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Delay dynamic double integral inequalities on time scales with applications

Abstract

In the article, we present the explicit bounds for three generalized delay dynamic Gronwall–Bellman type integral inequalities on time scales, which are the unification of continuous and discrete results. As applications, the boundedness for the solutions of delay dynamic integro-differential equations with initial conditions is discussed.

Introduction

Theory of time scales is the unification of both continuous and discrete analysis due to Stephen Hilger [1] in his PhD thesis, and it has wide applications in quantum calculus and difference and differential calculus. Due to its vast contributions in different branches of mathematics, it attracts the researchers and mathematicians to work on it. The role of inequalities cannot be forgot because they have huge contributions in the theory of differential equations [222], bivariate means [2331], calculation and optimization [3249], special functions [5069], probability and statistics [7075], and so on.

It is well known that the explicit bounds for integral inequalities on unknown functions are very useful in the qualitative and quantitative analysis for the solutions of differential, integral, and integro-differential equations [76100]. One of the most important inequalities in mathematics is the Gronwall inequality, it is an indispensable tool to obtain various estimates in the theory of ordinary and stochastic differential equations. The differential form of this inequality was proved by Gronwall [101] and its integral form was proved by Bellman [102]. Gronwall–Bellman integral inequalities have a lot of contributions to analyzing the behavior of solutions of the differential and integral equations. Recently, to dealt with the difficulties encountered in the solutions of differential and difference equations, the Gronwall–Bellman integral inequality on time scales has become one of the most important topics in mathematics research. A lot of work has been done in this area [103106]. To discuss the abstract analysis of the solutions of certain types of dynamic equations, Gronwall–Bellman delay type inequality on time scales can play a significant role, but it still cannot deal with the abstract analysis for the solutions of some more general differential and difference equations. This gives us the motivation to discuss some generalized delay integral inequalities on time scales. The main purpose of the article is to provide the explicit bounds for some delay double integral inequalities on time scales, present the characteristics of the solutions of certain integro-differential equations, and establish their discrete inequalities with initial conditions by use of the analytical and numerical methods in our obtained results.

Preliminaries

The time scale \(\mathbb{T}\) is a nonempty closed subset of the real numbers set \(\mathbb{R}\), its forward jump operator \(\sigma : \mathbb{T}\rightarrow \mathbb{T}\) for \(t\in \mathbb{T}\) is defined by \(\sigma (t):=\inf \{ r\in \mathbb{T}: r>t\}\) and its backward jump operator \(\rho :\mathbb{T}\rightarrow \mathbb{T}\) for \(t\in \mathbb{T}\) is defined by \(\rho (t):=\sup \{ r\in \mathbb{T}: r< t\}\). The derived set \(\mathbb{T}^{k}\) is defined as follows: if \(\mathbb{T}\) has a left-scattered maximum m, then \(\mathbb{T}^{k}= \mathbb{T}-\{m\}\); otherwise, \(\mathbb{T}^{k}=\mathbb{T}\).

Lemma 2.1

(see [107])

Let \(a, x\in \mathbb{T}^{k}\)with \(x>a\), \(f:\mathbb{T} \times \mathbb{T}^{k}\rightarrow \mathbb{R}\)be continuous at \((x,x)\), \(f^{\Delta }(x,\cdot )\)be the derivative offwith respect to its first variable such that it is right-dense continuous on \([a,\sigma (x)]\), and \(g(x)=\int _{a}^{x}f(x,\tau )\Delta \tau \). If, for each \(\epsilon >0\), there exists a neighborhoodUofxindependent of \(\tau \in [a,\sigma (x)]\)such that

$$ \bigl\vert f\bigl(\sigma (x),\tau \bigr)-f(y,\tau )-f^{\Delta }(x,\tau ) \bigl(\sigma (x)-y\bigr) \bigr\vert \leq \epsilon \bigl\vert \sigma (x)-y \bigr\vert $$

for all \(y\in U\), then

$$ g^{\Delta }(x)= \int _{a}^{x}f^{\Delta }(x,\tau )\Delta \tau +f\bigl(\sigma (x),x\bigr). $$

Theorem 2.2

(see [108])

Let \({f}_{1}: \mathbb{R\rightarrow \mathbb{R}}\)be continuous and differentiable and \({f}_{2}:\mathbb{T}\rightarrow \mathbb{R}\)be delta differentiable. Then \({f}_{1}^{\circ }{f}_{2}: \mathbb{T}\rightarrow \mathbb{R}\)is delta differentiable and

$$ \bigl({f}_{1}^{\circ }{f}_{2} \bigr)^{\Delta }({x})= \biggl\{ \int _{0}^{1}{f} _{1}^{\prime } \bigl({f}_{2}({x}) +h\mu ({x}){f}_{2}^{\Delta }({x}) \bigr)\,dh \biggr\} {f}_{2}^{\Delta }({x}). $$

Theorem 2.3

(see [108])

Let \({h}_{1}:\mathbb{T}\rightarrow \mathbb{R}\)be strictly increasing such that \(\overline{\mathbb{T}}={h}_{1}(\mathbb{T})\)is a time scale, and \({h}_{2}:\overline{{\mathbb{T}}}\rightarrow \mathbb{R}\). Then

$$ \bigl({h}_{2}^{\circ }{h}_{1} \bigr)^{\Delta }=\bigl({h}_{2}^{\overline{\Delta }} {^{\circ }} {h}_{1}\bigr){h}_{1}^{\Delta } $$

if \({h}_{1}^{\Delta }({x})\)and \({h}_{2}^{\overline{\Delta }}({h}_{1}( {x}))\)exist for \({x}\in \mathbb{T}^{k}\).

Main results

Throughout the section, \(\mathbb{R}\) represents the set of real numbers, \(\mathbb{R}_{0}^{+}=[0,\infty )\), \(\mathbb{R}_{1}^{+}=[1,\infty )\), \(\mathbb{Z}\) represents the set of integers, \(\mathbb{N}_{0}\) represents the set of nonnegative integers, \(\mathbb{A}_{j}\subseteq \mathbb{N} _{0}\) \((1\leq j\leq 2)\), \({x}_{0j}\in \mathbb{T}\), \(\mathbb{T}_{j}=[ {x}_{0j},\infty )_{\mathbb{T}}\subseteq {\mathbb{T}}^{k}\) is the time scale, \(\rho _{ji}\) is the backward jump operator, \({X}^{\Delta y_{i}}(y _{1},y_{2},\ldots ,y_{n})\) \((1\leq i\leq n)\) is the partial delta-derivative of X with respect to its ith variable and \({\Delta y_{i}}{X}(y_{1},y_{2},\ldots ,y_{n})\) is the forward difference of X with respect to its ith variable.

Theorem 3.1

Let \(u, r_{i}, a_{j}:\mathbb{T}_{1}\times \mathbb{T}_{2}\rightarrow \mathbb{R}_{0}^{+}\)and \(f_{i}, g_{i}, f^{\Delta {x}_{1}}_{i}: \mathbb{T}_{1}^{2}\times \mathbb{T}_{2}^{2}\rightarrow \mathbb{R}_{0} ^{+}\)be nonnegative and right-dense continuous such that \(a_{j}\)is nondecreasing with respect to its each variable, let \(\gamma _{ji}: \mathbb{T}_{j}\rightarrow \mathbb{R}_{0}^{+}\)be nonnegative, nondecreasing, and right-dense continuous such that \(\gamma _{ji}( {x}_{j})\leq {x}_{j}\)and \(\gamma ^{\Delta }_{ji}({x}_{j})>0\), \(\mu _{ji}:\mathbb{T}_{j}\rightarrow \mathbb{T}\)such that \(\mu _{ji}( {x}_{j})\leq {x}_{j}\)and \(-\infty <\mathfrak{p}_{j}=\inf \{\min (\mu _{ji}({x}_{j}) ),{x}_{j}\in \mathbb{T}_{j}\}\leq {x}_{0j}\), \(\mathfrak{a}:([\mathfrak{p_{1}},{x}_{01}]\times [ \mathfrak{p_{2}},{x}_{02}])_{\mathbb{T}^{2}}\rightarrow \mathbb{R} _{0}^{+}\)be nonnegative and right-dense continuous, \(w, w_{j}: \mathbb{R}_{0}^{+}\rightarrow \mathbb{R}_{0}^{+}\)be nonnegative, nondecreasing, and continuous such that \(w(p)>0\), \(w_{j}(p)>0\)for \(p>0\)and

$$\begin{aligned}& w\bigl(u({x}_{1},{x}_{2})\bigr) \\& \quad \leq a_{1}({x}_{1},{x}_{2})+a_{2}({x}_{1}, {x}_{2}) \sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}( {x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} w _{1}\bigl(u\bigl( \mu _{1i}({t}_{1}),\mu _{2i}({t}_{2}) \bigr)\bigr) \\& \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{2}\bigl(u\bigl( \mu _{1i}({t}_{1}), \mu _{2i}({t}_{2})\bigr)\bigr) + \int _{{\gamma }_{1i}({x}_{01})} ^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}} g_{i}({t}_{1}, {m}_{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(u\bigl(\mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1} \biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2} \Delta {t}_{1} \end{aligned}$$
(3.1)

for \(({x}_{1},{x}_{2})\in \mathbb{T}_{1}\times \mathbb{T}_{2}\)with the initial condition

$$ \textstyle\begin{cases} w(u({x}_{1},{x}_{2}))=\mathfrak{a}({x}_{1},{x}_{2}), \quad {x} _{1}\in [\mathfrak{p}_{1},{x}_{01}]_{\mathbb{T}} \textit{ or } {x} _{2}\in [\mathfrak{p}_{2},{x}_{02}]_{\mathbb{T}}, \\ \mathfrak{a}(\mu _{1i}({x}_{1}),\mu _{2i}({x}_{2}))\leq a_{1}({x}_{1}, {x}_{2}) , \quad \mu _{1i}({x}_{1})\leq {x}_{01} \textit{ or } \mu _{2i}({x}_{2})\leq {x}_{02}. \end{cases} $$
(3.2)

Then

$$ u({x}_{1},{x}_{2})\leq w^{-1} \bigl({G}_{1}^{-1}\bigl({G}_{2}^{-1} \bigl({G}_{2}\bigl({b} _{1}({x}_{1},{x}_{2}) \bigr) +a_{2}({x}_{1},{x}_{2}){c}({x}_{1},{x}_{2}) \bigr)\bigr)\bigr) $$
(3.3)

for all \({x}_{01}\leq {x}_{1}\leq \tilde{x}_{1}\)and \({x}_{02}\leq {x}_{2}\leq \tilde{x}_{2}\)if

$$\begin{aligned}& {b}_{1}({x}_{1},{x}_{2})= {G}_{1} \bigl(a_{1}({x}_{1},{x}_{2}) \bigr) +a_{2}( {x}_{1},{x}_{2})\sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma } _{1i}({x_{1}})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x} _{2})}r_{i}({t}_{1},{t}_{2}) \Delta {t}_{2}\Delta {t}_{1}, \\& \begin{aligned} {c}({x}_{1},{x}_{2})&=\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})} ^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma } _{2i}({x}_{2})}f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\ &\quad {}\times \biggl(1+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}} g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2})\Delta {m}_{2}\Delta {m}_{1} \biggr)\Delta {t}_{2}\Delta {t}_{1}, \end{aligned} \\& {G}_{1}({s})= \int _{s_{1}}^{{s}}\frac{\Delta p}{w_{1}(w^{-1}(p))},\quad {s}>{s}_{1}>0 \textit{ with } {G}_{1}(\infty )=\infty , \\& {G}_{2}({s})= \int _{s_{2}}^{{s}}\frac{\Delta p}{w_{2}(w^{-1}({G}_{1} ^{-1}(p)))},\quad {s}>{s}_{2}>0 \textit{ with } {G}_{2}(\infty )=\infty , \end{aligned}$$

where \({G}_{1}^{-1}\)and \({G}_{2}^{-1}\)are respectively the inverse functions of \({G}_{1}\)and \({G}_{2}\), and \(\tilde{x}_{1}\), \(\tilde{x} _{2}\)are chosen such that

$$\begin{aligned}& {G}_{2}\bigl({b}_{1}({x}_{1},{x}_{2}) \bigr) +a_{2}({x}_{1},{x}_{2}){c}({x}_{1}, {x}_{2})\in \operatorname{Dom} \bigl({G}_{2}^{-1} \bigr), \\& {G}_{2}^{-1} \bigl({G}_{2} \bigl({b}_{1}({x}_{1},{x}_{2})\bigr) +a_{2}({x}_{1},{x} _{2}){c}({x}_{1},{x}_{2}) \bigr)\in \operatorname{Dom} \bigl({G}_{1}^{-1}\bigr), \\& {G}_{1}^{-1}\bigl({G}_{2}^{-1} \bigl({G}_{2}\bigl({b}_{1}({x}_{1},{x}_{2}) \bigr) +a_{2}( {x}_{1},{x}_{2}){c}({x}_{1},{x}_{2}) \bigr)\bigr)\in \operatorname{Dom} \bigl({w^{-1}}\bigr). \end{aligned}$$

Proof

For fixed numbers \(\bar{x}_{1}\in \mathbb{T}_{1}\), \(\bar{x}_{2}\in \mathbb{T}_{2}\) with \({x}_{01}\leq \bar{x}_{1}\leq \tilde{x}_{1}\) and \({x}_{02}\leq \bar{x}_{2}\leq \tilde{x}_{2}\), inequality (3.1) can be rewritten as follows:

$$\begin{aligned}& w\bigl(u({x}_{1},{x}_{2})\bigr) \\& \quad \leq a_{1}( \bar{x}_{1},\bar{x}_{2})+a_{2}( \bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})} ^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma } _{2i}({x}_{2})} w_{1}\bigl(u\bigl( \mu _{1i}({t}_{1}),\mu _{2i}({t}_{2}) \bigr)\bigr) \\& \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{2}\bigl(u\bigl( \mu _{1i}({t}_{1}), \mu _{2i}({t}_{2})\bigr)\bigr) + \int _{{\gamma }_{1i}({x}_{01})} ^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1}, {m}_{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(u\bigl(\mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1} \biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2} \Delta {t}_{1}. \end{aligned}$$

Let

$$\begin{aligned}& \xi _{1}({x}_{1},{x}_{2}) \\& \quad =a_{1}( \bar{x}_{1},\bar{x}_{2})+a_{2}( \bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})} ^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma } _{2i}({x}_{2})} w_{1}\bigl(u\bigl( \mu _{1i}({t}_{1}),\mu _{2i}({t}_{2}) \bigr)\bigr) \\& \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{2}\bigl(u\bigl( \mu _{1i}({t}_{1}), \mu _{2i}({t}_{2})\bigr)\bigr) + \int _{{\gamma }_{1i}({x}_{01})} ^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1}, {m}_{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(u\bigl(\mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1} \biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2} \Delta {t}_{1} \end{aligned}$$
(3.4)

for \(x_{1}\in [x_{01},\bar{x}_{1}]_{\mathbb{T}}\) and \(x_{2}\in [x_{02}, \bar{x}_{2}]_{\mathbb{T}}\). Then

$$ \xi _{1}({x}_{01},{x}_{2})=\xi _{1}({x}_{1},{x}_{02})=a_{1}( \bar{x}_{1}, \bar{x}_{2}) $$
(3.5)

and

$$ w\bigl(u({x}_{1},{x}_{2})\bigr)\leq \xi _{1}({x}_{1},{x}_{2}), \qquad u({x}_{1}, {x}_{2})\leq w^{-1}\bigl(\xi _{1}({x}_{1},{x}_{2}) \bigr). $$
(3.6)

If \(\mu _{1i}({x}_{1})\geq {x}_{01}\) and \(\mu _{2i}({x}_{2})\geq {x} _{02}\) for \(x_{1}\in [x_{01},\bar{x}_{1}]_{\mathbb{T}}\) and \(x_{2}\in [x_{02},\bar{x}_{2}]_{\mathbb{T}}\), then \(\mu _{1i}({x}_{1}) \in [x_{01},\bar{x}_{1}]_{\mathbb{T}}\), \(\mu _{2i}({x}_{1})\in [x_{02}, \bar{x}_{2}]_{\mathbb{T}}\), and

$$ u\bigl(\mu _{1i}({x}_{1}),\mu _{2i}({x}_{2}) \bigr) \leq w^{-1}\bigl(\xi _{1}\bigl(\mu _{1i}( {x}_{1}),\mu _{2i}({x}_{2}) \bigr)\bigr)\leq w^{-1}\bigl(\xi _{1}({x}_{1},{x}_{2}) \bigr). $$
(3.7)

On the other hand, if \(\mu _{1i}({x}_{1})\leq {x}_{01}\) or \(\mu _{2i}( {x}_{2})\leq {x}_{02}\), then from (3.2) we have

$$\begin{aligned} u\bigl(\mu _{1i}({x}_{1}),\mu _{2i}({x}_{2}) \bigr) =&w^{-1}\bigl(\mathfrak{a}\bigl(\mu _{1i}( {x}_{1}),\mu _{2i}({x}_{2})\bigr)\bigr) \\ \leq& w^{-1}\bigl(a_{1}({x}_{1},{x}_{2}) \bigr)\leq w^{-1}\bigl(\xi _{1}({x}_{1},{x} _{2})\bigr). \end{aligned}$$
(3.8)

It follows from (3.7) and (3.8) that

$$ u\bigl(\mu _{1i}({x}_{1}),\mu _{2i}({x}_{2}) \bigr)\leq w^{-1}\bigl(\xi _{1}({x}_{1}, {x}_{2})\bigr) $$
(3.9)

for \(x_{1}\in [x_{01},\bar{x}_{1}]_{\mathbb{T}}\) and \(x_{2}\in [x_{02}, \bar{x}_{2}]_{\mathbb{T}}\).

