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Certain new dynamic nonlinear inequalities in two independent variables and applications

Abstract

Several inequalities were proved in 2018 by Boudeliou, in 2015 by Abdeldain and El-Deeb and in 1998 by Pachpatte. It is our aim in this paper to generalize these inequalities to time scales. Beside that, we also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as special cases. Furthermore, we study the boundedness of some problems by applying our results.

Introduction

In 2018, Boudeliou [9] discussed the following inequalities.

Theorem 1.1

Suppose\(a\in C(\hat{\varOmega } ,\mathbb{R}_{+})\)is nondecreasing with respect to\((\breve{x},\breve{y}) \in \hat{\varOmega }=I_{1}\times I_{2} \), let\(\hat{\alpha }(\breve{x})\in C^{1}(I_{1},I_{2})\)and\(\hat{\beta }(\breve{y})\in C^{1}(I_{2},I_{2})\)be nondecreasing functions with\(\hat{\alpha }(\breve{x})\leq \breve{x}\)on\(I_{1}\), \(\hat{\beta }(\breve{y})\leq \breve{y}\), andg, u, p, \(f\in C(\hat{\varOmega } ,\mathbb{R}_{+})\). Furthermore, supposeψ̄, \(\bar{\varphi } \in C(\mathbb{R}_{+},\mathbb{R}_{+})\)are nondecreasing functions with\(\{ \bar{\psi } ,\bar{\varphi } \} (u)>0\)for\(u>0\), and\(\lim_{u\rightarrow +\infty } \bar{\psi } (u)=+\infty \). If\(u(\breve{x},\breve{y}) \)satisfies

$$\begin{aligned} \bar{\psi } \bigl( u(\breve{x},\breve{y}) \bigr) \leq &a( \breve{x},\breve{y}) + \int _{0}^{\hat{\alpha }(\breve{x})} \int _{0}^{ {\hat{\beta }}(\breve{y})} \bigl[ f(\breve{s},\breve{t})\bar{ \varphi } \bigl( u(\breve{s},\breve{t}) \bigr) +p(\breve{s},\breve{t}) \bigr] \,d \breve{t}\,d \breve{s} \\ &{}+ \int _{0}^{\hat{\alpha }(\breve{x})} \int _{0}^{\hat{\beta }( \breve{y})}f(\breve{s},\breve{t}) \biggl( \int _{0}^{\breve{s}}g( \breve{\tau },\breve{t})\bar{ \varphi } \bigl( u(\breve{\tau }, \breve{t}) \bigr) \,d \breve{\tau } \biggr) \,d \breve{t}\,d \breve{s}, \end{aligned}$$

for\((\breve{x},\breve{y}) \in \hat{\varOmega }\), then

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \biggl\{ \breve{G}^{-1} \biggl[ \breve{G} \bigl( q(\breve{x},\breve{y}) \bigr) + \int _{0} ^{\hat{\alpha }(\breve{x})} \int _{0}^{\hat{\beta }(\breve{y})}f( \breve{s},\breve{t}) \biggl( 1+ \int _{0}^{\breve{s}}g(\breve{\tau }, \breve{t})\,d \breve{ \tau } \biggr) \,d \breve{t}\,d \breve{s} \biggr] \biggr\} , $$

for\(0\leq \breve{x}\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\), where

$$\begin{aligned}& q(\breve{x},\breve{y}) =a(\breve{x},\breve{y}) + \int _{0}^{ \hat{\alpha }(\breve{x})} \int _{0}^{\hat{\beta }(\breve{y})}p( \breve{s},\breve{t})\,d \breve{t}\,d \breve{s}, \\& \breve{G}(r)= \int _{r_{0}}^{r}\frac{d \breve{s}}{\bar{\varphi } \circ \bar{ \psi } ^{-1}(\breve{s})},\quad r\geq r_{0}>0,\qquad \breve{G}(+\infty )= \int _{r_{0}}^{+\infty }\frac{d \breve{s}}{\bar{\varphi } \circ \bar{ \psi } ^{-1}(\breve{s})}=+\infty , \end{aligned}$$

and\(( \breve{x}_{1},\breve{y}_{1} ) \in \hat{\varOmega } \)is chosen so that

$$ \biggl( \breve{G} \bigl( q(\breve{x},\breve{y}) \bigr) + \int _{0} ^{\hat{\alpha }(\breve{x})} \int _{0}^{\hat{\beta }(\breve{y})}f( \breve{s},\breve{t}) \biggl( 1+ \int _{0}^{\breve{s}}g(\breve{\tau }, \breve{t})\,d \breve{ \tau } \biggr) \,d \breve{t}\,d \breve{s} \biggr) \in \operatorname{Dom} \bigl( G^{-1} \bigr). $$

