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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.

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 [111] for details). For some summary papers, the readers are refereed to [1118]. In papers [1924], the authors give qualitative analysis of some integro-differential equations using certain integral inequalities; in papers [2, 2528], the authors give some integral inequalities with more than two independent variables; papers [24, 2933] 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 [812] 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.

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. □

Inequalities with retardation

Let us define a class of functions in the -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). □

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Acknowledgements

The authors sincerely thank the referees for a number of constructive suggestions and corrections which have significantly improved the contents and the exposition of the paper.

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The data and material used to support the findings of this study are available within the paper.

Funding

This research was partially supported by the NSF of China (Grant 11671227), NSF of Shandong Province (Grant ZR2019MA034), and Project of Qufu Normal University (201602028012, 18jg06).

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Xing, L., Qiu, D. & Zheng, Z. Some new integral inequalities of Wendorff type for discontinuous functions with integral jump conditions. J Inequal Appl 2020, 171 (2020). https://doi.org/10.1186/s13660-020-02437-2

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Keywords

  • Integral inequality
  • Impulsive differential inequality
  • Discontinuous function
  • Integro-sum inequality