} \Biggr)^{m-1} \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b(\xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \Biggr]^{- \frac{m}{m-1}} \\ &\qquad{}+\beta _{k} \int _{t_{k}-\tau _{k}}^{t_{k}-\sigma _{k}} \int _{x_{k}- \delta _{k}}^{x_{k}-\lambda _{k}} \prod_{i=1}^{k}D_{i-1}^{m^{k-i}} \\&\qquad{}\times \Biggl[1-(m-1) \Biggl(\prod_{i=1}^{k}D_{i-1}^{m^{k-i}} \Biggr)^{m-1} \int _{t_{k-1}}^{\xi } \int _{x_{k-1}}^{\eta }b(\tau , s)a^{m-1}(\tau , s) \,\mathrm{d}\tau \,\mathrm{d}s \Biggr]^{-\frac{1}{m-1}}\,\mathrm{d}\xi \,\mathrm{d} \eta \\ &\qquad{}+ \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )W^{m}( \xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad \leq \prod_{i=1}^{k}D_{i-1}^{m^{k-i+1}} \Biggl\{ \bigl(1+\gamma _{k}a^{m-1}(t_{k}, x_{k})\bigr)\\&\qquad{}\times \Biggl[1-(m-1) \Biggl(\prod_{i=1}^{k}D_{i-1}^{m^{k-i}} \Biggr)^{m-1} \int _{t_{k-1}}^{t_{k}} \int _{x_{k-1}}^{x_{k}}b(\xi , \eta )a^{m-1}( \xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \Biggr]^{-\frac{m}{m-1}} \\ &\qquad{}+\beta _{k} \int _{t_{k}-\tau _{k}}^{t_{k}-\sigma _{k}} \int _{x_{k}- \delta _{k}}^{x_{k}-\lambda _{k}} \Biggl[1-(m-1) \Biggl(\prod _{i=1}^{k}D_{i-1}^{m^{k-i}} \Biggr)^{m-1}\\&\qquad{}\times \int _{t_{k-1}}^{\xi } \int _{x_{k-1}}^{\eta }b(\tau , s)a^{m-1}( \tau , s) \,\mathrm{d}\tau \,\mathrm{d}s \Biggr]^{-\frac{1}{m-1}} \,\mathrm{d}\xi \, \mathrm{d}\eta \Biggr\} \\ &\qquad{}+ \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )W^{m}( \xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad = \prod_{i=1}^{k}D_{i-1}^{m^{k-i+1}} \cdot D_{k}+ \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )W^{m}(\xi , \eta ) \,\mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad = \prod_{i=1}^{k+1}D_{i-1}^{m^{k-i+1}} + \int _{t_{k}}^{t} \int _{x_{k}}^{x}b( \xi , \eta )a^{m-1}(\xi , \eta )W^{m}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta . \end{aligned}$$

The right-hand side of the last inequality is defined as $U(t, x)$, then $U(t_{k}, x)=U(t, x_{k})=\prod_{i=1}^{k+1}D_{i-1}^{m^{k-i+1}}$ and $W(t, x)\leq U(t, x)$. Differentiating $U(t, x)$ with respect to t, and on account of the generalized first mean value theorem of integrals, we can get that

$$\begin{aligned}& \begin{aligned} U_{t}(t, x)&= \int _{x_{k}}^{x}b(t, \eta )a^{m-1}(t, \eta )W^{m}(t, \eta )\,\mathrm{d}\eta \\ &\leq \int _{x_{k}}^{x}b(t, \eta )a^{m-1}(t, \eta )\, \mathrm{d}\eta U^{m}(t, \eta ), \end{aligned} \\& \frac{U_{t}(t, x)}{U^{m}(t, x)}\leq \int _{x_{k}}^{x}b(t, \eta )a^{m-1}(t, \eta )\, \mathrm{d}\eta , \\& (1-m)\frac{U_{t}(t, x)}{U^{m}(t, x)}\leq (1-m) \int _{x_{k}}^{x}b(t, \eta )a^{m-1}(t, \eta )\, \mathrm{d}\eta , \end{aligned}$$

thus

$$\begin{aligned} \frac{d}{dt}U^{1-m}(t, x)\leq (1-m) \int _{x_{k}}^{x}b(t, \eta )a^{m-1}(t, \eta )\, \mathrm{d}\eta . \end{aligned}$$

Integrating this inequality from $t_{k}$ to t, we have

$$\begin{aligned} U^{1-m}(t, x)&\geq U^{1-m}(t_{k}, x)-(m-1) \int _{t_{k}}^{t} \int _{x_{k}}^{x}b( \xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &= \Biggl(\prod_{i=1}^{k+1}D_{i-1}^{m^{k-i+1}} \Biggr)^{1-m}-(m-1) \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta ) \,\mathrm{d}\xi \,\mathrm{d}\eta , \end{aligned}$$