From Lemma 2.1, (3.4), and (3.9), one has

$$\begin{aligned}& {\xi _{1}}^{\Delta {x}_{1}}({x}_{1},{x}_{2}) \\ & \quad =a_{2}( \bar{x}_{1},\bar{x} _{2})\sum _{i=1}^{n}{{\gamma }_{1i}^{\Delta }({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}w_{1}\bigl(u\bigl(\mu _{1i}\bigl(\gamma _{1i}({x}_{1})\bigr),\mu _{2i}({t}_{2})\bigr)\bigr) \\ & \qquad {}\times \biggl[f_{i}\bigl(\sigma ({x}_{1}),{\gamma }_{1i}({x}_{1}),{x}_{2}, {t}_{2}\bigr) \biggl\{ w_{2}\bigl(u\bigl(\mu _{1i}\bigl(\gamma _{1i}({x}_{1})\bigr),\mu _{2i}( {t}_{2})\bigr)\bigr) \\ & \qquad {}+ \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}} g_{i}\bigl({\gamma }_{1i}({x} _{1}),{m}_{1},{t}_{2},{m}_{2} \bigr) \\ & \qquad {}\times w_{2}\bigl(u\bigl(\mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1} \biggr\} +r_{i} \bigl({\gamma }_{1i}({x}_{1}),{t}_{2}\bigr) \biggr] \Delta {t}_{2} \\ & \qquad {}+a_{2}(\bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \biggl[ \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}w_{1}\bigl(u\bigl(\mu _{1i}({t}_{1}),\mu _{2i}({t}_{2}) \bigr)\bigr) \\ & \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{2}\bigl(u\bigl( \mu _{1i}({t}_{1}), \mu _{2i}({t}_{2})\bigr)\bigr) \\ & \qquad {}+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) w_{2}\bigl(u\bigl(\mu _{1i}({m}_{1}),\mu _{2i}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1} \biggr\} \\ & \qquad {}+r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2} \biggr]^{\Delta x_{1}} \Delta {t}_{1} \end{aligned}$$

and

$$\begin{aligned}& {\xi _{1}}^{\Delta {x}_{1}}({x}_{1},{x}_{2}) \\ & \quad \leq a_{2}(\bar{x}_{1}, \bar{x}_{2})\sum _{i=1}^{n}{{\gamma }_{1i}^{\Delta }({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}w_{1} \bigl(w^{-1}\bigl( \xi _{1}\bigl(\gamma _{1i}({x}_{1}),{t}_{2}\bigr)\bigr)\bigr) \\ & \qquad {}\times \biggl[f_{i}\bigl(\sigma ({x}_{1}),{\gamma }_{1i}({x}_{1}),{x}_{2}, {t}_{2}\bigr) \biggl\{ w_{2}\bigl(w^{-1} \bigl(\xi _{1}\bigl(\gamma _{1i}({x}_{1}),{t}_{2} \bigr)\bigr)\bigr) \\ & \qquad {}+ \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}} g_{i}\bigl({\gamma }_{1i}({x} _{1}),{m}_{1},{t}_{2},{m}_{2} \bigr) \\ & \qquad {}\times w_{2}\bigl(w^{-1}\bigl(\xi _{1}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} +r_{i}\bigl({\gamma }_{1i}({x}_{1}),{t}_{2} \bigr) \biggr]\Delta {t}_{2} \\ & \qquad {}+a_{2}(\bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \biggl[ \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}w_{1} \bigl(w^{-1}\bigl( \xi _{1}({t}_{1},{t}_{2}) \bigr)\bigr) \\ & \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{2}\bigl(w ^{-1}\bigl(\xi _{1}({t}_{1},{t}_{2}) \bigr)\bigr) \\ & \qquad {}+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) w_{2}\bigl(w^{-1}\bigl( \xi _{1}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} \\ & \qquad {}+r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2} \biggr]^{\Delta x_{1}} \Delta {t}_{1}. \end{aligned}$$

Using the fact that \(w_{1}\), \(w^{-1}\), and \(\xi _{1}\) are nondecreasing, we have

$$\begin{aligned}& {\xi _{1}}^{\Delta {x}_{1}}({x}_{1},{x}_{2}) \\& \quad \leq a_{2}(\bar{x}_{1},\bar{x}_{2}){w_{1} \bigl(w^{-1}\bigl(\xi _{1}({x}_{1}, {x}_{2})\bigr)\bigr)} \sum_{i=1}^{n}{{ \gamma }_{1i}^{\Delta }({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} \biggl[f_{i}\bigl( \sigma ({x}_{1}),{\gamma }_{1i}({x}_{1}),{x}_{2},{t}_{2} \bigr) \\& \qquad {}\times \biggl\{ w_{2}\bigl(w^{-1}\bigl(\xi _{1}\bigl({\gamma }_{1i}({x}_{1}),{t}_{2} \bigr)\bigr)\bigr) + \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}} g_{i}\bigl({\gamma }_{1i}({x} _{1}),{m}_{1},{t}_{2},{m}_{2} \bigr) \\& \qquad {}\times w_{2}\bigl(w^{-1}\bigl(\xi _{1}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} +r_{i}\bigl({\gamma }_{1i}({x}_{1}),{t}_{2} \bigr) \biggr]\Delta {t}_{2} \\& \qquad {}+a_{2}(\bar{x}_{1},\bar{x}_{2}){w_{1} \bigl(w^{-1}\bigl(\xi _{1}({x}_{1},{x}_{2}) \bigr)\bigr)} \sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x} _{1})} \biggl[ \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times \biggl\{ w_{2}\bigl(w^{-1}\bigl(\xi _{1}({t}_{1},{t}_{2})\bigr)\bigr)+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(w^{-1}\bigl(\xi _{1}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2} \biggr]^{ \Delta x_{1}}\Delta {t}_{1}. \end{aligned}$$

Dividing both sides by \({w_{1}(w^{-1}(\xi _{1}({x}_{1},{x}_{2})))}\), we obtain

$$\begin{aligned}& \frac{{\xi _{1}}^{\Delta {{x}_{1}}}({x}_{1},{x}_{2})}{w_{1}(w^{-1}(\xi _{1}({x}_{1},{x}_{2})))} \\& \quad \leq a_{2}(\bar{x}_{1}, \bar{x}_{2})\sum_{i=1} ^{n}{{\gamma }_{1i}^{\Delta }({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})} ^{{\gamma }_{2i}({x}_{2})} \biggl[f_{i}\bigl( \sigma ({x}_{1}),{\gamma }_{1i}( {x}_{1}),{x}_{2},{t}_{2} \bigr) \\& \qquad {}\times \biggl\{ w_{2}\bigl(w^{-1}\bigl(\xi _{1}\bigl({\gamma }_{1i}({x}_{1}),{t}_{2} \bigr)\bigr)\bigr) + \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}} g_{i}\bigl({\gamma }_{1i}({x} _{1}),{m}_{1},{t}_{2},{m}_{2} \bigr) \\& \qquad {}\times w_{2}\bigl(w^{-1}\bigl(\xi _{1}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} +r_{i}\bigl({\gamma }_{1i}({x}_{1}),{t}_{2} \bigr) \biggr]\Delta {t}_{2} \\& \qquad {}+a_{2}(\bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \biggl[ \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} \biggl[f_{i}( {x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times \biggl\{ w_{2}\bigl(w^{-1}\bigl(\xi _{1}({t}_{1},{t}_{2})\bigr)\bigr)+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(w^{-1}\bigl(\xi _{1}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr] \Delta {t}_{2} \biggr]^{ \Delta x_{1}}\Delta {t}_{1} \\& \quad =a_{2}(\bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \biggl[ \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} \biggl[f_{i}( {x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times \biggl\{ w_{2}\bigl(w^{-1}\bigl(\xi _{1}({t}_{1},{t}_{2})\bigr)\bigr)+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(w^{-1}\bigl(\xi _{1}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2}\Delta {t}_{1} \biggr]^{\Delta x_{1}}. \end{aligned}$$

Integrating over \([{x}_{01},{{x}_{1}}]\), then using the definition of \({G}_{1}\) and (3.5), we get

$$\begin{aligned}& {G}_{1}\bigl(\xi _{1}({x}_{1},{x}_{2}) \bigr) \\& \quad \leq {G}_{1}\bigl(a_{1}(\bar{x}_{1}, \bar{x}_{2})\bigr) +a_{2}(\bar{x}_{1}, \bar{x}_{2})\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma } _{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x} _{2})}r_{i}({t}_{1},{t}_{2}) \Delta {t}_{2}\Delta {t}_{1} \\& \qquad {}+a_{2}(\bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}f_{i}({x} _{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times \biggl\{ w_{2}\bigl(w^{-1}\bigl(\xi _{1}({t}_{1},{t}_{2})\bigr)\bigr)+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(w^{-1}\bigl(\xi _{1}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2}\Delta {t}_{1} \\& \quad \leq {G}_{1}\bigl(a_{1}(\bar{x}_{1}, \bar{x}_{2})\bigr) +a_{2}(\bar{x}_{1}, \bar{x}_{2})\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma } _{1i}(\bar{x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}( \bar{x}_{2})}r_{i}({t}_{1},{t}_{2}) \Delta {t}_{2}\Delta {t}_{1} \\& \qquad {}+a_{2}(\bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}f_{i}({x} _{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times \biggl\{ w_{2}\bigl(w^{-1}\bigl(\xi _{1}({t}_{1},{t}_{2})\bigr)\bigr)+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(w^{-1}\bigl(\xi _{1}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2}\Delta {t}_{1} \\& \quad ={b}_{1}(\bar{x}_{1},\bar{x}_{2})+a_{2}( \bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} f_{i}({x} _{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times \biggl\{ w_{2}\bigl(w^{-1}\bigl(\xi _{1}({t}_{1},{t}_{2})\bigr)\bigr)+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(w^{-1}\bigl(\xi _{1}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2}\Delta {t}_{1}. \end{aligned}$$
(3.10)

Let

$$\begin{aligned} \zeta _{1}({x}_{1},{x}_{2}) =&{b}_{1}( \bar{x}_{1},\bar{x}_{2})+a_{2}( \bar{x}_{1},\bar{x}_{2}) \sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})} ^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma } _{2i}({x}_{2})}f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\ &{}\times \biggl\{ w_{2}\bigl(w^{-1}\bigl(\xi _{1}({t}_{1},{t}_{2})\bigr)\bigr)+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\ &{}\times w_{2}\bigl(w^{-1}\bigl(\xi _{1}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2}\Delta {t}_{1}. \end{aligned}$$
(3.11)

Then we have

$$ \zeta _{1}({x}_{01},{x}_{2})=\zeta _{1}({x}_{1},{x}_{02})={b}_{1}( \bar{x}_{1},\bar{x}_{2}) $$
(3.12)

and

$$ \xi _{1}({x}_{1},{x}_{2})\leq {G}_{1}^{-1}\bigl(\zeta _{1}({x}_{1},{x}_{2}) \bigr). $$
(3.13)

It follows from Lemma 2.1 and (3.11) that

$$\begin{aligned} {\zeta _{1}}^{\Delta {{x}_{1}}}({x}_{1},{x}_{2}) =&a_{2}( \bar{x}_{1}, \bar{x}_{2}) \sum _{i=1}^{n}{{\gamma }^{\Delta }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} f_{i}\bigl( \sigma ({x}_{1}),{{\gamma }_{1i}({x}_{1})},{x}_{2},{t}_{2} \bigr) \\ &{}\times \biggl\{ w_{2}\bigl(w^{-1}\bigl(\xi _{1}\bigl({{\gamma }_{1i}({x}_{1})},{t}_{2} \bigr)\bigr)\bigr) + \int _{{\gamma }_{1i}({x}_{01})}^{{{\gamma }_{1i}({x}_{1})}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}\bigl({{\gamma }_{1i}({x} _{1})},{m}_{1},{t}_{2},{m}_{2} \bigr) \\ &{}\times w_{2}\bigl(w^{-1}\bigl(\xi _{1}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2} \\ &{}+a_{2}(\bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})} ^{{\gamma }_{1i}({x}_{1})} \biggl[ \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} f_{i}({x} _{1},{t}_{1},{x}_{2},{t}_{2}) \\ &{}\times \biggl\{ w_{2}\bigl(w^{-1}\bigl(\xi _{1}({t}_{1},{t}_{2})\bigr)\bigr)+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\ &{}\times w_{2}\bigl(w^{-1}\bigl(\xi _{1}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2} \biggr]^{\Delta x_{1}}\Delta {t}_{1}. \end{aligned}$$

Making use of (3.13) and the fact that \(w_{2}\), \(w^{-1}\), \(G_{1}^{-1}\), and \(\zeta _{1}\) are nondecreasing, we get

$$\begin{aligned}& \frac{{\zeta _{1}}^{\Delta {{x}_{1}}}({x}_{1},{x}_{2})}{w_{2}(w^{-1}( {G}_{1}^{-1} (\zeta _{1}({x}_{1},{x}_{2}))))} \\& \quad \leq a_{2}(\bar{x}_{1}, \bar{x}_{2})\sum_{i=1}^{n} \biggl[{{\gamma }^{\Delta }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} f_{i}\bigl( \sigma ({x}_{1}),{{\gamma }_{1i}({x}_{1})},{x}_{2},{t}_{2} \bigr) \\& \qquad {}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{{\gamma }_{1i}( {x}_{1})}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}\bigl({{\gamma } _{1i}({x}_{1})},{m}_{1},{t}_{2},{m}_{2} \bigr) \Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2} \\& \qquad {}+ \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \biggl[ \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} f_{i}({x} _{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2}) \Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2} \biggr]^{\Delta x_{1}} \Delta {t}_{1} \biggr] \\& \quad =a_{2}(\bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \biggl[ \int _{{\gamma }_{1i}({x}_{01})} ^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} f_{i}({x} _{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2}) \Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2} \Delta {t}_{1} \biggr]^{\Delta x_{1}}. \end{aligned}$$

Integrating over \([{x}_{01},{{x}_{1}}]\), then using the definition of \({G}_{2}\) and (3.12), we get

$$\begin{aligned} {G}_{2}\bigl(\zeta _{1}({x}_{1},{x}_{2}) \bigr) \leq& {G}_{2}\bigl({b}_{1}(\bar{x}_{1}, \bar{x}_{2})\bigr)+a_{2}(\bar{x}_{1}, \bar{x}_{2}) \sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} f_{i}({x} _{1},{t}_{1},{x}_{2},{t}_{2}) \\ &{}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2}) \Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2} \Delta {t}_{1} \end{aligned}$$
(3.14)
$$\begin{aligned} =&{G}_{2}\bigl({b}_{1}(\bar{x}_{1}, \bar{x}_{2})\bigr) +a_{2}(\bar{x}_{1}, \bar{x} _{2}){c}({x}_{1},{x}_{2}). \end{aligned}$$
(3.15)

A combination of (3.6), (3.13), and (3.15) gives the desired result (3.3). □

Remark 3.2

Let \(\mathbb{T}=\mathbb{Z}\), \(a_{1}({x}_{1},{x}_{2})=c\), \(a_{2}({x} _{1},{x}_{2})=w_{2}=1\), \(\gamma _{ji}=\mu _{ji}=I\), \(n=1\), \(f_{i}({x} _{1},{t}_{1},{x}_{2},{t}_{2})=f({t}_{1},{t}_{2})\), \(g_{i}=r_{i}=0\). Then Theorem 3.1 coincides with Theorem 2.1 in [109]. Moreover, if \(w(u)=u^{p}\), then it coincides with Theorem 2.1 in [110].

Remark 3.3

Let \(\mathbb{T}=\mathbb{R}\), \(a_{1}({x}_{1},{x}_{2})=c\), \(a_{2}({x} _{1},{x}_{2})=1\), \(w_{1}=\mu _{ji}=I\), \(f_{i}({x}_{1},{t}_{1},{x}_{2}, {t}_{2})=f_{i}({t}_{1},{t}_{2})\), \(g_{i}=0\) (\(1\leq i\leq n\)). Then Theorem 3.1 leads to Theorem 2.3 in [111]. Moreover, if \(r_{i}=0\), then it reduces to Theorem 2.2 in [111].

Remark 3.4

Let \(\mathbb{T}=\mathbb{R}\), \(w_{1}(u)=u^{q}\), \(\mu _{ji}=I\), \(f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2})=f_{i}({t}_{1},{t}_{2})\), \(g_{i}=0\) \((1\leq i\leq n)\). Then Theorem 3.1 becomes Theorem 2.1 in [112].

Remark 3.5

Let \(n=1\), \(\gamma _{ji}=I\), \(w_{2}=1\), \(f_{i}({x}_{1},{t}_{1},{x}_{2}, {t}_{2})=f({t}_{1},{t}_{2})\), and \(g_{i}=r_{i}=0\). Theorem 3.1 coincides with Theorem 1 in [113].