Theorem 1.2

Assume thatg, a, f, u, β̂, α̂, ,ψ̄andφ̄be as in Theorem1.1. If\(u(\breve{x},\breve{y}) \)satisfies

$$\begin{aligned} \bar{\psi } \bigl( u(\breve{x},\breve{y}) \bigr) \leq &a( \breve{x},\breve{y}) + \biggl( \int _{0}^{\hat{\alpha }(\breve{x})} \int _{0}^{\hat{\beta }(\breve{y})}f(\breve{s},\breve{t})\bar{\varphi } \bigl( u(\breve{s},\breve{t}) \bigr) \,d \breve{t}\,d \breve{s} \biggr) ^{2} \\ &{}+ \int _{0}^{\hat{\alpha }(\breve{x})} \int _{0}^{\hat{\beta }( \breve{y})}f(\breve{s},\breve{t})\bar{\varphi } \bigl( u(\breve{s}, \breve{t}) \bigr) \biggl( \int _{0}^{\breve{s}}g(\breve{\tau }, \breve{t})\bar{ \varphi } \bigl( u(\breve{\tau },\breve{t}) \bigr) \,d \breve{\tau } \biggr) \,d \breve{t}\,d \breve{s}, \end{aligned}$$

for\((\breve{x},\breve{y}) \in \hat{\varOmega } \), then

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \biggl\{ \breve{H}^{-1} \biggl[ \breve{H} \bigl( a ( \breve{x},\breve{y} ) \bigr) +\breve{B}(\breve{x},\breve{y}) + \biggl( \int _{0}^{\hat{\alpha }( \breve{x})} \int _{0}^{\hat{\beta }(\breve{y})}f(\breve{s},\breve{t})\,d \breve{t}\,d \breve{s} \biggr) ^{2} \biggr] \biggr\} , $$

for\(0\leq \breve{x} \leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\), where

$$\begin{aligned}& \breve{B}(\breve{x},\breve{y}) = \int _{0}^{\hat{\alpha }(\breve{x})} \int _{0}^{\hat{\beta }(\breve{y})}f(\breve{s},\breve{t}) \biggl( \int _{0}^{\breve{s}}g(\breve{\tau },\breve{t})\,d \breve{ \tau } \biggr) \,d \breve{t}\,d \breve{s}, \\& \breve{H}(r)= \int _{r_{0}}^{r}\frac{d \breve{s}}{ ( \bar{\varphi } \circ \bar{\psi } ^{-1} ) ^{2} ( \breve{s} ) },\quad r \geq r_{0}>0,\qquad \breve{H }(+\infty )= \int _{r_{0}}^{+\infty }\frac{d \breve{s}}{ ( \bar{\varphi } \circ \bar{\psi } ^{-1} ) ^{2} ( \breve{s} ) }=+\infty , \end{aligned}$$

and\(( \breve{x}_{1},\breve{y}_{1} ) \in \hat{\varOmega } \)is chosen so that

$$ \biggl( \breve{H} \bigl( a ( \breve{x},\breve{y} ) \bigr) +B(\breve{x}, \breve{y}) + \biggl( \int _{0}^{\hat{\alpha }(\breve{x})} \int _{0}^{\hat{\beta }(\breve{y})}f(\breve{s},\breve{t})\,d \breve{t}\,d \breve{s} \biggr) ^{2} \biggr) \in \operatorname{Dom} \bigl( \breve{H}^{-1} \bigr). $$

In 1988, Hilger [33] presented time scale theory to unify continuous and discrete analysis. For some Gronwall–Bellman-type integral, dynamic inequalities and other type inequalities on time scales, see Refs. [18, 13, 14, 1632, 3441]. For more details on time scales calculus see [15].

A time scale \(\mathbb{T}\) is an arbitrary nonempty closed subset of \(\mathbb{R}\). We suppose throughout the article that \(\mathbb{T}\) has the topology that it inherits from the standard topology on \(\mathbb{R}\). The forward jump operator \(\sigma : \mathbb{T}\to \mathbb{T}\) is defined for any \(t\in \mathbb{T}\) by

$$ \sigma (t):=\inf \{s\in \mathbb{T}: s>t\}, $$

and the backward jump operator \(\rho : \mathbb{T}\to \mathbb{T}\) is defined for any \(t\in \mathbb{T}\) by

$$ \rho (t):=\sup \{s\in \mathbb{T}: s< t\}. $$

In the previous two definitions, we set \(\inf \emptyset =\sup \mathbb{T}\) (i.e., if t is the maximum of \(\mathbb{T}\), then \(\sigma (t)=t\)) and \(\sup \emptyset =\inf \mathbb{T}\) (i.e., if t is the minimum of \(\mathbb{T}\), then \(\rho (t)=t\)), where is the empty set.