Then we can get that $\forall (t, x)\in \varOmega _{k+1, k+1}$ satisfying that

$$\begin{aligned}& \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta ) \,\mathrm{d}\xi \,\mathrm{d}\eta < \frac{1}{(m-1)(\prod_{i=1}^{k+1}D_{i-1}^{m^{k-i+1}})^{m-1}}, \\& \begin{aligned}[b] U(t, x)&\leq \Biggl[ \Biggl(\prod_{i=1}^{k+1}D_{i-1}^{m^{k-i+1}} \Biggr)^{1-m}-(m-1) \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )a^{m-1}( \xi , \eta ) \,\mathrm{d}\xi \,\mathrm{d}\eta \Biggr]^{-\frac{1}{m-1}} \\ &\leq \prod_{i=1}^{k+1}D_{i-1}^{m^{k-i+1}} \Biggl[1-(m-1) \Biggl( \prod_{i=1}^{k+1}D_{i-1}^{m^{k-i+1}} \Biggr)^{m-1}\\ &\quad {}\times \int _{t_{k}}^{t} \int _{x_{k}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \Biggr]^{-\frac{1}{m-1}}. \end{aligned} \end{aligned}$$

So when $m>1$, (20) stands. By mathematical induction, this completes the proof. □

Theorem 2.3

Suppose that there exists a nonnegative piecewise continuous function$\varphi (t, x)$determined in the domainΩ, with discontinuity of the first kind in the points$(t_{k}, x_{k})$ ($t_{0}< t_{1}< t_{2}<\cdots$ , $x_{0}< x_{1}< x_{2}<\cdots$ , $\lim_{i\rightarrow \infty } t_{i}=\infty$, $\lim_{i\rightarrow \infty } x_{i}=\infty$), and it satisfies the inequality

$$\begin{aligned} \varphi (t, x)&\leq a(t, x)+g(t, x) \int _{t_{0}}^{t} \int _{x_{0}}^{x}b( \xi , \eta )\varphi ^{m}( \xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad {} + \sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)} \gamma _{i}\varphi ^{m}\bigl(t_{i}^{-}, x_{i}^{-}\bigr) \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int ^{t_{i}- \tau _{i}}_{t_{i}-\sigma _{i}} \int ^{x_{i}-\delta _{i}}_{x_{i}- \lambda _{i}} \varphi (\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta , \end{aligned}$$

(26)

$m>0$, $m\neq 1$, wherea, b, $\gamma _{i}$, $\beta _{i}$satisfy the conditions of Theorem 2.1. Then, for$(t, x)\in \varOmega$, $k=1, 2, \ldots$ , the following estimates hold:

$$\begin{aligned}& \begin{aligned}[b] \varphi (t, x)&\leq a(t, x)g(t, x)\prod_{i=1}^{k-1}C_{i} \\&\quad {}\cdot \biggl[1+(1-m) \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )g^{m}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \biggr]^{ \frac{1}{1-m}}, \end{aligned} \\& \begin{aligned} C_{i}&=\bigl(1+\gamma _{i} a^{m-1}(t_{i}, x_{i})\bigr)g^{m}(t_{i}, x_{i})\\ &\quad {}\cdot \biggl[1+(1-m) \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}}b(\xi , \eta )a^{m-1}(\xi , \eta )g^{m}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d} \eta \biggr]^{\frac{1}{1-m}} \\ &\quad{}+\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}- \delta _{i}}^{x_{i}-\lambda _{i}} \biggl[1+(1-m) \int _{t_{i-1}}^{\xi } \int _{x_{i-1}}^{ \eta }b(\tau , s)a^{m-1}(\tau , s)g^{m}(\tau , s) \,\mathrm{d}\tau \,\mathrm{d}s\biggr]^{\frac{1}{1-m}} \,\mathrm{d}\xi \,\mathrm{d}\eta , \end{aligned} \\& C_{0}=1, \end{aligned}$$

(27)

if$0< m<1$; and

$$\begin{aligned} \varphi (t, x)&\leq a(t, x)g(t, x)\sum_{i=1}^{k-1} \biggl[\bigl(1+\gamma _{i}g(t_{i}, x_{i})\bigr) \exp \biggl[ \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}} b(\xi , \eta )g(\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d}\eta \biggr] \\ &\quad{}+\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}- \delta _{i}}^{x_{i}-\lambda _{i}} \exp \biggl[ \int _{t_{i-1}}^{\xi } \int _{x_{i-1}}^{ \eta } b(\tau , s)g(\tau , s)\,\mathrm{d} \tau \,\mathrm{d}s\biggr]\,\mathrm{d} \xi \,\mathrm{d}\eta \biggr], \end{aligned}$$