Remark 3.6

Let \(\mathbb{T}=\mathbb{Z}\), \(w_{2}=1\), \(\gamma _{ji}=\mu _{ji}=I\), \(n=1\), \(f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2})=f({t}_{1},{t}_{2})\), and \(g_{i}=r_{i}=0\). Then Theorem 3.1 leads to Theorem 1 in [114].

Remark 3.7

Let \(\mathbb{T}=\mathbb{R}\), \(\mu _{ji}=I\), \(f_{i}({x}_{1},{t}_{1}, {x}_{2},{t}_{2})=f_{i}({t}_{1},{t}_{2})\), and \(g_{i}=0\). Then Theorem 3.1 reduces to Theorem 1 in [115].

Theorem 3.8

Let \(a_{j}\), \(r_{i}\), w, \(w_{j}\), \(f_{i}\), \(g_{i}\), \(\gamma _{ji}\), \(\mu _{ji}\), and \(\mathfrak{a}\)be defined as in Theorem 3.1such that \(w_{j+2}\)has the same conditions of \(w_{j}\), and let \(u:\mathbb{T} _{1}\times \mathbb{T}_{2}\rightarrow \mathbb{R}_{1}^{+}\)be a right-dense continuous function such that

$$\begin{aligned}& w\bigl(u({x}_{1},{x}_{2})\bigr) \\& \quad \leq a_{1}({x}_{1},{x}_{2})+a_{2}({x}_{1}, {x}_{2})\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma } _{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}( {x}_{2})} w_{1}\bigl(u\bigl( \mu _{1i}({t}_{1}),\mu _{2i}({t}_{2}) \bigr)\bigr) \\& \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2})w_{2} \bigl(u\bigl(\mu _{1i}( {t}_{1}),\mu _{2i}({t}_{2}) \bigr)\bigr) \biggl\{ w_{3}\bigl(u\bigl(\mu _{1i}({t}_{1}), \mu _{2i}( {t}_{2})\bigr)\bigr) \\& \qquad {}+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \times w_{3}\bigl(u\bigl( \mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} \\& \qquad {}+r_{i}({t}_{1},{t}_{2})w_{4} \bigl(\log \bigl(u\bigl(\mu _{1i}({t}_{1}),\mu _{2i}({t} _{2})\bigr)\bigr)\bigr) \biggr]\Delta {t}_{2}\Delta {t}_{1} \end{aligned}$$
(3.16)

for \(({x}_{1},{x}_{2})\in \mathbb{T}_{1}\times \mathbb{T}_{2}\)with initial condition (3.2). Then the following statements are true:

(1) If \(w_{2}({u})\geq w_{4}(\log ({u}))\), then

$$ u({x}_{1},{x}_{2})\leq w^{-1} \bigl({G}_{1}^{-1}\bigl({G}_{2}^{-1} \bigl({Q}_{1}^{-1} \bigl( {Q}_{1} \bigl({d}_{1}({x}_{1},{x}_{2}) \bigr)+a_{2}({x}_{1},{x}_{2}){c}({x}_{1}, {x}_{2})\bigr)\bigr)\bigr)\bigr) $$
(3.17)

for all \({x}_{01}\leq {x}_{1}\leq \tilde{x}_{1}\)and \({x}_{02}\leq {x}_{2}\leq \tilde{x}_{2}\);

(2) If \(w_{2}({u})< w_{4}(\log ({u}))\), then

$$ u({x}_{1},{x}_{2})\leq w^{-1} \bigl({G}_{1}^{-1}\bigl({G}_{3}^{-1} \bigl({Q}_{2}^{-1} \bigl( {Q}_{2} \bigl({d}_{2}({x}_{1},{x}_{2}) \bigr)+a_{2}({x}_{1},{x}_{2}){c}({x}_{1}, {x}_{2})\bigr)\bigr)\bigr)\bigr) $$
(3.18)

for all \({x}_{01}\leq {x}_{1}\leq \tilde{x}_{3}\)and \({x}_{02}\leq {x}_{2}\leq \tilde{x}_{4}\), where

$$\begin{aligned}& {d}_{j}({x}_{1},{x}_{2})={G}_{j+1} \bigl({G}_{1}\bigl(a_{1}({x}_{1},{x}_{2}) \bigr)\bigr)+a _{2}({x}_{1},{x}_{2}) \sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})} ^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma } _{2i}({x}_{2})} r_{i}({t}_{1},{t}_{2}) \Delta {t}_{2}\Delta {t}_{1}, \\& {G}_{3}(s)= \int _{s_{3}}^{s} \frac{\Delta p}{w_{4}(w^{-1}({G}_{1}^{-1}(p)))}, \quad s> s_{3}>0 \textit{ with } {G}_{3}(\infty )=\infty , \\& {Q}_{j}(s)= \int _{s_{3+j}}^{s}\frac{\Delta p}{w_{3}(w^{-1}({G}_{1}^{-1}( {G}_{j+1}^{-1}(p))))},\quad s> s_{3+j}>0 \textit{ with } {Q}_{j}(\infty )=\infty , \end{aligned}$$

c, \({G}_{1}\), and \({G}_{2}\)are defined as in Theorem 3.1, \({G}_{1}^{-1}\), \({G}_{2}^{-1}\), \({G}_{3}^{-1}\), and \({G}_{4}^{-1}\)are respectively the inverse functions of \({G}_{1}\), \({G}_{2}\), \({G}_{3}\), and \({G}_{4}\), and \(\tilde{x}_{1}\), \(\tilde{x}_{2}\), \(\tilde{x}_{3}\), and \(\tilde{x}_{4}\)are chosen such that

$$\begin{aligned}& {Q}_{j}\bigl({d}_{j}({x}_{1},{x}_{2}) \bigr) +a_{2}({x}_{1},{x}_{2}){c}({x}_{1}, {x}_{2})\in \operatorname{Dom} \bigl({Q}_{j}^{-1} \bigr), \\& {Q}_{j}^{-1} \bigl({Q}_{j} \bigl({d}_{j}({x}_{1},{x}_{2})\bigr) +a_{2}({x}_{1},{x} _{2}){c}({x}_{1},{x}_{2}) \bigr)\in \operatorname{Dom} \bigl({G}_{j+1}^{-1}\bigr), \\& {G}_{j+1}^{-1}\bigl({Q}_{j}^{-1} \bigl({Q}_{j}\bigl({d}_{j}({x}_{1},{x}_{2}) \bigr) +a_{2}( {x}_{1},{x}_{2}){c}({x}_{1},{x}_{2}) \bigr)\bigr)\in \operatorname{Dom} \bigl({G}_{1}^{-1} \bigr), \\& {G}_{1}^{-1}\bigl({G}_{j+1}^{-1} \bigl({Q}_{j}^{-1} \bigl({Q}_{j} \bigl({d}_{j}({x}_{1}, {x}_{2})\bigr) +a_{2}({x}_{1},{x}_{2}){c}({x}_{1},{x}_{2}) \bigr)\bigr)\bigr)\in \operatorname{Dom} \bigl({w^{-1}}\bigr). \end{aligned}$$

Proof

Let \(\bar{x}_{1}\in \mathbb{T}_{1}\) and \(\bar{x}_{2}\in \mathbb{T} _{2}\) with \({x}_{01}\leq \bar{x}_{1}\leq \tilde{x}_{1}\) and \({x}_{02}\leq \bar{x}_{2}\leq \tilde{x}_{2}\). Then inequality (3.16) can be rewritten as

$$\begin{aligned}& w\bigl(u({x}_{1},{x}_{2})\bigr) \\& \quad \leq a_{1}( \bar{x}_{1},\bar{x}_{2})+a_{2}( \bar{x}_{1},\bar{x}_{2}) \sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})} ^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma } _{2i}({x}_{2})}w_{1}\bigl(u\bigl(\mu _{1i}({t}_{1}),\mu _{2i}({t}_{2}) \bigr)\bigr) \\& \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2})w_{2} \bigl(u\bigl(\mu _{1i}( {t}_{1}),\mu _{2i}({t}_{2}) \bigr)\bigr) \biggl\{ w_{3}\bigl(u\bigl(\mu _{1i}({t}_{1}), \mu _{2i}({t}_{2})\bigr)\bigr) \\& \qquad {}+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i} ({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \times w_{3}\bigl(u\bigl( \mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} \\& \qquad {}+r_{i}({t}_{1},{t}_{2})w_{4} \bigl(\log \bigl(u\bigl(\mu _{1i}({t}_{1}),\mu _{2i}({t} _{2})\bigr)\bigr)\bigr) \biggr]\Delta {t}_{2}\Delta {t}_{1} \end{aligned}$$

for \({x}_{01}\leq {x}_{1}\leq \bar{x}_{1}\) and \({x}_{02}\leq {x}_{2} \leq \bar{x}_{2}\).

Let

$$\begin{aligned}& \xi _{2}({x}_{1},{x}_{2}) \\& \quad =a_{1}( \bar{x}_{1},\bar{x}_{2})+a_{2}( \bar{x}_{1},\bar{x}_{2}) \sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}w_{1}\bigl(u\bigl(\mu _{1i}({t}_{1}),\mu _{2i}({t}_{2}) \bigr)\bigr) \\& \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2})w_{2} \bigl(u\bigl(\mu _{1i}( {t}_{1}),\mu _{2i}({t}_{2}) \bigr)\bigr) \biggl\{ w_{3}\bigl(u\bigl(\mu _{1i}({t}_{1}), \mu _{2i}({t}_{2})\bigr)\bigr) \\& \qquad {}+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i} ({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \times w_{3}\bigl(u\bigl( \mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} \\& \qquad {}+r_{i}({t}_{1},{t}_{2})w_{4} \bigl(\log \bigl(u\bigl(\mu _{1i}({t}_{1}),\mu _{2i}({t} _{2})\bigr)\bigr)\bigr) \biggr]\Delta {t}_{2}\Delta {t}_{1}. \end{aligned}$$

Then one has

$$ \xi _{2}({x}_{01},{x}_{2})=\xi _{2}({x}_{1},{x}_{02})=a_{1}( \bar{x}_{1}, \bar{x}_{2}) $$

and

$$ u({x}_{1},{x}_{2})\leq w^{-1}\bigl(\xi _{2}({x}_{1},{x}_{2})\bigr). $$
(3.19)

On taking the identical steps as in (3.7)–(3.10), we get

$$\begin{aligned}& {G}_{1}\bigl(\xi _{2}({x}_{1},{x}_{2}) \bigr) \\& \quad \leq {G}_{1}\bigl(a_{1}(\bar{x}_{1}, \bar{x}_{2})\bigr)+a_{2}(\bar{x}_{1}, \bar{x}_{2}) \sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma } _{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x} _{2})} \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times w_{2}\bigl(w^{-1}\bigl(\xi _{2}({t}_{1},{t}_{2}) \bigr)\bigr) \biggl\{ w_{3}\bigl(w^{-1}\bigl(\xi _{2}({t}_{1},{t}_{2})\bigr)\bigr) + \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2}) \\& \qquad {}\times w_{3}\bigl(w^{-1}\bigl(\xi _{2}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} \\& \qquad {}+r_{i}({t}_{1},{t}_{2}) w_{4} \bigl(\log \bigl(w^{-1}\bigl(\xi _{2}( {t}_{1},{t}_{2})\bigr)\bigr)\bigr) \biggr] \Delta {t}_{2}\Delta {t}_{1}. \end{aligned}$$
(3.20)

For \(w_{2}(\mathfrak{u})\geq w_{4}(\log (\mathfrak{u}))\), inequality (3.20) can be rewritten as follows:

$$\begin{aligned}& {G}_{1}\bigl(\xi _{2}({x}_{1},{x}_{2}) \bigr) \\& \quad \leq {G}_{1}\bigl(a_{1}(\bar{x}_{1}, \bar{x}_{2})\bigr)+a_{2}(\bar{x}_{1}, \bar{x}_{2})\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma } _{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x} _{2})} w_{2} \bigl(w^{-1}\bigl(\xi _{2}({t}_{1},{t}_{2}) \bigr)\bigr) \\& \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{3}\bigl(w ^{-1}\bigl(\xi _{2}({t}_{1},{t}_{2}) \bigr)\bigr)+ \int _{{\gamma }_{1i}({x}_{01})}^{ {t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m} _{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{3}\bigl(w^{-1}\bigl(\xi _{2}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2}\Delta {t}_{1}. \end{aligned}$$
(3.21)

Let

$$\begin{aligned} \zeta _{2}({x}_{1},{x}_{2}) =&{G}_{1} \bigl(a_{1}(\bar{x}_{1},\bar{x}_{2})\bigr) +a _{2}(\bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}w_{2} \bigl(w^{-1}\bigl( \xi _{2}({t}_{1},{t}_{2}) \bigr)\bigr) \\ &{}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{3}\bigl(w ^{-1}\bigl(\xi _{2}({t}_{1},{t}_{2}) \bigr)\bigr) + \int _{{\gamma }_{1i}({x}_{01})}^{ {t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m} _{1},{t}_{2},{m}_{2}) \\ &{}\times w_{3}\bigl(w^{-1}\bigl(\xi _{2}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2}\Delta {t}_{1}. \end{aligned}$$

Then

$$ \zeta _{2}({x}_{01},{x}_{2})=\zeta _{2}({x}_{1},{x}_{02})={G}_{1} \bigl(a_{1}( \bar{x}_{1},\bar{x}_{2})\bigr) $$

and

$$ \xi _{2}({x}_{1},{x}_{2})\leq {G}_{1}^{-1}\bigl(\zeta _{2}({x}_{1},{x}_{2}) \bigr). $$
(3.22)

On taking the identical steps as those in (3.13)–(3.14), one has

$$\begin{aligned}& {G}_{2}\bigl(\zeta _{2}({x}_{1},{x}_{2}) \bigr) \\& \quad \leq {G}_{2}\bigl({G}_{1}\bigl(a_{1}( \bar{x}_{1},\bar{x}_{2})\bigr)\bigr)+a_{2}( \bar{x} _{1},\bar{x}_{2}) \sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{ {\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma } _{2i}({x}_{2})} \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times w_{3}\bigl(w^{-1}\bigl({G}_{1}^{-1} \bigl(\zeta _{2}({t}_{1},{t}_{2})\bigr) \bigr)\bigr) \\& \qquad {}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2}) \Delta {m}_{2}\Delta {m}_{1} \biggr\} +r_{i}({t}_{1}, {t}_{2}) \biggr]\Delta {t}_{2}\Delta {t}_{1} \\& \quad \leq {d}_{1}(\bar{x}_{1},\bar{x}_{2})+a_{2}( \bar{x}_{1},\bar{x}_{2}) \sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}f_{i}({x} _{1},{t}_{1},{x}_{2},{t}_{1}) \\& \qquad {}\times w_{3}\bigl(w^{-1}\bigl({G}_{1}^{-1} \bigl(\zeta _{2}({t}_{1},{t}_{2})\bigr) \bigr)\bigr) \\& \qquad {}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2}) \Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2} \Delta {t}_{1}. \end{aligned}$$

Let

$$\begin{aligned} \vartheta ({x}_{1},{x}_{2}) =&{d}_{1}( \bar{x}_{1},\bar{x}_{2})+a_{2}( \bar{x}_{1},\bar{x}_{2}) \sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})} ^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma } _{2i}({x}_{2})} f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{1}) \\ &{}\times w_{3}\bigl(w^{-1}\bigl({G}_{1}^{-1} \bigl(\zeta _{2}({t}_{1},{t}_{2})\bigr) \bigr)\bigr) \\ &{}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2}) \Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2} \Delta {t}_{1}. \end{aligned}$$
(3.23)

Then

$$ \vartheta ({x}_{01},{x}_{2})=\vartheta ({x}_{1},{x}_{02})={d}_{1}( \bar{x}_{1},\bar{x}_{2}) $$
(3.24)

and

$$ \zeta _{2}({x}_{1},{x}_{2})\leq {G}_{2}^{-1} \bigl(\vartheta ({x}_{1}, {x}_{2}) \bigr). $$
(3.25)

It follows from Lemma 2.1 and (3.23) that

$$\begin{aligned}& {\vartheta }^{\Delta {{x}_{1}}}({x}_{1},{x}_{2}) \\& \quad =a_{2}(\bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \biggl[{{\gamma }^{\Delta }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}( {x}_{2})}f_{i}\bigl(\sigma ({x}_{1}),{{\gamma }_{1i}({x}_{1})},{x}_{2}, {t}_{2}\bigr) \\& \qquad {}\times w_{3}\bigl(w^{-1}\bigl({G}_{1}^{-1} \bigl(\zeta _{2}\bigl({{\gamma }_{1i}({x}_{1})}, {t}_{2}\bigr)\bigr)\bigr)\bigr) \\& \qquad {}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{{\gamma }_{1i}( {x}_{1})}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}\bigl({{\gamma } _{1i}({x}_{1})},{m}_{1},{t}_{2},{m}_{2} \bigr) \Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2} \\& \qquad {}+ \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \biggl[ \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} f_{i}({x} _{1},{t}_{1},{x}_{2},{t}_{1})w_{3} \bigl(w^{-1}\bigl({G}_{1}^{-1} \bigl(\zeta _{2}( {t}_{1},{t}_{2})\bigr)\bigr)\bigr) \\& \qquad {}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2}) \Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2} \biggr]^{\Delta x_{1}} \Delta {t}_{1} \biggr]. \end{aligned}$$