A point \(t\in \mathbb{T}\) with \(\inf \mathbb{T}< t<\sup \mathbb{T}\) is said to be right-scattered if \(\sigma (t)>t\), right-dense if \(\sigma (t)=t\), left-scattered if \(\rho (t)< t\), and left-dense if \(\rho (t)=t\). Points that are simultaneously right-dense and left-dense are called dense points. Points that are simultaneously right-scattered and left-scattered are called isolated points.

We define the forward graininess function \(\mu :\mathbb{T}\to [0, \infty )\) for any \(t \in \mathbb{T}\) by \(\mu (t):= \sigma (t)-t\).

Let \(f : \mathbb{T}\to \mathbb{R}\) be a function. Then the function \(f^{\sigma } : \mathbb{T}\to \mathbb{R}\) is defined by \(f^{\sigma }(t)=f( \sigma (t))\), \(\forall t\in \mathbb{T}\), that is, \(f^{\sigma }=f \circ \sigma \). In a similar manner, the function \(f^{\rho } : \mathbb{T} \to \mathbb{R}\) is defined by \(f^{\rho }(t)=f(\rho (t))\), \(\forall t \in \mathbb{T}\), that is, \(f^{\rho }=f\circ \rho \).

We introduce the set \(\mathbb{T}^{\kappa }\) as follows: If \(\mathbb{T}\) has a left-scattered maximum m, then \(\mathbb{T}^{ \kappa }=\mathbb{T}-\{m\}\), otherwise \(\mathbb{T}^{\kappa } = \mathbb{T}\).

The interval \([a,b]\) in \(\mathbb{T}\) is defined by

$$ [a,b]_{\mathbb{T}}=\{t\in \mathbb{T}:a\leq t\leq b\}. $$

Open intervals and half-closed interval are defined similarly.

Suppose \(f : \mathbb{T}\to \mathbb{R}\) is a function and \(t\in \mathbb{T}^{\kappa }\). Then we say that \(f^{\Delta }(t)\in \mathbb{R}\) is the delta derivative of f at t if for any \(\varepsilon > 0\) there exists a neighborhood U of t such that, for all \(s\in U\), we have

$$ \bigl|\bigl[ f\bigl(\sigma (t)\bigr)-f(s)\bigr]-f^{\Delta }(t)\bigl[\sigma (t)-s\bigr]\bigr| \leq \varepsilon \bigl\vert \sigma (t)-s \bigr\vert . $$

Furthermore, f is said to be delta differentiable on \(\mathbb{T} ^{\kappa }\) if it is delta differentiable at each \(t\in \mathbb{T} ^{\kappa }\).

If f, \(g:\mathbb{T}\to \mathbb{R}\) are delta differentiable functions at \(t\in \mathbb{T}^{\kappa }\), then

  1. 1.

    \((f+g)^{\Delta }(t)=f^{\Delta }(t)+g^{\Delta }(t)\);

  2. 2.

    \((fg)^{\Delta }(t)=f^{\Delta }(t)g(t)+f(\sigma (t))g^{\Delta }(t)=f(t)g ^{\Delta }(t)+f^{\Delta }(t)g(\sigma (t))\);

  3. 3.

    \(( \frac{f}{g} )^{\Delta }(t)=\frac{f^{\Delta }(t)g(t)-f(t)g^{ \Delta }(t)}{g(t)g(\sigma (t))}\) provided \(g(t)g(\sigma (t))\neq 0\).

A function \(g : \mathbb{T}\to \mathbb{R}\) is called right-dense continuous (rd-continuous) if g is continuous at the right-dense points in \(\mathbb{T}\) and its left-sided limits exist at all left-dense points in \(\mathbb{T}\).

A function \(F : \mathbb{T}\to \mathbb{R}\) is said to be a delta antiderivative of \(f : \mathbb{T}\to \mathbb{R}\) if \(F^{\Delta }(t)=f(t)\) for all \(t\in \mathbb{T}^{\kappa }\). In this case, the definite delta integral of f is defined by

$$ \int _{a}^{b}f(\eta )\Delta \eta =F(b)-F(a) \quad \text{for all } a,b\in \mathbb{T}. $$

If \(g \in C_{\mathrm{rd}}(\mathbb{T})\) and t, \(t_{0}\in \mathbb{T}\), then the definite integral \(G(t) :=\int _{t_{0}}^{t} g(s)\Delta s\) exists, and \(G^{\Delta } (t) = g(t)\) holds.