(28)

if$m=1$; and

$$\begin{aligned}& \begin{aligned}[b] \varphi (t, x)&\leq a(t, x)g(t, x)\prod_{i=1}^{k} D_{i}^{m^{k-i}} \Biggl[1-(m-1) \Biggl(\prod _{i=1}^{k}D_{i-1}^{m^{k-i}} \Biggr)^{m-1}\\ &\quad {}\times \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )g^{m}(\xi , \eta ) \,\mathrm{d}\xi \,\mathrm{d}\eta \Biggr]^{\frac{-1}{m-1}}, \end{aligned} \\& \begin{aligned} D_{i}&=(1+\gamma _{i} \bigl(a(t_{i}, x_{i})g(t_{i}, x_{i}) \bigr)^{m-1}\\ &\quad {}\cdot \Biggl[1-(m-1) \Biggl(\prod _{j=1}^{i}D_{j-1}^{m^{i-j}} \Biggr)^{m-1} \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}}b(\xi , \eta )a^{m-1}( \xi , \eta )g^{m}(\xi , \eta ) \,\mathrm{d}\xi \,\mathrm{d}\eta \Biggr]^{- \frac{m}{m-1}} \\ &\quad{}+\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}- \delta _{i}}^{x_{i}-\lambda _{i}} \Biggl[1-(m-1) \Biggl(\prod _{j=1}^{i}D_{j-1}^{m^{i-j}} \Biggr)^{m-1}\\&\quad{}\times \int _{t_{i-1}}^{\xi } \int _{x_{i-1}}^{\eta }b(\tau , s)a^{m-1}( \tau , s)g^{m}(\tau , s) \,\mathrm{d}\tau \,\mathrm{d}s\Biggr]^{-\frac{1}{m-1}} \,\mathrm{d}\xi \,\mathrm{d}\eta , \end{aligned} \end{aligned}$$

(29)

if$m>1$with$D_{0}=1$and$(t, x)\in \varOmega$satisfying

$$\begin{aligned} \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )g^{m}(\xi , \eta ) \,\mathrm{d}\xi \,\mathrm{d}\eta \leq \frac{1}{(m-1)(\prod_{i=1}^{k}D_{i-1}^{m^{k-i}})^{m-1}}. \end{aligned}$$

Proof

Since $a(t, x)>0$, $g(t, x)>0$, we get

$$\begin{aligned} \frac{\varphi (t, x)}{a(t, x)}&\leq g(t, x) \biggl[1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta ) \frac{\varphi ^{m}(\xi , \eta )}{a(\xi , \eta )}\, \mathrm{d}\xi \,\mathrm{d}\eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i} \frac{\varphi ^{m}(t_{i}^{-}, x_{i}^{-})}{a(t_{i}^{-}, x_{i}^{-})} \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}}\frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr]. \end{aligned}$$

(30)

Set

$$\begin{aligned} W(t, x)&= 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta ) \frac{\varphi ^{m}(\xi , \eta )}{a(\xi , \eta )}\, \mathrm{d}\xi \,\mathrm{d}\eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i} \frac{\varphi ^{m}(t_{i}^{-}, x_{i}^{-})}{a(t_{i}^{-}, x_{i}^{-})} \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}}\frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\,\mathrm{d}\xi \, \mathrm{d}\eta , \end{aligned}$$

(31)

$W(t_{0}, x)=W(t, x_{0})=1$, $\varphi (t, x)\leq a(t, x)g(t, x)W(t, x)$, then

$$\begin{aligned} W(t, x)&\leq 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )a^{m-1}( \xi , \eta )g^{m}(\xi , \eta ) \biggl[ \frac{\varphi (\xi , \eta )}{a(\xi , \eta )} \biggr]^{m} \,\mathrm{d} \xi \,\mathrm{d}\eta \,\mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i}a^{m-1}(t_{i}, x_{i})g^{m}(t_{i}, x_{i}) \biggl[ \frac{\varphi (t_{i}^{-}, x_{i}^{-})}{a(t_{i}^{-}, x_{i}^{-})} \biggr]^{m} \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}} \frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\,\mathrm{d} \xi \, \mathrm{d}\eta \\ &\leq 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )g^{m}(\xi , \eta )W^{m}(\xi , \eta )\,\mathrm{d}\xi \, \mathrm{d} \eta \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i}a^{m-1}(t_{i}, x_{i})g^{m}(t_{i}, x_{i})\bigl[W \bigl(t_{i}^{-}, x_{i}^{-}\bigr) \bigr]^{m} \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}} g(\xi , \eta )W(\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d} \eta . \end{aligned}$$

(32)