From (3.25) and the fact that \(w_{3}\), \(w^{-1}\), \(G_{1}^{-1}\), \(G_{2}^{-1}\), and ϑ are nondecreasing, we have

$$\begin{aligned}& \frac{{\vartheta }^{\Delta {{x}_{1}}}({x}_{1},{x}_{2})}{w_{3}(w^{-1}( {G}_{1}^{-1}({G}_{2}^{-1}(\vartheta ({x}_{1},{x}_{2})))))} \\& \quad \leq a_{2}(\bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \biggl[{{\gamma }^{ \Delta }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma } _{2i}({x}_{2})} f_{i}\bigl(\sigma ({x}_{1}),{{\gamma }_{1i}({x}_{1})}, {x}_{2},{t}_{2}\bigr) \\& \qquad {}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{{\gamma }_{1i}( {x}_{1})}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}\bigl({{\gamma } _{1i}({x}_{1})},{m}_{1},{t}_{2},{m}_{2} \bigr) \Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2} \\& \qquad {}+ \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \biggl[ \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} f_{i}({x} _{1},{t}_{1},{x}_{2},{t}_{1}) \\& \qquad {}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2}) \Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2} \biggr]^{\Delta x_{1}} \Delta {t}_{1} \biggr] \\& \quad =a_{2}(\bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \biggl[ \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}f_{i}({x} _{1},{t}_{1},{x}_{2},{t}_{1}) \\& \qquad {}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2}) \Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2} \Delta {t}_{1} \biggr]^{\Delta x_{1}}. \end{aligned}$$

Using the definition of \({Q}_{1}\) and (3.24), integrating over \([{x}_{01},{{x}_{1}}]\), we get

$$ {Q}_{1}\bigl(\vartheta ({x}_{1},{x}_{2}) \bigr)\leq {Q}_{1}\bigl({d}_{1}(\bar{x}_{1}, \bar{x}_{2})\bigr) +a_{2}(\bar{x}_{1}, \bar{x}_{2}){c}({x}_{1},{x}_{2}). $$
(3.26)

It follows from (3.19), (3.22), (3.25), and (3.26) that

$$ u({x}_{1},{x}_{2})\leq w^{-1} \bigl({G}_{1}^{-1}\bigl({G}_{2}^{-1} \bigl({Q}_{1}^{-1} \bigl( {Q}_{1} \bigl({d}_{1}(\bar{x}_{1},\bar{x}_{2}) \bigr)+a_{2}(\bar{x}_{1},\bar{x} _{2}){c}({x}_{1},{x}_{2}) \bigr)\bigr)\bigr)\bigr). $$

Let \({x}_{1}=\bar{x}_{1}\) and \({x}_{2}=\bar{x}_{2}\). Then

$$ u(\bar{x}_{1},\bar{x}_{2})\leq w^{-1} \bigl({G}_{1}^{-1}\bigl({G}_{2}^{-1} \bigl({Q} _{1}^{-1} \bigl({Q}_{1} \bigl({d}_{1}(\bar{x}_{1},\bar{x}_{2}) \bigr)+a_{2}(\bar{x} _{1},\bar{x}_{2}){c}( \bar{x}_{1},\bar{x}_{2})\bigr)\bigr)\bigr)\bigr). $$
(3.27)

Since \({x}_{01}\leq \bar{x}_{1}\leq \tilde{x}_{1}\) and \({x}_{02} \leq \bar{x}_{2}\leq \tilde{x}_{2}\) are chosen arbitrarily, so after substituting \(\bar{x}_{1}\) and respectively \(\bar{x}_{2}\) with \({x}_{1}\) and \({x}_{2}\), we obtain the desired result (3.17).

For \(w_{2}(\mathfrak{u})< w_{4}(\log (\mathfrak{u}))\), inequality (3.20) can be rewritten as follows:

$$\begin{aligned}& {G}_{1}\bigl(\xi _{2}({x}_{1},{x}_{2}) \bigr) \\& \quad \leq {G}_{1}\bigl(a_{1}(\bar{x}_{1}, \bar{x}_{2})\bigr)+a_{2}(\bar{x}_{1}, \bar{x}_{2})\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma } _{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x} _{2})}w_{4}\bigl(\log \bigl(w^{-1}\bigl(\xi _{2}({t}_{1},{t}_{2}) \bigr)\bigr)\bigr) \\& \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{3}\bigl(w ^{-1}\bigl(\xi _{2}({t}_{1},{t}_{2}) \bigr)\bigr) + \int _{{\gamma }_{1i}({x}_{01})}^{ {t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m} _{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{3}\bigl(w^{-1}\bigl(\xi _{2}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2}\Delta {t}_{1} \\& \quad \leq {G}_{1}\bigl(a_{1}(\bar{x}_{1}, \bar{x}_{2})\bigr)+a_{2}(\bar{x}_{1}, \bar{x}_{2})\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma } _{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x} _{2})}w_{4} \bigl(w^{-1}\bigl(\xi _{2}({t}_{1},{t}_{2}) \bigr)\bigr) \\& \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{3}\bigl(w ^{-1}\bigl(\xi _{2}({t}_{1},{t}_{2}) \bigr)\bigr)+ \int _{{\gamma }_{1i}({x}_{01})}^{ {t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m} _{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{3}\bigl(w^{-1}\bigl(\xi _{2}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1} \biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2}\Delta {t}_{1}. \end{aligned}$$

On taking the identical steps as those (3.21)–(3.27), then after substituting \(\bar{x}_{1}\) and \(\bar{x}_{2}\) with \({x}_{1}\) and \({x}_{2}\), we obtain the desired result (3.18). □

Remark 3.9

Let \(\mathbb{T}=\mathrm{R}\), \(w_{3}=1\), \(\mu _{ji}=I\), \(f_{i}({x}_{1}, {t}_{1},{x}_{2},{t}_{2})=f_{i}({t}_{1},{t}_{2})\), and \(g_{i}=0\). Then Theorem 3.8 leads to Theorem 2 in [115].

Theorem 3.10

Let \(a_{j}\), \(r_{i}\), w, \(w_{j}\), \(f_{i}\), \(g_{i}\), \(\gamma _{ji}\), \(\mu _{ji}\), and \(\mathfrak{a}\)be defined as in Theorem 3.1, and \(\mathrm{L}, \mathrm{M}:\mathbb{T}_{1}\times \mathbb{T}_{2}\times \mathbb{R}_{0}^{+}\rightarrow \mathbb{R}_{0}^{+}\)be right-dense continuous on \(\mathbb{T}_{1}\times \mathbb{T}_{2}\)and continuous on \(\mathbb{R}_{0}^{+}\)such that

$$ 0\leq \mathrm{L}({x}_{1},{x}_{2},\mathfrak{u})- \mathrm{L}({x}_{1}, {x}_{2},\mathfrak{v}) \leq \mathrm{M}({x}_{1},{x}_{2},\mathfrak{v}) ( \mathfrak{u}- \mathfrak{v}) $$

for \(\mathfrak{u}>\mathfrak{v}\geq 0\)and \(({x}_{1},{x}_{2})\in \mathbb{T}_{1}\times \mathbb{T}_{2}\), and

$$\begin{aligned}& w\bigl(u({x}_{1},{x}_{2})\bigr) \\& \quad \leq a_{1}({x}_{1},{x}_{2})+a_{2}({x}_{1},{x} _{2}) \sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} w_{1}\bigl(u\bigl( \mu _{1i}({t}_{1}),\mu _{2i}({t}_{2}) \bigr)\bigr) \\& \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{2}\bigl(u\bigl( \mu _{1i}({t}_{1}), \mu _{2i}({t}_{2})\bigr)\bigr) + \int _{{\gamma }_{1i}({x}_{01})} ^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}} g_{i}({t}_{1}, {m}_{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(u\bigl(\mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1} \biggr\} \\& \qquad {}+r_{i}({t}_{1},{t}_{2})\mathrm{L} \bigl({t}_{1},{t}_{2},w_{2} \bigl(u({t}_{1}, {t}_{2})\bigr)\bigr) \biggr]\Delta {t}_{2}\Delta {t}_{1} \end{aligned}$$
(3.28)

for \(({x}_{1},{x}_{2})\in \mathbb{T}_{1}\times \mathbb{T}_{2}\)with initial condition (3.2). Then

$$\begin{aligned} u({x}_{1},{x}_{2}) \leq& w^{-1} \Biggl({G}_{1}^{-1}\Biggl({G}_{2}^{-1} \Biggl({G}_{2} \bigl( {b}_{2}({x}_{1},{x}_{2}) \bigr)+a_{2}({x}_{1},{x}_{2})\Biggl\{ {c}({x}_{1},{x}_{2}) \\ &{}+\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x} _{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}r_{i}( {t}_{1},{t}_{2})\mathrm{M} ({t}_{1},{t}_{2},0) \Delta {t}_{2}\Delta {t}_{1}\Biggr\} \Biggr)\Biggr)\Biggr) \end{aligned}$$
(3.29)

for all \({x}_{01}\leq {x}_{1}\leq \tilde{x}_{1}\)and \({x}_{02}\leq {x}_{2}\leq \tilde{x}_{2}\), where

$$\begin{aligned} {b}_{2}({x}_{1},{x}_{2}) =&{G}_{1} \bigl(a_{1}({x}_{1},{x}_{2}) \bigr)+a_{2}({x} _{1},{x}_{2}) \\ &{}\times \sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}( {x_{1}})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}r _{i}({t}_{1},{t}_{2}) \mathrm{L}({t}_{1},{t}_{2},0)\Delta {t}_{2} \Delta {t}_{1}, \end{aligned}$$

c, \({G}_{1}\), and \({G}_{2}\)are as defined in Theorem 3.1, and \(\tilde{x}_{1}\)and \(\tilde{x}_{2}\)are chosen such that

$$\begin{aligned}& {G}_{2}\bigl({b}_{2}({x}_{1},{x}_{2}) \bigr)+a_{2}({x}_{1},{x}_{2})\Biggl\{ {c}({x}_{1}, {x}_{2}) \\& \quad {}+\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x} _{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}r_{i}( {t}_{1},{t}_{2})\mathrm{M} ({t}_{1},{t}_{2},0) \Delta {t}_{2}\Delta {t}_{1}\Biggr\} \in \operatorname{Dom}\bigl({G}_{2}^{-1}\bigr), \\& {G}_{2}^{-1}\Biggl({G}_{2} \bigl({b}_{2}({x}_{1},{x}_{2}) \bigr)+a_{2}({x}_{1},{x}_{2}) \Biggl\{ {c}({x}_{1},{x}_{2}) \\& \quad {}+\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x} _{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}r_{i}( {t}_{1},{t}_{2})\mathrm{M} ({t}_{1},{t}_{2},0) \Delta {t}_{2}\Delta {t}_{1}\Biggr\} \Biggr)\in \operatorname{Dom}\bigl({G}_{1}^{-1}\bigr), \\& {G}_{1}^{-1}\Biggl({G}_{2}^{-1} \Biggl({G}_{2}\bigl({b}_{2}({x}_{1},{x}_{2}) \bigr)+a_{2}( {x}_{1},{x}_{2})\Biggl\{ {c}({x}_{1},{x}_{2}) \\& \quad {}+\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x} _{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}r_{i}( {t}_{1},{t}_{2})\mathrm{M} ({t}_{1},{t}_{2},0) \Delta {t}_{2}\Delta {t}_{1}\Biggr\} \Biggr)\Biggr)\in \operatorname{Dom}\bigl(w^{-1}\bigr). \end{aligned}$$

Proof

Let \(\bar{x}_{1}\in \mathbb{T}_{1}\) and \(\bar{x}_{2}\in \mathbb{T} _{2}\) with \({x}_{01}\leq \bar{x}_{1}\leq \tilde{x}_{1}\) and \({x}_{02}\leq \bar{x}_{2}\leq \tilde{x}_{2}\). Then inequality (3.28) can be rewritten as follows:

$$\begin{aligned}& w\bigl(u({x}_{1},{x}_{2})\bigr) \\& \quad \leq a_{1}( \bar{x}_{1},\bar{x}_{2})+a_{2}( \bar{x}_{1},\bar{x}_{2}) \sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}w_{1}\bigl(u\bigl(\mu _{1i}({t}_{1}),\mu _{2i}({t}_{2}) \bigr)\bigr) \\& \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{2}\bigl(u\bigl(\mu _{1i}( {t}_{1}),\mu _{2i}({t}_{2})\bigr)\bigr) + \int _{{\gamma }_{1i}({x}_{01})}^{{t} _{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}} g_{i}({t}_{1},{m}_{1}, {t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(u\bigl(\mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1}\biggr\} +r_{i}({t}_{1},{t}_{2}) \mathrm{L}\bigl({t}_{1},{t}_{2},w _{2} \bigl(u({t}_{1},{t}_{2})\bigr)\bigr)\biggr]\Delta {t}_{2}\Delta {t}_{1} \end{aligned}$$

for \({x}_{01}\leq {x}_{1}\leq \bar{x}_{1}\) and \({x}_{02}\leq {x}_{2} \leq \bar{x}_{2}\). Let

$$\begin{aligned}& \xi _{3}({x}_{1},{x}_{2}) \\& \quad =a_{1}( \bar{x}_{1},\bar{x}_{2})+a_{2}(\bar{x} _{1},\bar{x}_{2}) \sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{ {\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma } _{2i}({x}_{2})} w_{1}\bigl(u\bigl( \mu _{1i}({t}_{1}),\mu _{2i}({t}_{2}) \bigr)\bigr) \\& \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{2}\bigl(u\bigl(\mu _{1i}( {t}_{1}),\mu _{2i}({t}_{2})\bigr)\bigr) + \int _{{\gamma }_{1i}({x}_{01})}^{{t} _{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}} g_{i}({t}_{1},{m}_{1}, {t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(u\bigl(\mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1}\biggr\} +r_{i}({t}_{1},{t}_{2}) \mathrm{L}\bigl({t}_{1},{t}_{2},w _{2} \bigl(u({t}_{1},{t}_{2})\bigr)\bigr) \biggr]\Delta {t}_{2}\Delta {t}_{1}. \end{aligned}$$

Then

$$ \xi _{3}({x}_{01},{x}_{2})=\xi _{3}({x}_{1},{x}_{02})=a_{1}( \bar{x}_{1}, \bar{x}_{2}) $$

and

$$ u({x}_{1},{x}_{2})\leq w^{-1}\bigl(\xi _{3}({x}_{1},{x}_{2})\bigr). $$
(3.30)

On taking the identical steps as in (3.17)–(3.10), one has

$$\begin{aligned}& {G}_{1}\bigl(\xi _{3}({x}_{1},{x}_{2}) \bigr) \\& \quad \leq {G}_{1}\bigl(a_{1}(\bar{x}_{1}, \bar{x}_{2})\bigr)+a_{2}(\bar{x}_{1}, \bar{x}_{2})\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma } _{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x} _{2})}\biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times \biggl\{ w_{2}\bigl(w^{-1}\bigl(\xi _{3}({t}_{1},{t}_{2})\bigr)\bigr)+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(w^{-1}\bigl(\xi _{3}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1}\biggr\} \\& \qquad {}+r_{i}({t}_{1},{t}_{2})\mathrm{L} \bigl({t}_{1},{t}_{2},w_{2} \bigl(w^{-1}\bigl(\xi _{3}({t}_{1},{t}_{2}) \bigr)\bigr)\bigr)\biggr]\Delta {t}_{2}\Delta {t}_{1} \\& \quad \leq {G}_{1}\bigl(a_{1}(\bar{x}_{1}, \bar{x}_{2})\bigr)+a_{2}(\bar{x}_{1}, \bar{x}_{2}) \sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma } _{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x} _{2})}\biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times \biggl\{ w_{2}\bigl(w^{-1}\bigl(\xi _{3}({t}_{1},{t}_{2})\bigr)\bigr)+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(w^{-1}\bigl(\xi _{3}({m}_{1},{m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1}\biggr\} \\& \qquad {}+r_{i}({t}_{1},{t}_{2})\bigl\{ \mathrm{L}({t}_{1},{t}_{2},0)+\mathrm{M}( {t}_{1},{t}_{2},0) w_{2} \bigl(w^{-1}\bigl(\xi _{3}({t}_{1},{t}_{2}) \bigr)\bigr)\bigr\} \biggr]\Delta {t}_{2}\Delta {t}_{1} \\& \quad \leq {G}_{1}\bigl(a_{1}(\bar{x}_{1}, \bar{x}_{2})\bigr)+a_{2}(\bar{x}_{1}, \bar{x}_{2}) \sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma } _{1i}((\bar{x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}( \bar{x}_{2})}r_{i}({t}_{1},{t}_{2}) \mathrm{L}({t}_{1},{t}_{2},0) \Delta {t}_{2}\Delta {t}_{1} \\& \qquad {}+a_{2}(\bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}w_{2} \bigl(w^{-1}\bigl( \xi _{3}({t}_{1},{t}_{2}) \bigr)\bigr) \\& \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ 1 + \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \Delta {m}_{2} \Delta {m}_{1}\biggr\} \\& \qquad {}+r_{i}({t}_{1},{t}_{2}) \mathrm{M}({t}_{1},{t}_{2},0)\biggr]\Delta {t}_{2} \Delta {t}_{1} \\& \quad =b_{2}(\bar{x}_{1},\bar{x}_{2})+a_{2}( \bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}w_{2} \bigl(w^{-1}\bigl( \xi _{3}({t}_{1},{t}_{2}) \bigr)\bigr) \\& \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ 1 + \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \Delta {m}_{2} \Delta {m}_{1}\biggr\} \\& \qquad {}+r_{i}({t}_{1},{t}_{2}) \mathrm{M}({t}_{1},{t}_{2},0)\biggr]\Delta {t}_{2} \Delta {t}_{1}. \end{aligned}$$