Using the procedure for $W(t, x)$ in Theorem 2.2, it is possible to obtain for $W(t, x)$ the following estimates:

$$\begin{aligned}& W(t, x)\leq \prod_{i=1}^{k-1}C_{i} \cdot \biggl[1+(1-m) \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )g^{m}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta \biggr]^{\frac{1}{1-m}}, \\& \begin{aligned} C_{i}&=\bigl(1+\gamma _{i} a^{m-1}(t_{i}, x_{i})\bigr)g^{m}(t_{i}, x_{i})\\&\quad{}\cdot \biggl[1+(1-m) \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}}b(\xi , \eta )a^{m-1}(\xi , \eta )g^{m}(\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d} \eta \biggr]^{\frac{1}{1-m}} \\ &\quad{}+\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}- \delta _{i}}^{x_{i}-\lambda _{i}} g(\tau , s)\\&\quad{}\times \biggl[1+(1-m) \int _{t_{i-1}}^{ \xi } \int _{x_{i-1}}^{\eta }b(\tau , s)a^{m-1}(\tau , s)g^{m}(\tau , s) \,\mathrm{d}\tau \,\mathrm{d}s\biggr]^{\frac{1}{1-m}} \,\mathrm{d}\xi \,\mathrm{d} \eta , \quad C_{0}=1, \end{aligned} \end{aligned}$$

(33)

if $0< m\leq 1$;

$$\begin{aligned} W(t, x)&\leq \sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)} \biggl[\bigl(1+ \gamma _{i}g(t_{i}, x_{i})\bigr)\exp \biggl[ \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}} b(\xi , \eta )g(\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d}\eta \biggr] \\ &\quad{}+\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}- \delta _{i}}^{x_{i}-\lambda _{i}} g(\tau , s)\exp \biggl[ \int _{t_{i-1}}^{ \xi } \int _{x_{i-1}}^{\eta } b(\tau , s)g(\tau , s)\,\mathrm{d} \tau \,\mathrm{d}s\biggr]\,\mathrm{d}\xi \,\mathrm{d}\eta \biggr], \end{aligned}$$

(34)

if $m=1$;

$$\begin{aligned}& \begin{aligned}[b] W(t, x)&\leq \prod_{i=1}^{k} D_{i}^{m^{k-i}} \Biggl[1-(m-1) \Biggl( \prod _{i=1}^{k}D_{i-1}^{m^{k-i}} \Biggr)^{m-1}\\&\quad{}\times \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )g^{m}(\xi , \eta ) \,\mathrm{d}\xi \,\mathrm{d}\eta \Biggr]^{-\frac{1}{m-1}}, \end{aligned} \\& \begin{aligned} D_{i} &=\bigl(1+\gamma _{i} \bigl(a(t_{i}, x_{i})g(t_{i}, x_{i})\bigr)\bigr)^{m-1} \\&\quad{}\cdot \Biggl[1-(m-1) \Biggl(\prod_{j=1}^{i}D_{j-1}^{m^{i-j}} \Biggr)^{m-1} \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}}b(\xi , \eta )a^{m-1}(\xi , \eta )g^{m}(\xi , \eta ) \,\mathrm{d}\xi \,\mathrm{d}\eta \Biggr]^{-\frac{m}{m-1}} \\ &\quad{}+\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}- \delta _{i}}^{x_{i}-\lambda _{i}} g(\tau , s) \Biggl[1-(m-1) \Biggl( \prod_{j=1}^{i}D_{j-1}^{m^{i-j}} \Biggr)^{m-1}\\&\quad{}\times \int _{t_{i-1}}^{\xi } \int _{x_{i-1}}^{\eta }b(\tau , s)a^{m-1}( \tau , s)g^{m}(\xi , \eta ) \,\mathrm{d}\tau \,\mathrm{d}s \Biggr]^{ \frac{-1}{m-1}}\,\mathrm{d}\xi \,\mathrm{d}\eta , \end{aligned} \end{aligned}$$

(35)

if $m>1$ with $D_{0}=1$ and $\forall (t, x)\in \varOmega$:

$$\begin{aligned} \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x}b(\xi , \eta )a^{m-1}(\xi , \eta )g^{m}(\xi , \eta ) \,\mathrm{d}\xi \,\mathrm{d}\eta \leq \frac{1}{(m-1) (\prod_{i=1}^{k}D_{i-1}^{m^{k-i}} )^{m-1}}. \end{aligned}$$

From (33)–(35) estimates (28)–(29) for the function φ will follow. □

3 Inequalities with retardation

Let us define a class of functions in the ℑ-class of continuous functions $\sigma (t)$ as retardation, and for $\sigma (t)$ the following estimates hold:

$(a_{1})$:

$\sigma (t)\leq t$, $\forall t\in R_{+}$, $R_{+}:=[0, +\infty )$;

$(a_{2})$:

$\lim_{t\rightarrow +\infty }\sigma (t)=+\infty$;

$(a_{3})$:

$\sigma (t)$ is nondecreasing.