Let

$$\begin{aligned} \zeta _{3}({x}_{1},{x}_{2}) =&b_{2}( \bar{x}_{1},\bar{x}_{2})+a_{2}( \bar{x}_{1},\bar{x}_{2})\sum _{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})} ^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma } _{2i}({x}_{2})} w_{2} \bigl(w^{-1}\bigl(\xi _{3}({t}_{1},{t}_{2}) \bigr)\bigr) \\ &{}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ 1 + \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \Delta {m}_{2} \Delta {m}_{1}\biggr\} \\ &{}+r_{i}({t}_{1},{t}_{2}) \mathrm{M}({t}_{1},{t}_{2},0)\biggr]\Delta {t}_{2} \Delta {t}_{1}. \end{aligned}$$

Then

$$ \zeta _{3}({x}_{01},{x}_{2})=\zeta _{3}({x}_{1},{x}_{02})=b_{2}( \bar{x} _{1},\bar{x}_{2}) $$

and

$$ \xi _{3}({x}_{1},{x}_{2})\leq {G}_{1}^{-1} \bigl(\zeta _{3}({x}_{1}, {x}_{2}) \bigr). $$
(3.31)

On taking the identical steps as in (3.13) to (3.14), one has

$$\begin{aligned}& {G}_{2}\bigl(\zeta _{3}({x}_{1},{x}_{2}) \bigr) \\& \quad \leq {G}_{2}\bigl(b_{2}(\bar{x}_{1}, \bar{x}_{2})\bigr)+a_{2}(\bar{x}_{1}, \bar{x}_{2}) \sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma } _{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x} _{2})} \biggl\{ f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times \biggl(1+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2}) \Delta {m}_{2}\Delta {m}_{1}\biggr) \\& \qquad {}+r_{i}({t}_{1},{t}_{2}) \mathrm{M}({t}_{1},{t}_{2},0)\biggr\} \Delta {t}_{2} \Delta {t}_{1} \\& \quad ={G}_{2}\bigl({b}_{2}(\bar{x}_{1}, \bar{x}_{2})\bigr)+a_{2}(\bar{x}_{1},\bar{x} _{2})\Biggl\{ {c}({x}_{1},{x}_{2}) \\& \qquad {}+\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x} _{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}(\mathfrak{x} _{2})} r_{i}({t}_{1},{t}_{2}) \mathrm{M}({t}_{1},{t}_{2},0)\Delta {t}_{2} \Delta {t}_{1}\Biggr\} . \end{aligned}$$
(3.32)

It follows from (3.30)–(3.32) that

$$\begin{aligned} u({x}_{1},{x}_{2}) \leq& w^{-1} \Biggl({G}_{1}^{-1}\Biggl({G}_{2}^{-1} \Biggl({G}_{2} \bigl( {b}_{2}(\bar{x}_{1}, \bar{x}_{2})\bigr)+a_{2}(\bar{x}_{1}, \bar{x}_{2}) \Biggl\{ {c}({x}_{1},{x}_{2}) \\ &{}+\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x} _{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} r _{i}({t}_{1},{t}_{2}) \mathrm{M}({t}_{1},{t}_{2},0)\Delta {t}_{2} \Delta {t}_{1}\Biggr\} \Biggr)\Biggr)\Biggr). \end{aligned}$$

Let \({x}_{1}=\bar{x}_{1}\) and \({x}_{2}=\bar{x}_{2}\). Then we have

$$\begin{aligned} u(\bar{x}_{1},\bar{x}_{2}) \leq& w^{-1} \Biggl({G}_{1}^{-1}\Biggl({G}_{2}^{-1} \Biggl({G} _{2} \bigl({b}_{2}(\bar{x}_{1}, \bar{x}_{2})\bigr)+a_{2}(\bar{x}_{1}, \bar{x}_{2}) \Biggl\{ {c}(\bar{x}_{1}, \bar{x}_{2}) \\ &{}+\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}( \bar{x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}(\bar{x} _{2})} r_{i}({t}_{1},{t}_{2}) \mathrm{M}({t}_{1},{t}_{2},0)\Delta {t}_{2} \Delta {t}_{1}\Biggr\} \Biggr)\Biggr)\Biggr). \end{aligned}$$

Since \({x}_{01}\leq \bar{x}_{1}\leq \tilde{x}_{1}\) and \({x}_{02} \leq \bar{x}_{2}\leq \tilde{x}_{2}\) are chosen arbitrarily, we obtain

$$\begin{aligned} u({x}_{1},{x}_{2}) \leq& w^{-1} \Biggl({G}_{1}^{-1}\Biggl({G}_{2}^{-1} \Biggl({G}_{2} \bigl( {b}_{2}({x}_{1},{x}_{2}) \bigr)+a_{2}({x}_{1},{x}_{2})\Biggl\{ {c}({x}_{1},{x}_{2}) \\ &{}+\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x} _{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}r_{i}( {t}_{1},{t}_{2})\mathrm{M}({t}_{1},{t}_{2},0) \Delta {t}_{2}\Delta {t}_{1}\Biggr\} \Biggr)\Biggr) \Biggr). \end{aligned}$$

 □

Remark 3.11

Let \(\mathbb{T}=\mathbb{R}\), \(w_{2}=\mu _{ji}=I\), \(f_{i}({x}_{1},{t} _{1},{x}_{2},{t}_{2})=f_{i}({t}_{1},{t}_{2})\), and \(g_{i}=0\). Then Theorem 3.10 coincides with Theorem 3 in [115].

Corollary 3.12

Letu, \(r_{i}\), \(w_{2}\), \(f_{i}\), \(g_{i}\), \(\gamma _{ji}\), and \(\mu _{ji}\)be defined as in Theorem 3.1, \(\bar{\mathfrak{a}}({x}_{1}, {x}_{2})\)be a nonnegative and right-dense continuous function defined on \(([\overline{\mathfrak{p}}_{1},{x}_{01}]\times [\overline{ \mathfrak{p}}_{2},{x}_{02}])_{\mathbb{T}^{2}}\), and \(q_{1}>q_{2}>0\)and \(\mathfrak{C}\geq 0\)be constants such that

$$\begin{aligned} u^{q_{1}}({x}_{1},{x}_{2}) \leq& \mathfrak{C}+ \frac{q_{1}}{q_{1}-q_{2}} \sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}u^{q_{2}}\bigl( \mu _{1i}({t}_{1}),\mu _{2i}({t}_{2}) \bigr) \\ &{}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{2}\bigl(u\bigl(\mu _{1i}( {t}_{1}),\mu _{2i}({t}_{2})\bigr)\bigr) \\ &{} + \int _{{\gamma }_{1i}({x}_{01})}^{{t} _{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1}, {t}_{2},{m}_{2}) \\ &{}\times w_{2}\bigl(u\bigl(\mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1}\biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2}\Delta {t}_{1} \end{aligned}$$
(3.33)

for \(({x}_{1},{x}_{2})\in \mathbb{T}_{1}\times \mathbb{T}_{2}\)with the initial condition

$$ \textstyle\begin{cases} u({x}_{1},{x}_{2})=\overline{\mathfrak{a}}({x}_{1},{x}_{2}), \quad {x}_{1}\in [\overline{\mathfrak{p}}_{1},{x}_{01}]_{\mathbb{T}} \textit{ or } {x}_{2}\in [\overline{\mathfrak{p}}_{2},{x}_{02}]_{ \mathbb{T}} , \\ \overline{\mathfrak{a}}(\mu _{1i}({x}_{1}),\mu _{2i}({x}_{2}))\leq \sqrt[q _{1}]{\mathfrak{C}} , \quad \mu _{1i}({x}_{1})\leq {x}_{01} \textit{ or } \mu _{2i}({x}_{2})\leq {x}_{02}. \end{cases} $$
(3.34)

Then

$$ u({x}_{1},{x}_{2})\leq \sqrt[q_{1}-q_{2}]{{{H}_{1}}^{-1} \bigl({H}_{1}\bigl( \overline{b}_{1}({x}_{1},{x}_{2}) \bigr) +{c}({x}_{1},{x}_{2})\bigr)} $$
(3.35)

for all \({x}_{01}\leq {x}_{1}\leq \tilde{x}_{1}\)and \({x}_{02}\leq {x}_{2}\leq \tilde{x}_{2}\), where

$$\begin{aligned}& \overline{b}_{1}({x}_{1},{x}_{2}):= \mathfrak{C}^{\frac{q_{1}-q_{2}}{q _{1}}} +\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}( {x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}r _{i}({t}_{1},{t}_{2}) \Delta {t}_{2}\Delta {t}_{1}, \\& {H}_{1}(s)= \int _{{s}_{6}}^{s}\frac{\Delta p}{w_{2}( \sqrt[q_{1}-q_{2}]{p})},\quad s> s_{6}>0 \textit{ with } {H}_{1}(\infty )=\infty , \end{aligned}$$

cis defined as in Theorem 3.1, \({H}_{1}^{-1}\)is the inverse function of \({H}_{1}\), and \(\tilde{x}_{1}\)and \(\tilde{x}_{2}\)are chosen such that

$$ {H}_{1}\bigl(\overline{b}_{1}({x}_{1},{x}_{2}) \bigr) +{c}({x}_{1},{x}_{2}) \in \operatorname{Dom} \bigl({{H}_{1}}^{-1}\bigr). $$

Proof

Let

$$\begin{aligned} \overline{\xi }_{1}({x}_{1},{x}_{2}) =& \mathfrak{C}+\frac{q_{1}}{q_{1}-q _{2}} \sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}( {x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} u ^{q_{2}}\bigl(\mu _{1i}({t}_{1}),\mu _{2i}({t}_{2}) \bigr) \\ &{}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{2}\bigl(u\bigl(\mu _{1i}( {t}_{1}),\mu _{2i}({t}_{2})\bigr)\bigr) \\ &{} + \int _{{\gamma }_{1i}({x}_{01})}^{{t} _{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1}, {t}_{2},{m}_{2}) \\ &{}\times w_{2}\bigl(u\bigl(\mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1}\biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2}\Delta {t}_{1}. \end{aligned}$$
(3.36)

Then

$$ \bar{\xi }_{1}({x}_{01},{x}_{2})=\bar{\xi }_{1}({x}_{1},{x}_{02})= \mathfrak{C} $$
(3.37)

and

$$ u({x}_{1},{x}_{2})\leq \sqrt[q_{1}]{ \overline{\xi }_{1}({x}_{1},{x} _{2})} \quad \text{for } {x}_{1}\in [x_{01},\bar{x}_{1}]_{\mathbb{T}} \text{ and } {x}_{2}\in [x_{02}, \bar{x}_{2}]_{\mathbb{T}}, $$
(3.38)

where \(\bar{x}_{1}\in \mathbb{T}_{1}\) and \(\bar{x}_{2}\in \mathbb{T} _{2}\) are fixed numbers with \({x}_{01}\leq \bar{x}_{1}\leq \tilde{x} _{1}\) and \({x}_{02}\leq \bar{x}_{2}\leq \tilde{x}_{2}\).

For \({x}_{1}\in [x_{01},\bar{x}_{1}]_{\mathbb{T}}\) and \({x}_{2}\in [x _{02},\bar{x}_{2}]_{\mathbb{T}}\), if \(\mu _{1i}({x}_{1})\geq {x}_{01}\) and \(\mu _{2i}({x}_{2})\geq {x}_{02}\), then \(\mu _{1i}({x}_{1})\in [x _{01},\bar{x}_{1}]_{\mathbb{T}}\), \(\mu _{2i}({x}_{1})\in [x_{02}, \bar{x}_{2}]_{\mathbb{T}}\), and

$$ u\bigl(\mu _{1i}({x}_{1}),\mu _{2i}({x}_{2}) \bigr)\leq \sqrt[q_{1}]{\overline{ \xi }_{1}\bigl(\mu _{1i}({x}_{1}),\mu _{2i}({x}_{2}) \bigr)} \leq \sqrt[q_{1}]{\overline{ \xi }_{1}({x}_{1},{x}_{2})}. $$
(3.39)

If \(\mu _{1i}({x}_{1})\leq {x}_{01}\) or \(\mu _{2i}({x}_{2})\leq {x}_{02}\), then from (3.34) we obtain

$$ u\bigl(\mu _{1i}({x}_{1}),\mu _{2i}({x}_{2}) \bigr)=\overline{\mathfrak{a}}\bigl(\mu _{1i}({x}_{1}), \mu _{2i}({x}_{2})\bigr) \leq \sqrt[q_{1}]{ \mathfrak{C}} \leq \sqrt[q_{1}]{\overline{\xi }_{1}({x}_{1},{x}_{2})}. $$
(3.40)

It follows from (3.39) and (3.40) that

$$ u\bigl(\mu _{1i}({x}_{1}),\mu _{2i}({x}_{2}) \bigr)\leq \sqrt[q_{1}]{\overline{ \xi }_{1}({x}_{1},{x}_{2})} $$
(3.41)

for \({x}_{1}\in [x_{01},\bar{x}_{1}]_{\mathbb{T}}\) and \({x}_{2}\in [x _{02},\bar{x}_{2}]_{\mathbb{T}}\).

From Lemma 2.1 and (3.36) we know that

$$\begin{aligned}& {\overline{\xi }_{1}}^{\Delta {{x}_{1}}}({x}_{1},{x}_{2}) \\& \quad = \frac{q _{1}}{q_{1}-q_{2}}\sum_{i=1}^{n}{{ \gamma }_{1i}^{\Delta }({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}u^{q_{2}}\bigl( \mu _{1i}\bigl(\gamma _{1i}({x}_{1})\bigr),\mu _{2i}({t}_{2})\bigr) \\& \qquad {}\times \biggl[f_{i}\bigl(\sigma ({x}_{1}),{\gamma }_{1i}({x}_{1}),{x}_{2},{t} _{2}\bigr)\biggl\{ w_{2}\bigl(u\bigl(\mu _{1i}\bigl(\gamma _{1i}({x}_{1})\bigr),\mu _{2i}({t}_{2})\bigr)\bigr) \\& \qquad {}+ \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}} g_{i}\bigl({\gamma }_{1i}({x} _{1}),{m}_{1},{t}_{2},{m}_{2} \bigr) \\& \qquad {}\times w_{2}\bigl(u\bigl(\mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1}\biggr\} +r_{i} \bigl({\gamma }_{1i}({x}_{1}),{t}_{2}\bigr) \biggr]\Delta {t} _{2} \\& \qquad {}+\frac{q_{1}}{q_{1}-q_{2}}\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})} ^{{\gamma }_{1i}({x}_{1})} \biggl[ \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}u^{q_{2}}\bigl(\mu _{1i}({t}_{1}),\mu _{2i}({t}_{2}) \bigr) \\& \qquad {}\times \biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{2}\bigl(u\bigl(\mu _{1i}( {t}_{1}),\mu _{2i}({t}_{2})\bigr)\bigr) + \int _{{\gamma }_{1i}({x}_{01})}^{{t} _{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1}, {t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(u\bigl(\mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1}\biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2}\biggr]^{\Delta x _{1}}\Delta {t}_{1}. \end{aligned}$$

Making use of (3.41) and the fact that \({\overline{\xi }_{1}}\) is nondecreasing, we have

$$\begin{aligned}& \frac{{\overline{\xi }_{1}}^{\Delta {{x}_{1}}}({x}_{1},{x}_{2})}{ {\overline{\xi }}^{\frac{q_{2}}{q_{1}}}_{1}({x}_{1},{x}_{2})} \\& \quad \leq \frac{q _{1}}{q_{1}-q_{2}}\sum _{i=1}^{n}{{\gamma }_{1i}^{\Delta }({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} \biggl[f_{i}\bigl( \sigma ({x}_{1}),{\gamma }_{1i}({x}_{1}),{x}_{2},{t}_{2} \bigr) \\& \qquad {}\times \biggl\{ w_{2}\bigl(\sqrt[q_{1}]{\overline{\xi }_{1}\bigl({\gamma }_{1i}({x} _{1}),{t}_{2} \bigr)}\bigr) + \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}( {x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}} g_{i}\bigl({\gamma } _{1i}({x}_{1}),{m}_{1},{t}_{2},{m}_{2} \bigr) \\& \qquad {}\times w_{2}\bigl(\sqrt[q_{1}]{\overline{\xi }_{1}({m}_{1},{m}_{2})}\bigr) \Delta {m}_{2}\Delta {m}_{1}\biggr\} +r_{i}\bigl({ \gamma }_{1i}({x}_{1}),{t}_{2}\bigr)\biggr] \Delta {t}_{2} \\& \qquad {}+\frac{q_{1}}{q_{1}-q_{2}}\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})} ^{{\gamma }_{1i}({x}_{1})} \biggl[ \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}\biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{2}\bigl(\sqrt[q _{1}]{\overline{\xi }_{1}({t}_{1},{t}_{2})}\bigr) \\& \qquad {}+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) w_{2}\bigl(\sqrt[q _{1}]{\overline{\xi }_{1}({m}_{1},{m}_{2})}\bigr)\Delta {m}_{2}\Delta {m}_{1}\biggr\} \\& \qquad {}+r_{i}({t}_{1},{t}_{2})\biggr]\Delta {t}_{2}\biggr]^{\Delta x_{1}}\Delta {t}_{1} \\& \quad =\frac{q_{1}}{q_{1}-q_{2}}\sum_{i=1}^{n} \biggl[ \int _{{\gamma }_{1i}({x}_{01})} ^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma } _{2i}({x}_{2})}\biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{2}\bigl(\sqrt[q _{1}]{\overline{\xi }_{1}({t}_{1},{t}_{2})}\bigr) \\& \qquad {}+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) w_{2}\bigl(\sqrt[q _{1}]{\overline{\xi }_{1}({m}_{1},{m}_{2})}\bigr)\Delta {m}_{2}\Delta {m}_{1}\biggr\} \\& \qquad {}+r_{i}({t}_{1},{t}_{2})\biggr]\Delta {t}_{2}\Delta {t}_{1}\biggr]^{\Delta x_{1}}. \end{aligned}$$
(3.42)

Theorem 2.2 and \({\overline{\xi }_{1}}^{\Delta _{{x}_{1}}}({x}_{1}, {x}_{2})\geq 0\) lead to the conclusion that

$$\begin{aligned}& \biggl(\frac{q_{1}}{q_{1}-q_{2}}{\overline{\xi }_{1}}^{\frac{q_{1}-q _{2}}{q_{1}}} ({x}_{1},{x}_{2}) \biggr)^{\Delta {{x}_{1}}} \\& \quad ={\overline{\xi }_{1}}^{\Delta {{x}_{1}}}({x}_{1},{x}_{2}) \int _{0} ^{1}\bigl\{ {\overline{\xi }_{1}}({x}_{1},{x}_{2}) +h\mu ({x}_{1},{x}_{2}) {\overline{\xi }_{1}}^{\Delta {{x}_{1}}}({x}_{1},{x}_{2}) \bigr\} ^{-\frac{q _{2}}{q_{1}}}\,dh \\& \quad =\frac{{\overline{\xi }_{1}}^{\Delta {{x}_{1}}}({x}_{1},{x}_{2})}{ {\overline{\xi }_{1}}^{\frac{q_{2}}{q_{1}}}({x}_{1},{x}_{2})} \int _{0} ^{1}\biggl\{ 1 +h\mu ({x}_{1},{x}_{2}) \frac{{\overline{\xi }_{1}}^{\Delta {{x}_{1}}}({x}_{1},{x}_{2})}{{\overline{\xi }_{1}}({x}_{1},{x}_{2})} \biggr\} ^{-\frac{q_{2}}{q_{1}}}\,dh \\& \quad =\frac{{\overline{\xi }_{1}}^{\Delta {{x}_{1}}}({x}_{1},{x}_{2})}{ {\overline{\xi }_{1}}^{\frac{q_{2}}{q_{1}}}({x}_{1},{x}_{2})}\frac{\{1+ \mu ({x}_{1},{x}_{2}) \frac{{\overline{\xi }_{1}}^{\Delta {{x}_{1}}}( {x}_{1},{x}_{2})}{{\overline{\xi }_{1}}({x}_{1},{x}_{2})}\}^{-\frac{q _{2}}{q_{1}}+1}-1}{\mu ({x}_{1},{x}_{2})\frac{{\overline{\xi }_{1}} ^{\Delta {{x}_{1}}}({x}_{1},{x}_{2})}{{\overline{\xi }_{1}}({x}_{1}, {x}_{2})}(1-\frac{q_{2}}{q_{1}})} \leq \frac{{\overline{\xi }_{1}} ^{\Delta {{x}_{1}}}({x}_{1},{x}_{2})}{{\overline{\xi }_{1}}^{\frac{q _{2}}{q_{1}}}({x}_{1},{x}_{2})}. \end{aligned}$$
(3.43)

From (3.42) and (3.43) we have

$$\begin{aligned}& \biggl(\frac{q_{1}}{q_{1}-q_{2}}{\overline{\xi }_{1}}^{\frac{q_{1}-q _{2}}{q_{1}}} ({x}_{1},{x}_{2}) \biggr)^{\Delta {{x}_{1}}} \\& \quad \leq \frac{q _{1}}{q_{1}-q_{2}} \sum_{i=1}^{n} \biggl[ \int _{{\gamma }_{1i}({x}_{01})}^{ {\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma } _{2i}({x}_{2})}\biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times \biggl\{ w_{2}\bigl(\sqrt[q_{1}]{\overline{\xi }_{1}({t}_{1},{t}_{2})}\bigr) + \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(\sqrt[q_{1}]{\overline{\xi }_{1}({m}_{1},{m}_{2})}\bigr) \Delta {m}_{2}\Delta {m}_{1}\biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2} \Delta {t}_{1}\biggr]^{\Delta x_{1}}. \end{aligned}$$

Making use of (3.37) and integrating over \([{x}_{01},{{x}_{1}}]\), we get

$$\begin{aligned} {\overline{\xi }_{1}}^{\frac{q_{1}-q_{2}}{q_{1}}}({x}_{1},{x}_{2}) \leq& {\mathfrak{C}}^{\frac{q_{1}-q_{2}}{q_{1}}} +\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} r_{i}({t} _{1},{t}_{2})\Delta {t}_{2} \Delta {t}_{1} \\ &{}+\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x} _{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} f_{i}( {x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\ &{}\times \biggl\{ w_{2}\bigl(\sqrt[q_{1}]{\overline{\xi }_{1}({t}_{1},{t}_{2})}\bigr)+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\ &{}\times w_{2}\bigl(\sqrt[q_{1}]{\overline{\xi }_{1}({m}_{1},{m}_{2})}\bigr) \Delta {m}_{2}\Delta {m}_{1}\biggr\} \Delta {t}_{2}\Delta {t}_{1} \\ \leq& {\mathfrak{C}}^{\frac{q_{1}-q_{2}}{q_{1}}}+\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}(\bar{x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}(\bar{x}_{2})}r_{i}( {t}_{1},{t}_{2})\Delta {t}_{2} \Delta {t}_{1} \\ &{}+\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x} _{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}f_{i}( {x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\ &{}\times \biggl\{ w_{2}\bigl(\sqrt[q_{1}]{\overline{\xi }_{1}({t}_{1},{t}_{2})}\bigr) + \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})} ^{{t}_{2}}g_{i} ({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\ &{}\times w_{2}\bigl(\sqrt[q_{1}]{\overline{\xi }_{1}({m}_{1},{m}_{2})}\bigr) \Delta {m}_{2}\Delta {m}_{1}\biggr\} \Delta {t}_{2}\Delta {t}_{1} \\ =&\overline{b}_{1}(\bar{x}_{1},\bar{x}_{2})+ \sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})} ^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}f_{i}({x} _{1},{t}_{1},{x}_{2},{t}_{2}) \\ &{}\times \biggl\{ w_{2}\bigl(\sqrt[q_{1}]{\overline{\xi }_{1}({t}_{1},{t}_{2})}\bigr)+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\ &{}\times w_{2}\bigl(\sqrt[q_{1}]{\overline{\xi }_{1}({m}_{1},{m}_{2})}\bigr) \Delta {m}_{2}\Delta {m}_{1}\biggr\} \Delta {t}_{2} \Delta {t}_{1}. \end{aligned}$$

Let

$$\begin{aligned} \overline{\zeta }_{1}({x}_{1},{x}_{2}) =& \overline{b}_{1}(\bar{x}_{1}, \bar{x}_{2})+ \sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}( {x}_{2})}f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\ &{}\times \biggl\{ w_{2}\bigl(\sqrt[q_{1}]{\overline{\xi }_{1}({t}_{1},{t}_{2})}\bigr)+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x} _{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\ &{}\times w_{2}\bigl(\sqrt[q_{1}]{\overline{\xi }_{1}({m}_{1},{m}_{2})}\bigr) \Delta {m}_{2}\Delta {m}_{1}\biggr\} \Delta {t}_{2}\Delta {t}_{1}. \end{aligned}$$
(3.44)

Then we have

$$ \overline{\zeta }_{1}({x}_{01},{x}_{2})= \overline{\zeta }_{1}({x}_{1}, {x}_{02})= \overline{b}_{1}(\bar{x}_{1},\bar{x}_{2}) $$
(3.45)

and

$$ \overline{\xi }_{1}({x}_{1},{x}_{2})\leq { \overline{\zeta }_{1}}^{\frac{q _{1}}{q_{1}-q_{2}}}({x}_{1},{x}_{2}). $$
(3.46)

It follows from Lemma 2.1 and (3.44) that

$$\begin{aligned}& {\overline{\zeta }_{1}}^{\Delta {{x}_{1}}}({x}_{1},{x}_{2}) \\& \quad =\sum_{i=1} ^{n}{{\gamma }^{\Delta }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})} ^{{\gamma }_{2i}({x}_{2})} f_{i}\bigl(\sigma ({x}_{1}),{{\gamma }_{1i}( {x}_{1})},{x}_{2},{t}_{2} \bigr) \biggl\{ w_{2}\bigl(\sqrt[q_{1}]{\overline{\xi }_{1}\bigl( {{\gamma }_{1i}({x}_{1})},{t}_{2} \bigr)}\bigr) \\& \qquad {}+ \int _{{\gamma }_{1i}({x}_{01})}^{{{\gamma }_{1i}({x}_{1})}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}\bigl({{\gamma }_{1i}({x} _{1})},{m}_{1},{t}_{2},{m}_{2} \bigr) \times w_{2}\bigl(\sqrt[q_{1}]{\overline{ \xi }_{1}({m}_{1},{m}_{2})}\bigr)\Delta {m}_{2}\Delta {m}_{1}\biggr\} \Delta {t}_{2} \\& \qquad {}+\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x} _{1})} \biggl[ \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}f _{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \times \biggl\{ w_{2}\bigl(\sqrt[q_{1}]{\overline{ \xi }_{1}({t}_{1},{t}_{2})}\bigr) \\& \qquad {}+ \int _{{\gamma }_{1i}({x}_{01})}^{{t} _{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1}, {t}_{2},{m}_{2}) w_{2}\bigl(\sqrt[q_{1}]{\overline{\xi }_{1}({m}_{1},{m}_{2})}\bigr) \Delta {m}_{2}\Delta {m}_{1}\biggr\} \Delta {t}_{2}\biggr]^{\Delta x_{1}}\Delta {t}_{1}. \end{aligned}$$

It follows from (3.46) and the fact that \(w_{2}\) and \({\overline{ \zeta }_{1}}\) are nondecreasing that

$$\begin{aligned}& \frac{{\overline{\zeta }_{1}}^{\Delta {{x}_{1}}}({x}_{1},{x}_{2})}{w _{2}(\sqrt[q_{1}-q_{2}]{\overline{\zeta }_{1}({x}_{1},{x}_{2})})} \\& \quad \leq \sum_{i=1}^{n}[{{ \gamma }^{\Delta }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} f_{i}\bigl( \sigma ({x}_{1}),{{\gamma }_{1i}({x}_{1})},{x}_{2},{t}_{2} \bigr) \\& \qquad {}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{{\gamma }_{1i}({x}_{1})}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}\bigl({{\gamma }_{1i}({x} _{1})},{m}_{1},{t}_{2},{m}_{2} \bigr) \Delta {m}_{2}\Delta {m}_{1}\biggr\} \Delta {t}_{2} \\& \qquad {}+\sum_{i=1}^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x} _{1})} \biggl[ \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}f _{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2}) \Delta {m}_{2}\Delta {m}_{1}\biggr\} \Delta {t}_{2}\biggr]^{\Delta x_{1}} \Delta {t}_{1} \\& \quad =\sum_{i=1}^{n}\biggl[ \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x} _{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})}f_{i}( {x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\& \qquad {}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2}) \Delta {m}_{2}\Delta {m}_{1}\biggr\} \Delta {t}_{2}\Delta {t}_{1} \biggr]^{\Delta x_{1}}. \end{aligned}$$

Making use of (3.45) and the definition of \(H_{1}\), integrating over \([{x}_{01},{{x}_{1}}]\) gives

$$\begin{aligned} {H}_{1} \bigl(\overline{\zeta }_{1}({x}_{1},{x}_{2}) \bigr) \leq& {H}_{1} \bigl(\overline{b}_{1}( \bar{x}_{1},{x}_{2}) \bigr) +\sum _{i=1} ^{n} \int _{{\gamma }_{1i}({x}_{01})}^{{\gamma }_{1i}({x}_{1})} \int _{{\gamma }_{2i}({x}_{02})}^{{\gamma }_{2i}({x}_{2})} f_{i}({x} _{1},{t}_{1},{x}_{2},{t}_{2}) \\ &{}\times \biggl\{ 1+ \int _{{\gamma }_{1i}({x}_{01})}^{{t}_{1}} \int _{{\gamma }_{2i}({x}_{02})}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2}) \Delta {m}_{2}\Delta {m}_{1} \biggr\} \Delta {t}_{2} \Delta {t}_{1} \\ =&{H}_{1} \bigl(\overline{b}_{1}(\bar{x}_{1},{x}_{2}) \bigr)+{c}( {x}_{1},{x}_{2}). \end{aligned}$$
(3.47)

Combining (3.38), (3.46), and (3.47), we get

$$ u({x}_{1},{x}_{2})\leq \sqrt[q_{1}-q_{2}]{{{H}_{1}}^{-1} \bigl({H}_{1}\bigl( \overline{b}_{1}(\bar{x}_{1}, \bar{x}_{2})\bigr) +{c}({x}_{1},{x}_{2}) \bigr)}. $$

Letting \({x}_{1}=\bar{x}_{1}\) and \({x}_{2}=\bar{x}_{2}\) in (3.3), and considering \(\bar{x}_{1}\in \mathbb{T}_{1}\) and \(\bar{x}_{2}\in \mathbb{T}_{2}\) are arbitrary, after substituting \(\bar{x}_{1}\) and \(\bar{x}_{2}\) with \({x}_{1}\) and \({x}_{2}\), we obtain the desired result (3.35). □

Let \(\mathbb{T}=\mathrm{Z}\). Then Theorem 3.1 leads to Corollary 3.13 immediately.

Corollary 3.13

Let \(u, r_{i}, a_{j}:A_{1}\times A_{2}\rightarrow \mathbb{R}_{0}^{+}\)and \(f_{i}, g_{i}, f_{i}^{{\Delta {x_{1}}}}:A_{1}^{2}\times A_{2}^{2} \rightarrow \mathbb{R}_{0}^{+}\)be nonnegative real-valued functions such that \(a_{j}\)is nondecreasing with respect to its each variable, let \(\gamma _{ji}:A_{j}\rightarrow \mathbb{R}_{0}^{+}\)be a nonnegative and nondecreasing function, \(\tilde{\mathfrak{a}}: \{-\rho _{1i}, \ldots ,-1,0\}\times \{-\rho _{2i},\ldots ,-1,0\}\rightarrow \mathbb{R}_{0}^{+}\)be a nonnegative function, \(-\infty < \tilde{\mathfrak{p}}_{j}=\inf \{\min ({x}_{j}-\rho _{ji} ), {x}_{j}\in {A}_{j}\}\leq {0}\), wand \(w_{j}\)be as defined in Theorem 3.1. If \(u({x}_{1},{x}_{2})\)satisfies the following discrete inequality

$$\begin{aligned}& w\bigl(u({x}_{1},{x}_{2})\bigr) \\& \quad \leq a_{1}({x}_{1},{x}_{2})+a_{2}({x}_{1},{x}_{2}) \sum_{i=1}^{n} \sum _{{t}_{1}={\gamma }_{1i}(0)}^{{\gamma }_{1i}({x}_{1})-1} \sum_{{t}_{2}={\gamma }_{2i}(0)}^{{\gamma }_{2i}({x}_{2})-1}w_{1} \bigl(u( {t}_{1}-\rho _{1i},{t}_{2}-\rho _{2i})\bigr) \\& \qquad {}\times \Biggl[f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \Biggl\{ w_{2}\bigl(u({t}_{1}-\rho _{1i},{t}_{2}- \rho _{2i})\bigr) +\prod_{l=1}^{2} \sum_{{m}_{l}={\gamma }_{li}(0)} ^{{t}_{l}-1} g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\& \qquad {}\times w_{2}\bigl(u({m}_{1}-\rho _{1i},{m}_{2}- \rho _{2i})\bigr)\Biggr\} +r_{i}({t}_{1}, {t}_{2})\Biggr] \end{aligned}$$
(3.48)

with the following initial condition

$$ \textstyle\begin{cases} w (u({x}_{1},{x}_{2}) )= \tilde{\mathfrak{a}}({x}_{1}, {x}_{2}), \quad {x}_{1}\in [\tilde{\mathfrak{p}}_{1},0] \textit{ or } {x}_{2}\in [\tilde{\mathfrak{p}}_{2},0], \\ \tilde{\mathfrak{a}} ({x}_{1}-\rho _{1i},{x}_{2}- \rho _{2i} ) \leq a_{1}({x}_{1},{x}_{2}) , \quad {x}_{1}\leq \rho _{1i}, \textit{ or } {x}_{2}\leq \rho _{2i}, \end{cases} $$
(3.49)

then

$$ u({x}_{1},{x}_{2})\leq w^{-1} \bigl({H}_{2}^{-1}\bigl({H}_{3}^{-1} \bigl({H}_{3} \bigl( \tilde{b}_{1}({x}_{1},{x}_{2}) \bigr)+a_{2}({x}_{1},{x}_{2})\tilde{c}({x} _{1},{x}_{2})\bigr)\bigr)\bigr) $$
(3.50)

for all \(0\leq {x}_{1}\leq \tilde{x}_{1}\)and \(0\leq {x}_{2}\leq \tilde{x}_{2}\), where

$$\begin{aligned}& \begin{aligned} \tilde{c}({x}_{1},{x}_{2})&=\sum _{i=1}^{n}\sum_{{t}_{1}={\gamma }_{1i}(0)} ^{{\gamma }_{1i}({x}_{1})-1} \sum_{{t}_{2}={\gamma }_{2i}(0)}^{{\gamma } _{2i}({x}_{2})-1}f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\ &\quad {}\times \Biggl(1+\sum_{{m}_{1}={\gamma }_{1i}(0)}^{{t}_{1}-1} \sum _{{m}_{2}={\gamma }_{2i}(0)}^{{t}_{2}-1} g_{i}({t}_{1},{m}_{1}, {t}_{2},{m}_{2})\Biggr), \end{aligned} \\& \tilde{b}_{1}({x}_{1},{x}_{2})= {H}_{2}\bigl(a_{1}({x}_{1},{x}_{2}) \bigr) +a _{2}({x}_{1},{x}_{2})\sum _{i=1}^{n}\sum_{{t}_{1}={\gamma }_{1i}(0)} ^{{\gamma }_{1i}({x}_{1})-1} \sum_{{t}_{2}={\gamma }_{2i}(0)}^{{\gamma } _{2i}({x}_{2})-1} r_{i}({t}_{1},{t}_{2}), \\& {H}_{2}(s)= \int _{s_{7}}^{s}\frac{dp}{w_{1}(w^{-1}(p))},\quad s> s_{7}>0 \textit{ with } {H}_{2}(\infty )=\infty , \\& {H}_{3}(s)= \int _{s_{8}}^{s}\frac{dp}{w_{2}(w^{-1}({H}_{2}^{-1}(p)))},\quad s> s_{8}>0 \textit{ with } {H}_{3}(\infty )=\infty , \end{aligned}$$