Theorem 3.1

Let$\sigma \in \Im$and$\varphi (t, x)$satisfy certain inequality

$$\begin{aligned} \varphi (t, x)&\leq a(t, x)+g(t, x) \int _{t_{0}}^{t} \int _{x_{0}}^{x}b( \xi , \eta )\varphi \bigl(\sigma ( \xi ), \sigma (\eta )\bigr)\,\mathrm{d}\xi \,\mathrm{d}\eta +\sum _{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i} \varphi ^{m} \bigl(t_{i}^{-}, x_{i}^{-}\bigr) \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i}\cdot \int ^{t_{i}- \tau _{i}}_{t_{i}-\sigma _{i}} \int ^{x_{i}-\delta _{i}}_{x_{i}- \lambda _{i}} \varphi (\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta , \end{aligned}$$

(36)

with$m>0$, and the functionsφ, a, g, bsatisfy the conditions of Theorem 2.3, $\gamma _{i}, \beta _{i}=\mathrm{const}\geq 0$. Then, for$k=1, 2, \ldots$ , $\varphi (t, x)$, the following estimates are valid:

$$\begin{aligned}& \varphi (t, x)\leq a(t, x)g(t, x)\prod_{i=1}^{k-1}S_{i} \exp \biggl[ \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x} \mathcal{F}\bigl(\sigma (\xi ), \sigma (\eta )\bigr)\,\mathrm{d}\xi \,\mathrm{d}\eta \biggr], \\& \begin{aligned} S_{i}&=\bigl(1+\gamma _{i}a^{m-1}(t_{i}, x_{i})g^{m}(t_{i}, x_{i})\bigr) \exp \biggl[ \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}}\mathcal{F}\bigl( \sigma (\xi ), \sigma (\eta )\bigr) \,\mathrm{d}\xi \,\mathrm{d}\eta \biggr] \\ &\quad{}+\beta _{i}\cdot \int ^{t_{i}-\tau _{i}}_{t_{i}-\sigma _{i}} \int ^{x_{i}- \delta _{i}}_{x_{i}-\lambda _{i}} g(\tau , s)\exp \biggl[ \int _{t_{i-1}}^{ \xi } \int _{x_{i-1}}^{\eta } \mathcal{F}\bigl(\sigma (\tau ), \sigma (s)\bigr) \,\mathrm{d}\tau \,\mathrm{d}s \biggr]\,\mathrm{d}\xi \, \mathrm{d}\eta ,\quad S_{0}=1, \end{aligned} \end{aligned}$$

(37)

if$0< m\leq 1$;

$$\begin{aligned}& \varphi (t, x)\leq a(t, x)g(t, x)\prod_{i=1}^{k-1}T_{i}^{m^{k-i-1}} \exp \biggl[ \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x} \mathcal{F}\bigl( \sigma (\xi ), \sigma (\eta )\bigr)\,\mathrm{d}\xi \,\mathrm{d}\eta \biggr], \\& \begin{aligned} T_{i}&=\bigl(1+\gamma _{i}a^{m-1}(t_{i}, x_{i})g^{m}(t_{i}, x_{i})\bigr) \exp \biggl[m \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}}\mathcal{F}\bigl( \sigma (\xi ), \sigma (\eta )\bigr) \,\mathrm{d}\xi \,\mathrm{d}\eta \biggr] \\ &\quad{}+\beta _{i}\cdot \int ^{t_{i}-\tau _{i}}_{t_{i}-\sigma _{i}} \int ^{x_{i}- \delta _{i}}_{x_{i}-\lambda _{i}} g(\tau , s)\exp \biggl[ \int _{t_{i-1}}^{ \xi } \int _{x_{i-1}}^{\eta } \mathcal{F}\bigl(\sigma (\tau ), \sigma (s)\bigr) \,\mathrm{d}\tau \,\mathrm{d}s \biggr]\,\mathrm{d}\xi \, \mathrm{d}\eta ,\quad T_{0}=1, \end{aligned} \end{aligned}$$

(38)

if$m\geq 1$. Here, the function$\mathcal{F}(\sigma (\xi ), \sigma (\eta ))$is defined by

$$\begin{aligned} \mathcal{F}\bigl(\sigma (\xi ), \sigma (\eta )\bigr)= \frac{b(\xi , \eta )a(\sigma (\xi ), \sigma (\eta )) g(\sigma (\xi ), \sigma (\eta ))}{a(\xi , \eta )}. \end{aligned}$$