\({H}_{2}^{-1}\)and \({H}_{3}^{-1}\)are respectively the inverse functions of \({H}_{2}\)and \({H}_{3}\), and \(\tilde{x}_{1}\)and \(\tilde{x}_{2}\)are chosen such that

$$\begin{aligned}& {H}_{3} \bigl(\tilde{b}_{1}({x}_{1},{x}_{2}) \bigr)+a_{2}({x}_{1},{x}_{2}) \tilde{c}({x}_{1},{x}_{2})\in \operatorname{Dom} \bigl({H}_{3}^{-1}\bigr), \\& {H}_{3}^{-1}\bigl({H}_{3} \bigl( \tilde{b}_{1}({x}_{1},{x}_{2}) \bigr)+a_{2}({x}_{1}, {x}_{2}) \tilde{c}({x}_{1},{x}_{2})\bigr)\in \operatorname{Dom} \bigl({H}_{2}^{-1}\bigr), \\& {H}_{2}^{-1}\bigl({H}_{3}^{-1} \bigl({H}_{3} \bigl(\tilde{b}_{1}({x}_{1},{x}_{2}) \bigr) +a _{2}({x}_{1},{x}_{2}) \tilde{c}({x}_{1},{x}_{2})\bigr)\bigr)\in \operatorname{Dom} \bigl(w ^{-1}\bigr). \end{aligned}$$

Applications

Example 4.1

Consider the following integro-differential equation with several arguments:

$$\begin{aligned} \bigl[u({x}_{1},{x}_{2})\bigr]^{\Delta {x}_{1}\Delta {x}_{2}} =& \mathrm{F}\biggl[{x} _{1},{t}_{1},{x}_{2},{t}_{2}, u\bigl(\mu _{11}({t}_{1}),\mu _{21}({t}_{2}) \bigr), \ldots , u\bigl(\mu _{1n}({t}_{1}),\mu _{2n}({t}_{2})\bigr), \\ &\int _{{x}_{01}}^{{t}_{1}} \int _{{x}_{02}}^{{t}_{2}}\mathbf{Q}\bigl({t}_{1}, {m}_{1},{t}_{2},{m}_{2}, u\bigl(\mu _{11}({m}_{1}),\mu _{21}({m}_{2}) \bigr), \\ &\ldots , u\bigl(\mu _{1n}({m}_{1}),\mu _{2n}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1}\biggr] \end{aligned}$$
(4.1)

with the initial condition

$$ \textstyle\begin{cases} [u({x}_{1},{x}_{02})]^{\Delta {x}_{2}}=\mathfrak{a}_{1}^{\Delta }( {x}_{1}),\qquad u({x}_{01},{x}_{2})=\mathfrak{a}_{2}({x}_{2}), \\ u({x}_{1},{x}_{2})=\mathfrak{a}({x}_{1},{x}_{2}), \quad {x}_{1}\in [\mathfrak{p}_{1},{x}_{01}]_{\mathbb{T}} \text{ or } {x}_{2}\in [\mathfrak{p}_{2},{x}_{02}]_{\mathbb{T}}, \\ |\mathfrak{a}(\mu _{1i}({x}_{1}),\mu _{2i}({x}_{2}))|\leq |a_{1}({x} _{1},{x}_{2})| , \quad \mu _{1i}({x}_{1})\leq {x}_{01} \text{ or } \mu _{2i}({x} _{2})\leq {x}_{02} \end{cases} $$
(4.2)

for \(\mathrm{F}:\mathbb{T}^{2}_{1}\times \mathbb{T}^{2}_{2}\times \mathrm{R}^{n+1}\rightarrow \mathbb{R}\) is right-dense continuous on \(\mathbb{T}^{2}_{1}\times \mathbb{T}^{2}_{2}\) and continuous on \(\mathbb{R}^{n+1}\), \(\mathbf{Q}:\mathbb{T}_{1}^{2}\times \mathbb{T} _{2}^{2}\times \mathbb{R}^{n}\rightarrow \mathbb{R}\) is right-dense continuous on \(\mathbb{T}^{2}_{1}\times \mathbb{T}^{2}_{2}\) and continuous on \(\mathbb{R}^{n}\), \(u:\mathbb{T}_{1}\times \mathbb{T} _{2}\rightarrow \mathbb{R} /\{0\}\), \(\mathfrak{a}_{j}:\mathbb{T}_{j} \rightarrow \mathbb{R}\), \(\mathfrak{a}: ([\mathfrak{p}_{1},{x}_{01}] \times [\mathfrak{p}_{2},{x}_{02}])_{{\mathbb{T}}^{2}}\rightarrow \mathbb{R}\), \(a_{1}:\mathbb{T}_{1}\times \mathbb{T}_{2}\rightarrow \mathbb{R}\) are right-dense continuous functions and \(\mu _{ji}\) is as defined in Theorem 3.1.

Theorem 4.2

Assume that

$$ \left . \textstyle\begin{array}{l} |\mathrm{F}({x}_{1},{t}_{1},{x}_{2},{t}_{2}, \mathfrak{k}_{1}, \mathfrak{k}_{2},\ldots , \mathfrak{k}_{n},k)|\leq a_{2}({x}_{1},{x} _{2})\sum_{i=1}^{n}w_{1}( \vert \mathfrak{k}_{i} \vert ) \\ \hphantom{|\mathrm{F}({x}_{1},{t}_{1},{x}_{2},{t}_{2}, \mathfrak{k}_{1}, \mathfrak{k}_{2},\ldots , \mathfrak{k}_{n},k)|\leq{}}{}\times [f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \{w_{2}( \vert \mathfrak{k} _{i} \vert ) + \vert k \vert \}+r_{i}({t}_{1},{t}_{2})], \\ |\mathbf{Q}({x}_{1},{t}_{1},{x}_{2},{t}_{2},\mathfrak{k}_{1},\ldots , \mathfrak{k}_{n})|\leq g_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2})w_{2}( \vert \mathfrak{k}_{i} \vert ), \end{array}\displaystyle \right \} $$
(4.3)

where \(f_{i}\), \(g_{i}\), \(r_{i}\), \(a_{2}\)are as defined in Theorem 3.1, \(a_{1}({x}_{1},{x}_{2})=\sum_{j=1}^{2}\mathfrak{a}_{j}({x}_{j})\), \(w_{1}(\eta )=\sqrt[3]{{\sigma ^{2}}(\eta )}+\sqrt[3]{{\sigma }(\eta ) \eta }+\sqrt[3]{\eta ^{2}}\), \(w_{2}(\eta )=\sqrt{{\sigma }(\sqrt[3]{ \eta })}+\sqrt[6]{\eta }\)for \(\eta \in \mathbb{R}_{0}^{+}\), and \(u({x}_{1},{x}_{2})\)is a solution of equation (4.1) with initial condition (4.2), then

$$ \bigl|u({x}_{1},{x}_{2})\bigr|\leq \bigl(\sqrt{ \mathfrak{b}_{3}({x}_{1},{x}_{2})} +a _{2}({x}_{1},{x}_{2})\mathfrak{b}_{4}({x}_{1},{x}_{2}) \bigr)^{6}, $$

where

$$\begin{aligned} &\mathfrak{b}_{3}({x}_{1},{x}_{2})= \sqrt[3]{ \bigl\vert a_{1}({x}_{1},{x}_{2}) \bigr\vert } +a _{2}({x}_{1},{x}_{2})\sum _{i=1}^{n} \int _{{x}_{01}}^{{x_{1}}} \int _{{x}_{02}}^{{x}_{2}}r_{i}({t}_{1},{t}_{2}) \Delta {t}_{2}\Delta {t}_{1}, \\ &\mathfrak{b}_{4}({x}_{1},{x}_{2})=\sum _{i=1}^{n} \int _{{x}_{01}}^{ {x}_{1}} \int _{{x}_{02}}^{{x}_{2}} f_{i}({x}_{1},{t}_{1},{x}_{2},{t} _{2}) \\ &\hphantom{\mathfrak{b}_{4}({x}_{1},{x}_{2})={}}{}\times \biggl(1+ \int _{{x}_{01}}^{{t}_{1}} \int _{{x}_{02}}^{{t}_{2}} g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \Delta {m}_{2}\Delta {m}_{1}\biggr) \Delta {t}_{2}\Delta {t}_{1}. \end{aligned}$$
(4.4)

Proof

Let \(\tilde{\mathbb{T}}=\varrho (\mathbb{T}_{1},{x}_{2})\) and \(\tilde{G}_{j}(\eta )=\sqrt[4-j]{\eta }\), where \(\varrho ({x}_{1}, {x}_{2})\) is strictly increasing with respect to \({x}_{1}\in \mathbb{T}_{1}\). Then it follows from Theorem 2.3 that

$$\begin{aligned}& \begin{aligned} \bigl[\tilde{G}_{1}\bigl(\varrho ({x}_{1},{x}_{2}) \bigr)\bigr]^{\Delta {x}_{1}} &= \tilde{G}_{1}^{\tilde{\Delta }}(\varrho ){\varrho }^{\Delta {x}_{1}}( {x}_{1},{x}_{2}) \\ &=\frac{{\varrho }^{\Delta {x}_{1}}({x}_{1},{x}_{2})}{\sqrt[3]{{\sigma ^{2}}(\varrho ({x}_{1},{x}_{2}))} +\sqrt[3]{{\sigma }(\varrho ({x} _{1},{x}_{2}))\varrho ({x}_{1},{x}_{2})}+\sqrt[3]{{\varrho ^{2}({x} _{1},{x}_{2})}}} \\ &=\frac{{\varrho }^{\Delta {x}_{1}}({x}_{1},{x}_{2})}{w_{1}({\varrho }( {x}_{1},{x}_{2}))} =\frac{{\varrho }^{\Delta {x}_{1}}({x}_{1},{x}_{2})}{w _{1}(w^{-1}(\varrho ({x}_{1},{x}_{2})))}, \end{aligned} \\& \begin{aligned} \bigl[\tilde{G}_{2}\bigl(\varrho ({x}_{1},{x}_{2}) \bigr)\bigr]^{\Delta {x}_{1}}&= \tilde{G}_{2}^{\tilde{\Delta }}(\varrho ){\varrho }^{\Delta {x}_{1}}( {x}_{1},{x}_{2}) \\ &=\frac{{\varrho }^{\Delta {x}_{1}}({x}_{1},{x}_{2})}{\sqrt{\sigma ( {{\varrho ({x}_{1},{x}_{2})}})} +\sqrt{{\varrho ({x}_{1},{x}_{2})}}} \\ &=\frac{{\varrho }^{\Delta {x}_{1}}({x}_{1},{x}_{2})}{w_{2}({\varrho } ^{3}({x}_{1},{x}_{2}))} =\frac{{\varrho }^{\Delta {x}_{1}}({x}_{1}, {x}_{2})}{w_{2}(w^{-1}(\tilde{{G}}_{1}^{-1}({\varrho }({x}_{1},{x} _{2}))))}. \end{aligned} \end{aligned}$$

From (4.1) and (4.2) we get

$$\begin{aligned} u({x}_{1},{x}_{2}) =&a_{1}({x}_{1},{x}_{2})+ \int _{{x}_{01}}^{{x}_{1}} \int _{{x}_{02}}^{{x}_{2}}\mathrm{F}\biggl[{x}_{1},{t}_{1},{x}_{2},{t}_{2}, u\bigl( \mu _{11}({t}_{1}),\mu _{21}({t}_{2}) \bigr), \\ &\ldots , u\bigl(\mu _{1n}({t}_{1}),\mu _{2n}({t}_{2})\bigr), \int _{{x}_{01}}^{ {t}_{1}} \int _{{x}_{02}}^{{t}_{2}}\mathbf{Q}\bigl({t}_{1},{m}_{1},{t}_{2}, {m}_{2}, u\bigl(\mu _{11}({m}_{1}),\mu _{21}({m}_{2})\bigr), \\ &\ldots , u\bigl(\mu _{1n}({m}_{1}),\mu _{2n}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1}\biggr] \Delta {t}_{2}\Delta {t}_{1}. \end{aligned}$$
(4.5)

From (4.3) we know that (4.5) has the form

$$\begin{aligned} \bigl|u({x}_{1},{x}_{2})\bigr| \leq& \bigl|a_{1}({x}_{1},{x}_{2}) \bigr\vert + \int _{{x}_{01}} ^{{x}_{1}} \int _{{x}_{02}}^{{x}_{2}} \biggl\vert \mathrm{F} \biggl[{x}_{1},{t}_{1},{x} _{2},{t}_{2}, u\bigl(\mu _{11}({t}_{1}),\mu _{21}({t}_{2}) \bigr), \\ &\ldots , u\bigl(\mu _{1n}({t}_{1}),\mu _{2n}({t}_{2})\bigr), \int _{{x}_{01}}^{ {t}_{1}} \int _{{x}_{02}}^{{t}_{2}}\mathbf{Q}\bigl({t}_{1},{m}_{1},{t}_{2}, {m}_{2}, u\bigl(\mu _{11}({m}_{1}),\mu _{21}({m}_{2})\bigr), \\ &\ldots , u\bigl(\mu _{1n}({m}_{1}),\mu _{2n}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1}\biggr] \biggr|\Delta {t}_{2}\Delta {t}_{1} \\ \leq& \bigl|a_{1}({x}_{1},{x}_{2})\bigr|+a_{2}({x}_{1},{x}_{2}) \sum_{i=1}^{n} \int _{{x}_{01}}^{{x}_{1}} \int _{{x}_{02}}^{{x}_{2}}w_{1}\bigl(\bigl|u\bigl( \mu _{1i}( {t}_{1}),\mu _{2i}({t}_{2}) \bigr)\bigr|\bigr) \\ &{}\times \biggl[{f}_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \biggl\{ w_{2}\bigl(\bigl|u\bigl(\mu _{1i}( {t}_{1}),\mu _{2i}({t}_{2})\bigr) \bigr\vert \bigr) + \int _{{x}_{01}}^{{t}_{1}} \int _{{x}_{02}} ^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t}_{2},{m}_{2}) \\ &{}\times w_{2}\bigl(\bigl|u\bigl(\mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr) \bigr\vert \bigr)\Delta {m}_{2} \Delta {m}_{1}\biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2}\Delta {t}_{1}. \end{aligned}$$
(4.6)