Proof

Because of $a(t, x)>0$, we get

$$\begin{aligned} \frac{\varphi (t, x)}{a(t, x)}&\leq g(t, x) \biggl[1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta ) \frac{\varphi (\sigma (\xi ), \sigma (\eta ))}{a(\xi , \eta )} \, \mathrm{d}\xi \,\mathrm{d}\eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i} \frac{\varphi ^{m}(t_{i}^{-}, x_{i}^{-})}{a(t_{i}^{-}, x_{i}^{-})} \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}}\frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr]. \end{aligned}$$

(39)

Set

$$\begin{aligned} W(t, x)&= 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta ) \frac{\varphi (\sigma (\xi ), \sigma (\eta ))}{a(\xi , \eta )} \, \mathrm{d}\xi \,\mathrm{d}\eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i} \frac{\varphi ^{m}(t_{i}^{-}, x_{i}^{-})}{a(t_{i}^{-}, x_{i}^{-})} \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}}\frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\,\mathrm{d}\xi \, \mathrm{d}\eta , \end{aligned}$$

(40)

$W(t_{0}, x)=W(t, x_{0})=1$, $\varphi (t, x)\leq a(t, x)g(t, x)W(t, x)$, then

$$\begin{aligned} W(t, x)&\leq 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x} \frac{b(\xi , \eta )a(\sigma (\xi ), \sigma (\eta ))g(\sigma (\xi ), \sigma (\eta ))}{a(\xi , \eta )} \cdot W\bigl(\sigma (\xi ), \sigma (\eta )\bigr)\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i}a^{m-1}(t_{i}, x_{i})g^{m}(t_{i}, x_{i})\bigl[W \bigl(t_{i}^{-}, x_{i}^{-}\bigr) \bigr]^{m} \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}} g(\xi , \eta )W(\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d} \eta . \end{aligned}$$

(41)

Using the result of Theorem 2.1 for (41), $k=1, 2, \ldots$ , we could obtain certain estimates:

$$\begin{aligned}& W(t, x)\leq \prod_{i=1}^{k-1}S_{i} \exp \biggl[ \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x} \mathcal{F}\bigl(\sigma (\xi ), \sigma (\eta )\bigr) \,\mathrm{d}\xi \,\mathrm{d}\eta \biggr], \\& \begin{aligned} S_{i}&=\bigl(1+\gamma _{i}a^{m-1}(t_{i}, x_{i})g^{m}(t_{i}, x_{i})\bigr) \exp \biggl[ \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}}\mathcal{F}\bigl( \sigma (\xi ), \sigma (\eta )\bigr) \,\mathrm{d}\xi \,\mathrm{d}\eta \biggr] \\ &\quad{}+\beta _{i}\cdot \int ^{t_{i}-\tau _{i}}_{t_{i}-\sigma _{i}} \int ^{x_{i}- \delta _{i}}_{x_{i}-\lambda _{i}} g(\tau , s)\exp \biggl[ \int _{t_{i-1}}^{ \xi } \int _{x_{i-1}}^{\eta } \mathcal{F}\bigl(\sigma (\tau ), \sigma (s)\bigr) \,\mathrm{d}\tau \,\mathrm{d}s \biggr]\,\mathrm{d}\xi \, \mathrm{d}\eta ,\quad S_{0}=1, \end{aligned} \end{aligned}$$

(42)

if $0< m\leq 1$;

$$\begin{aligned}& W(t, x)\leq \prod_{i=1}^{k-1}T_{i}^{m^{k-i-1}} \exp \biggl[ \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x} \mathcal{F}\bigl(\sigma (\xi ), \sigma (\eta )\bigr) \,\mathrm{d}\xi \,\mathrm{d}\eta \biggr], \\& \begin{aligned} T_{i}&=\bigl(1+\gamma _{i}a^{m-1}(t_{i}, x_{i})g^{m}(t_{i}, x_{i})\bigr) \exp \biggl[m \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}}\mathcal{F}\bigl( \sigma (\xi ), \sigma (\eta )\bigr) \,\mathrm{d}\xi \,\mathrm{d}\eta \biggr] \\ &\quad{}+\beta _{i}\cdot \int ^{t_{i}-\tau _{i}}_{t_{i}-\sigma _{i}} \int ^{x_{i}- \delta _{i}}_{x_{i}-\lambda _{i}} g(\tau , s)\exp \biggl[ \int _{t_{i-1}}^{ \xi } \int _{x_{i-1}}^{\eta } \mathcal{F}\bigl(\sigma (\tau ), \sigma (s)\bigr) \,\mathrm{d}\tau \,\mathrm{d}s \biggr]\,\mathrm{d}\xi \, \mathrm{d}\eta ,\quad T_{0}=1, \end{aligned} \end{aligned}$$