Therefore, the desired result follows easily from (3.3) and (4.6). □

Example 4.3

Consider another integro-differential equation with several arguments:

$$\begin{aligned}& \bigl[u^{\frac{2}{5}}({x}_{1},{x}_{2})u^{\Delta {x}_{1}}({x}_{1},{x}_{2}) \bigr]^{ \Delta {x}_{2}} \mathrm{F}\biggl[{x}_{1},{t}_{1},{x}_{2},{t}_{2}, u\bigl(\mu _{11}( {t}_{1}),\mu _{21}({t}_{2}) \bigr), \\& \quad \ldots , u\bigl(\mu _{1n}({t}_{1}),\mu _{2n}({t}_{2})\bigr), \int _{{x}_{01}}^{ {t}_{1}} \int _{{x}_{02}}^{{t}_{2}}\mathbf{Q}\bigl({t}_{1},{m}_{1},{t}_{2}, {m}_{2}, u\bigl(\mu _{11}({m}_{1}),\mu _{21}({m}_{2})\bigr), \\& \quad \ldots , u\bigl(\mu _{1n}({m}_{1}),\mu _{2n}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1}\biggr] \end{aligned}$$
(4.7)

with the initial condition

$$ \textstyle\begin{cases} u^{\frac{2}{5}}({x}_{1},{x}_{02})u^{\Delta {x}_{1}}({x}_{1},{x}_{02}) =\frac{5\mathfrak{f}_{1}^{\Delta }({x}_{1})}{3},\qquad u^{\frac{3}{5}}({x}_{01},{x}_{2})=\mathfrak{f}_{2}({x}_{2}), \\ u({x}_{1},{x}_{2})={\mathfrak{a}}({x}_{1},{x}_{2}), \quad {x}_{1}\in [\mathfrak{p}_{1},{x}_{01}]_{\mathbb{T}} \text{ or } {x}_{2}\in [\mathfrak{p}_{2},{x}_{02}]_{\mathbb{T}}, \\ |{\mathfrak{a}}(\mu _{1i}({x}_{1}),\mu _{2i}({x}_{2}))|\leq \mathfrak{C}^{\frac{1}{5}} , \quad \mu _{1i}({x}_{1})\leq {x}_{01} \text{ or } \mu _{2i}({x} _{2})\leq {x}_{02} \end{cases} $$
(4.8)

for \(\mathrm{F}:\mathbb{T}^{2}_{1}\times \mathbb{T}^{2}_{2}\times \mathbb{R}^{n+1}\rightarrow \mathbb{R}\) is right-dense continuous on \(\mathbb{T}^{2}_{1}\times \mathbb{T}^{2}_{2}\) and continuous on \(\mathbb{R}^{n+1}\), \(\mathbf{Q}:\mathbb{T}_{1}^{2}\times \mathbb{T} _{2}^{2}\times \mathbb{R}^{n}\rightarrow \mathbb{R}\) is right-dense continuous on \(\mathbb{T}^{2}_{1}\times \mathbb{T}^{2}_{2}\) and continuous on \(\mathbb{R}^{n}\), \(u: \mathbb{T}_{1}\times \mathbb{T} _{2}\rightarrow \mathbb{R} / \{0\}\), \(\mathfrak{f}_{j}:\mathbb{T}_{j} \rightarrow \mathbb{R}\), \(\mathfrak{a}:([\mathfrak{p}_{1},{x}_{01}] \times [\mathfrak{p}_{2},{x}_{02}])_{{\mathbb{T}}^{2}}\rightarrow \mathbb{R}\) are right-dense continuous functions, \(\mathfrak{C}\) is a nonzero constant such that \(\mathfrak{C}\geq \sum_{j=1}^{2}| \mathfrak{f}_{j}({x}_{j})|\) and \(\mu _{ji}\) is as defined in Theorem 3.1.

Theorem 4.4

Assume that

$$ \left . \textstyle\begin{array}{l} |\mathrm{F}({x}_{1},{t}_{1},{x}_{2},{t}_{2}, \mathfrak{k}_{1}, \mathfrak{k}_{2},\ldots , \mathfrak{k}_{n},k)| \\ \quad \leq \sum_{i=1}^{n}\frac{ \vert \mathfrak{k}_{i} \vert ^{2}}{3}[f_{i}({x}_{1}, {t}_{1},{x}_{2},{t}_{2}) \{w_{2}( \vert \mathfrak{k}_{i} \vert ) + \vert k \vert \}+r_{i}( {t}_{1},{t}_{2})], \\ |\mathrm{Q}({x}_{1},{t}_{1},{x}_{2},{t}_{2},\mathfrak{k}_{1},\ldots , \mathfrak{k}_{n})|\leq g_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2})w_{2}( \vert \mathfrak{k}_{i} \vert ), \end{array}\displaystyle \right \} $$
(4.9)

where \(f_{i}\), \(g_{i}\), and \(r_{i}\)are as defined in Theorem 3.1, \(w_{2}(\eta )=\sqrt[3]{{\sigma ^{2}}(\eta ^{3})}+\sqrt[3]{{\sigma }(\eta ^{3})\eta ^{3}}+{\eta ^{2}}\)for \(\eta \in \mathbb{R}_{0}^{+}\). If \(u({x}_{1},{x}_{2})\)is a solution of equation (4.7) satisfying initial condition (4.8), then

$$ \bigl|u({x}_{1},{x}_{2})\bigr|\leq \sqrt[3]{\overline{ \mathfrak{b}}_{3}({x}_{1}, {x}_{2})} + \mathfrak{b}_{4}({x}_{1},{x}_{2}), $$

where \(\mathfrak{b}_{4}({x}_{1},{x}_{2})\)is defined by (4.4) and

$$ \overline{\mathfrak{b}}_{3}({x}_{1},{x}_{2}):= \mathfrak{C}^{ \frac{3}{5}} +\sum_{i=1}^{n} \int _{{x}_{01}}^{{x}_{1}} \int _{{x}_{02}} ^{{x}_{2}}r_{i}({t}_{1},{t}_{2}) \Delta {t}_{2}\Delta {t}_{1}. $$

Proof

Let \(\overline{\mathbb{T}}=\varpi (\mathbb{T}_{1},{x}_{2})\) and \(\overline{G}_{2}(\eta )=\sqrt[3]{\eta }\), where ϖ is strictly increasing with respect to \({x}_{1}\in \mathbb{T}_{1}\). Then by Theorem 2.3 we have

$$\begin{aligned} \bigl[\overline{G}_{2}\bigl(\varpi ({x}_{1},{x}_{2}) \bigr)\bigr]^{\Delta {x}_{1}} =& \overline{G}_{2}^{\overline{\Delta }}( \varpi ){\varpi }^{\Delta {x} _{1}}({x}_{1},{x}_{2}) \\ =&\frac{{\varpi }^{\Delta {x}_{1}}({x}_{1},{x}_{2})}{\sqrt[3]{{\sigma ^{2}(\varpi ({x}_{1},{x}_{2}))}} +\sqrt[3]{{\sigma }(\varpi ({x}_{1}, {x}_{2}))\varpi ({x}_{1},{x}_{2})} +\sqrt[3]{{\varpi }^{2}({x}_{1}, {x}_{2})}} \\ =&\frac{{\varpi }^{\Delta {x}_{1}}({x}_{1},{x}_{2})}{w_{2}(\sqrt[3]{ \varpi ({x}_{1},{x}_{2})})}. \end{aligned}$$

Integrating equation (4.7) over \([{x}_{02},{{x}_{2}}]\) gives

$$\begin{aligned}& u^{\frac{2}{5}}({x}_{1},{x}_{2})u^{\Delta {{x}_{1}}}({x}_{1},{x}_{2}) \\& \quad =u^{\frac{2}{5}}({x}_{1},{x}_{02})u^{\Delta {{x}_{1}}}({x}_{1},{x} _{02}) + \int _{{x}_{02}}^{{x}_{2}}\mathrm{F}\biggl[{x}_{1},{t}_{1},{x}_{2}, {t}_{2}, u\bigl(\mu _{11}({t}_{1}),\mu _{21}({t}_{2})\bigr),\ldots , \\& \qquad u\bigl(\mu _{1n}({t}_{1}),\mu _{2n}({t}_{2}) \bigr), \int _{{x}_{01}}^{{t}_{1}} \int _{{x}_{02}}^{{t}_{2}}\mathbf{Q}\bigl({t}_{1},{m}_{1},{t}_{2},{m}_{2}, u\bigl( \mu _{11}({m}_{1}),\mu _{21}({m}_{2}) \bigr), \ldots , \\& \qquad u\bigl(\mu _{1n}({m}_{1}),\mu _{2n}({m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1}\biggr] \Delta {t}_{2}. \end{aligned}$$
(4.10)

It follows from Theorem 2.2 and \(\frac{u^{\Delta {{x}_{1}}}({x}_{1}, {x}_{2})}{u({x}_{1},{x}_{2})}\geq 0\) that

$$\begin{aligned} \biggl(\frac{5}{3}u^{\frac{3}{5}}({x}_{1},{x}_{2}) \biggr)^{\Delta {{x}_{1}}} =&u^{\Delta {{x}_{1}}}({x}_{1},{x}_{2}) \int _{0}^{1} \bigl\{ u({x}_{1},{x}_{2}) +h\mu ({x}_{1},{x}_{2})u^{\Delta {{x}_{1}}}( {x}_{1},{x}_{2})\bigr\} ^{-\frac{2}{5}}\,dh \\ =&\frac{u^{\Delta {{x}_{1}}}({x}_{1},{x}_{2})}{u^{\frac{2}{5}}({x}_{1}, {x}_{2})} \int _{0}^{1}\biggl\{ 1+h\mu ({x}_{1},{x}_{2}) \frac{u^{\Delta {{x}_{1}}}({x}_{1},{x}_{2})}{u({x}_{1},{x}_{2})}\biggr\} ^{-\frac{2}{5}}\,dh \\ =&\frac{u^{\Delta {{x}_{1}}}({x}_{1},{x}_{2})}{u^{\frac{2}{5}}({x}_{1}, {x}_{2})} \times \frac{5\{1+\mu ({x}_{1},{x}_{2})\frac{u^{\Delta {{x}_{1}}}(\mathfrak{x}_{1},\mathfrak{x}_{2})}{u({x}_{1},{x}_{2})}\} ^{\frac{3}{5}}-5}{3\mu ({x}_{1},{x}_{2})\frac{u^{\Delta {{x}_{1}}}( {x}_{1},\mathfrak{x}_{2})}{u({x}_{1},{x}_{2})}} \\ \leq& \frac{u^{\Delta {{x}_{1}}}({x}_{1},{x}_{2})}{u^{\frac{2}{5}}( {x}_{1},{x}_{2})}. \end{aligned}$$
(4.11)

From (4.10) and (4.11) one has

$$\begin{aligned}& \biggl(\frac{5}{3}u^{\frac{3}{5}}({x}_{1},{x}_{2}) \biggr)^{\Delta {{x}_{1}}} \\& \quad \leq \frac{5\mathfrak{f}_{1}^{\Delta }({x}_{1})}{3}+ \int _{{x}_{02}}^{{x}_{2}}\mathrm{F}\biggl[{x}_{1},{t}_{1},{x}_{2},{t}_{2}, u\bigl( \mu _{11}({t}_{1}),\mu _{21}({t}_{2}) \bigr), \\& \qquad \ldots , u\bigl(\mu _{1n}({t}_{1}),\mu _{2n}({t}_{2})\bigr), \int _{{x}_{01}}^{ {t}_{1}} \int _{{x}_{02}}^{{t}_{2}}\mathbf{Q}\bigl({t}_{1},{m}_{1},{t}_{2}, {m}_{2}, u\bigl(\mu _{11}({m}_{1}),\mu _{21}({m}_{2})\bigr), \ldots , \\& \qquad u\bigl(\mu _{1n}({m}_{1}),\mu _{2n}({m}_{2}) \bigr)\bigr)\Delta {m}_{2}\Delta {m}_{1}\biggr] \Delta {t}_{2}. \end{aligned}$$

Integrating over \([\mathfrak{x}_{01},{\mathfrak{x}_{1}}]\) leads to

$$\begin{aligned} u^{\frac{3}{5}}({x}_{1},{x}_{2}) \leq& \mathfrak{f}_{1}({x}_{1})+ \mathfrak{f}_{2}({x}_{2})+ \frac{3}{5} \int _{{x}_{01}}^{{x}_{1}} \int _{{x}_{02}}^{{x}_{2}}\mathrm{F}\biggl[{x}_{1},{t}_{1},{x}_{2},{t}_{2}, u\bigl( \mu _{11}({t}_{1}),\mu _{21}({t}_{2}) \bigr), \\ &\ldots , u\bigl(\mu _{1n}({t}_{1}),\mu _{2n}({t}_{2})\bigr), \int _{{x}_{01}}^{ {t}_{1}} \int _{{x}_{02}}^{{t}_{2}}\mathbf{Q}\bigl({t}_{1},{m}_{1},{t}_{2}, {m}_{2}, u\bigl(\mu _{11}({m}_{1}),\mu _{21}({m}_{2})\bigr), \\ &\ldots , u\bigl(\mu _{1n}({m}_{1}),\mu _{2n}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1}\biggr] \Delta {t}_{2}\Delta {t}_{1}. \end{aligned}$$
(4.12)

From (4.9) we know that inequality (4.12) has the form

$$\begin{aligned} \bigl|u^{\frac{3}{5}}({x}_{1},{x}_{2})\bigr| \leq& \mathfrak{C}+ \frac{3}{5} \int _{{x}_{01}}^{{x}_{1}} \int _{{x}_{02}}^{{x}_{2}}\biggl|\mathrm{F}\biggl[{x} _{1},{t}_{1},{x}_{2},{t}_{2}, u \bigl(\mu _{11}({t}_{1}),\mu _{21}({t}_{2}) \bigr), \\ &\ldots , u\bigl(\mu _{1n}({t}_{1}),\mu _{2n}({t}_{2})\bigr), \int _{{x}_{01}}^{ {t}_{1}} \int _{{x}_{02}}^{{t}_{2}}\mathbf{Q}\bigl({t}_{1},{m}_{1},{t}_{2}, {m}_{2}, u\bigl(\mu _{11}({m}_{1}),\mu _{21}({m}_{2})\bigr), \\ &\ldots , u\bigl(\mu _{1n}({m}_{1}),\mu _{2n}({m}_{2})\bigr)\bigr)\Delta {m}_{2} \Delta {m}_{1}\biggr]\biggr|\Delta {t}_{2}\Delta {t}_{1} \\ \leq& \mathfrak{C}+\frac{1}{5}\sum_{i=1}^{n} \int _{{x}_{01}}^{{x}_{1}} \int _{{x}_{02}}^{{x}_{2}}\bigl|u\bigl(\mu _{1i}({t}_{1}), \mu _{2i}({t}_{2})\bigr)\bigr|^{2} \biggl[{f}_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2}) \\ &{}\times \biggl\{ w_{2}\bigl(\bigl|u\bigl(\mu _{1i}({t}_{1}), \mu _{2i}({t}_{2})\bigr) \bigr\vert \bigr)+ \int _{{x} _{01}}^{{t}_{1}} \int _{{x}_{02}}^{{t}_{2}}g_{i}({t}_{1},{m}_{1},{t} _{2},{m}_{2}) \\ &{}\times w_{2}\bigl(\bigl|u\bigl(\mu _{1i}({m}_{1}), \mu _{2i}({m}_{2})\bigr) \bigr\vert \bigr)\Delta {m}_{2} \Delta {m}_{1}\biggr\} +r_{i}({t}_{1},{t}_{2}) \biggr]\Delta {t}_{2}\Delta {t}_{1}. \end{aligned}$$
(4.13)

Therefore, the desired result follows from (3.5) and (4.13). □

Example 4.5

Consider the delay discrete inequality (4.8) satisfying initial condition (4.9) with \(u({x}_{1},{x}_{2})=27^{{x}_{1}{x}_{2}}\), \({\rho }_{ji}=ji\), \(a_{1}({x}_{1},{x}_{2})=27^{8}\), \(a_{2}({x}_{1}, {x}_{2})=\sqrt[10]{{x}_{1}{x}_{2}}\), \(f_{i}({x}_{1},{t}_{1},{x}_{2}, {t}_{2})=\arctan (\sqrt[i+1]{{x}_{1}+{t}_{1}+{x}_{2}+{t}_{2}})\), \(r_{i}({x}_{1},{x}_{2})=\sqrt[i+1]{\exp ({{x}_{1}}{{x}_{2}})}\), \(\gamma _{ji}=w=I\), \(g_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2})=10^{-i-1} \sqrt[i+1]{ {x}_{1}+{t}_{1}+{x}_{2}+{t}_{2}}\) (\(1\leq i\leq 2\)), and \({w}_{j}\) is defined as in Theorem 4.2.

We find that the numerical solution agrees with the analytical solution for some discrete inequalities by calculating the value of \(u(x,y)\) from (4.8) and (4.10) (see Table 1).

Table 1 The value of \(u(x; y)\) from (4.8) and (4.10)

Conclusion

In the article, we have presented several explicit bounds for the delay double integral inequalities on time scales and have given their applications to the solutions of certain integro-differential equations. Our results are the improvements and generalizations of some previously known results. Furthermore, we have found some new estimates for the integral inequalities in the form of exponential function.

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Acknowledgements

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

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The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485).

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Rafeeq, S., Kalsoom, H., Hussain, S. et al. Delay dynamic double integral inequalities on time scales with applications. Adv Differ Equ 2020, 40 (2020). https://doi.org/10.1186/s13662-020-2516-3

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MSC

  • 26D10
  • 34C11
  • 39A12

Keywords

  • Delay integral inequality
  • Time scale
  • Dynamic equation
  • Discrete inequality
  • Boundedness