(43)

if $m\geq 1$. From (42)–(43) and the inequality $\varphi (t, x)\leq a(t, x)g(t, x)W(t, x)$, the result of Theorem 3.1 follows. □

Theorem 3.2

Let us suppose that all the conditions of Theorem 3.1are fulfilled and the function$\varphi (t, x)$satisfies a certain inequality

$$\begin{aligned} \varphi (t, x)&\leq a(t, x)+g(t, x) \int _{t_{0}}^{t} \int _{x_{0}}^{x}b( \xi , \eta )\varphi ^{m} \bigl(\sigma (\xi ), \sigma (\eta )\bigr)\,\mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad {}+\sum _{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i} \varphi ^{m} \bigl(t_{i}^{-}, x_{i}^{-}\bigr) \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i}\cdot \int ^{t_{i}- \tau _{i}}_{t_{i}-\sigma _{i}} \int ^{x_{i}-\delta _{i}}_{x_{i}- \lambda _{i}} \varphi (\xi , \eta )\,\mathrm{d}\xi \,\mathrm{d}\eta , \end{aligned}$$

(44)

with$m>0$.

Then$\forall (t, x)\in \varOmega$, the following estimates hold:

$$\begin{aligned}& \varphi (t, x)\leq a(t, x)g(t, x)\prod_{i=1}^{k-1}X_{i} [\biggl[1+(1-m) \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x} \mathcal{K}\bigl(\sigma (\xi ), \sigma (\eta )\bigr)\,\mathrm{d}\xi \,\mathrm{d}\eta \biggr], \\& \begin{aligned} X_{i}&=\bigl(1+\gamma _{i} a^{m-1}(t_{i}, x_{i})g^{m}(t_{i}, x_{i})\bigr)\cdot \biggl[1+(1-m) \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}} \mathcal{K}\bigl(\sigma (\xi ), \sigma (\eta )\bigr) \,\mathrm{d}\xi \,\mathrm{d} \eta \biggr]^{\frac{1}{1-m}} \\ &\quad{}+\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}- \delta _{i}}^{x_{i}-\lambda _{i}} g(\xi , \eta ) \biggl[1+(1-m) \int _{t_{i-1}}^{ \xi } \int _{x_{i-1}}^{\eta }\mathcal{K}\bigl(\sigma (\tau ), \sigma (s)\bigr) \,\mathrm{d}\tau \,\mathrm{d}s \biggr]^{\frac{1}{1-m}}\,\mathrm{d} \xi \,\mathrm{d}\eta , \end{aligned} \\& X_{0}=1, \end{aligned}$$

(45)

if$0< m\leq 1$;

$$\begin{aligned} \varphi (t, x)&\leq a(t, x)g(t, x)\prod_{i=1}^{k-1} \biggl\{ \bigl(1+ \gamma _{i}g(t_{i}, x_{i}) \bigr) \\ &\quad {}\times\exp \biggl[ \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}} b(\xi , \eta )g\bigl(\sigma (\xi ), \sigma (\eta )\bigr) \frac{a(\sigma (\xi ), \sigma (\eta ))}{a(\xi , \eta )}\,\mathrm{d}\xi \,\mathrm{d}\eta \biggr] \\ &\quad{}+\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}- \delta _{i}}^{x_{i}-\lambda _{i}} \exp \biggl[ \int _{t_{i-1}}^{\xi } \int _{x_{i-1}}^{\eta } b(\tau , s)g\bigl(\sigma (\tau ), \sigma (s)\bigr) \\ &\quad {}\times \frac{a(\sigma (\tau ), \sigma (s))}{a(\tau , s)}\,\mathrm{d}\tau \,\mathrm{d}s \biggr] \, \mathrm{d}\xi \,\mathrm{d}\eta \biggr\} , \end{aligned}$$

(46)

if$m=1$;

$$\begin{aligned}& \begin{aligned}[b] \varphi (t, x)&\leq a(t, x)g(t, x)\prod_{i=1}^{k} Y_{i}^{m^{k-i}} \Biggl[1-(m-1) \Biggl(\prod _{i=1}^{k}Y_{i-1}^{m^{k-i}} \Biggr)^{m-1}\\&\quad{}\times \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x}\mathcal{K}\bigl(\sigma (\xi ), \sigma (\eta )\bigr) \,\mathrm{d}\xi \,\mathrm{d}\eta \Biggr]^{-\frac{1}{m-1}}, \end{aligned} \\& \begin{aligned} Y_{i}&=\bigl(1+\gamma _{i} a^{m-1}(t_{i}, x_{i})g^{m-1}(t_{i}, x_{i})\bigr)\\&\quad{} \cdot \Biggl[1-(m-1) \Biggl(\prod _{j=1}^{i}Y_{j-1}^{m^{i-j}} \Biggr)^{m-1} \int _{t_{i-1}}^{t_{i}} \int _{x_{i-1}}^{x_{i}}\mathcal{K}\bigl(\sigma ( \xi ), \sigma (\eta )\bigr) \,\mathrm{d}\xi \,\mathrm{d}\eta \Biggr]^{- \frac{m}{m-1}} \\ &\quad{}+\beta _{i} \int _{t_{i}-\tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}- \delta _{i}}^{x_{i}-\lambda _{i}} \Biggl[1-(m-1) \Biggl(\prod _{j=1}^{i}Y_{j-1}^{m^{i-j}} \Biggr)^{m-1}\\&\quad{}\times \int _{t_{i-1}}^{\xi } \int _{x_{i-1}}^{\eta } \mathcal{K}\bigl(\sigma (\tau ), \sigma (s)\bigr) \,\mathrm{d}\tau \,\mathrm{d}s\Biggr]^{- \frac{1}{m-1}}\,\mathrm{d} \xi \,\mathrm{d}\eta ,\quad Y_{0}=1, \end{aligned} \end{aligned}$$

(47)

if$m>1$, $\forall (t, x)\in \varOmega$:

$$\begin{aligned} \int _{t_{k-1}}^{t} \int _{x_{k-1}}^{x}b(\xi , \eta )\mathcal{K}\bigl( \sigma (\xi ), \sigma (\eta )\bigr) \,\mathrm{d}\xi \,\mathrm{d}\eta \leq \frac{1}{(m-1) (\prod_{i=1}^{k}D_{i-1}^{m^{k-i}} )^{m-1}}. \end{aligned}$$

Here, the function$\mathcal{K}(\sigma (\xi ), \sigma (\eta ))$is defined by

$$\begin{aligned} \mathcal{K}\bigl(\sigma (\xi ), \sigma (\eta )\bigr)= \frac{b(\xi , \eta )a^{m}(\sigma (\xi ), \sigma (\eta )) g^{m}(\sigma (\xi ), \sigma (\eta ))}{a(\xi , \eta )}. \end{aligned}$$

Proof

Due to $g(t, x)\geq 1$, the following inequality is valid:

$$\begin{aligned} \frac{\varphi (t, x)}{a(t, x)}&\leq g(t, x) \biggl[1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta ) \frac{\varphi ^{m}(\sigma (\xi ), \sigma (\eta ))}{a(\xi , \eta )} \, \mathrm{d}\xi \,\mathrm{d}\eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i} \frac{\varphi ^{m}(t_{i}^{-}, x_{i}^{-})}{a(t_{i}^{-}, x_{i}^{-})} \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}}\frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\,\mathrm{d}\xi \, \mathrm{d}\eta \biggr]. \end{aligned}$$

(48)

Set

$$\begin{aligned} W(t, x)&= 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta ) \frac{\varphi ^{m}(\sigma (\xi ), \sigma (\eta ))}{a(\xi , \eta )} \, \mathrm{d}\xi \,\mathrm{d}\eta +\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i} \frac{\varphi ^{m}(t_{i}^{-}, x_{i}^{-})}{a(t_{i}^{-}, x_{i}^{-})} \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}}\frac{\varphi (\xi , \eta )}{a(\xi , \eta )}\,\mathrm{d}\xi \, \mathrm{d}\eta , \end{aligned}$$

(49)

thus

$$\begin{aligned} \varphi (t, x)\leq a(t, x)g(t, x)W(t, x), \end{aligned}$$

(50)

and $W(t_{0}, x)=W(t, x_{0})=1$, then

$$\begin{aligned} W(t, x)&\leq 1+ \int _{t_{0}}^{t} \int _{x_{0}}^{x}b(\xi , \eta ) \frac{a^{m}(\sigma (\xi ), \sigma (\eta ))g^{m}(\sigma (\xi ), \sigma (\eta ))W^{m}(\sigma (\xi ), \sigma (\eta ))}{a(\xi , \eta )} \, \mathrm{d}\xi \,\mathrm{d}\eta \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\gamma _{i}a^{m-1}(t_{i}, x_{i})g^{m}(t_{i}, x_{i})W^{m} \bigl(t_{i}^{-}, x_{i}^{-}\bigr) \\ &\quad{}+\sum_{(t_{0}, x_{0})< (t_{i}, x_{i})< (t, x)}\beta _{i} \int _{t_{i}- \tau _{i}}^{t_{i}-\sigma _{i}} \int _{x_{i}-\delta _{i}}^{x_{i}- \lambda _{i}} g(\xi , \eta )W(\xi , \eta )\, \mathrm{d}\xi \,\mathrm{d} \eta . \end{aligned}$$

(51)

Using the result of Theorem 2.3 for inequality (51) and taking into account estimate (50), we obtain estimates (45)–(47). □