Assume that a, b, \(c\in \mathbb{T}\), \(\alpha \in \mathbb{R}\), and f, g be continuous functions on \([a,b]_{\mathbb{T}}\). Then

  1. 1.

    \(\int _{a}^{b} [f(t)+g(t) ]\Delta \eta =\int _{a}^{b}f(\eta ) \Delta \eta +\int _{a}^{b}g(\eta )\Delta \eta \);

  2. 2.

    \(\int _{a}^{b}\alpha f(\eta )\Delta \eta =\alpha \int _{a}^{b}f(\eta ) \Delta \eta \);

  3. 3.

    \(\int _{a}^{b}f(\eta )\Delta \eta =\int _{a}^{c}f(\eta )\Delta \eta + \int _{c}^{b}f(\eta )\Delta \eta \);

  4. 4.

    \(\int _{a}^{b}f(\eta )\Delta \eta =-\int _{b}^{a}f(\eta )\Delta \eta \);

  5. 5.

    \(\int _{a}^{a}f(\eta )\Delta \eta =0\);

  6. 6.

    if \(f(t)\geq g(t)\) on \([a,b]_{\mathbb{T}}\), then \(\int _{a}^{b}f( \eta )\Delta \eta \geq \int _{a}^{b}g(\eta )\Delta \eta \).

We will need the following important relations between calculus on time scales \(\mathbb{T}\) and either continuous calculus on \(\mathbb{R}\) or discrete calculus on \(\mathbb{Z}\). Note that:

  1. 1.

    If \(\mathbb{T}=\mathbb{R}\), then

    $$ \sigma (t)=t,\qquad \mu (t)=0,\qquad f^{\Delta }(t)=f^{\prime }(t), \qquad \int _{a}^{b}f(\eta )\Delta \eta = \int _{a}^{b}f(t)\,{d}t. $$
  2. 2.

    If \(\mathbb{T}=\mathbb{Z}\), then

    $$ \sigma (t)=t+1,\qquad \mu (t)=1,\qquad f^{\Delta }(t)=f(t+1)-f(t), \qquad \int _{a}^{b}f(\eta )\Delta \eta =\sum _{t=a}^{b-1}f(t). $$

In the following, we present the basic theorems that will be needed in the proofs of our main results.

Theorem 1.3

Ifis Δ̂-integrable on\([a,b]\), then so is\(\vert \hat{f} \vert \), and

$$ \biggl\vert \int _{a}^{b}\hat{f}(\check{t})\hat{\Delta } \check{t} \biggr\vert \leq \int _{a}^{b} \bigl\vert \hat{f}(\check{t}) \bigr\vert \hat{\Delta } \check{t}. $$

Theorem 1.4

(Chain rule on time scales [15])

Assume\(\hat{g}:\mathbb{R}\rightarrow \mathbb{R}\)is continuous, \(\hat{g}:\breve{\mathbb{T}}\rightarrow \mathbb{R}\)is Δ̂-differentiable on\(\mathbb{T^{\kappa }}\), and\(\hat{f}:\mathbb{R}\rightarrow \mathbb{R}\)is continuously differentiable. Then there exists\(c\in [\check{t},\sigma (\check{t})]\)with

$$ (\hat{f}\circ \hat{g})^{\hat{\Delta }}(\check{t})= \hat{f}'\bigl(\hat{g}(c)\bigr) \hat{g}^{\hat{\Delta }}(\check{t}). $$
(1.1)

Theorem 1.5

(Chain rule on time scales [15])

Let\(\hat{f}: \mathbb{R}\rightarrow \mathbb{R}\)be continuously differentiable and suppose\(\hat{g}: \breve{\mathbb{T}}\rightarrow \mathbb{R}\)is Δ̂-differentiable. Then\(f\circ \hat{g} : \breve{\mathbb{T}}\rightarrow \mathbb{R}\)is Δ̂-differentiable and the formula

$$ (\hat{f}\circ \hat{g})^{\hat{\Delta }}(\check{t}) = \biggl\{ \int _{0} ^{1} \bigl[\hat{f}' \bigl(h\hat{g}^{\sigma }(\check{t})+(1-h)\hat{g}( \check{t})\bigr) \bigr]\,dh \biggr\} \hat{g}^{\hat{\hat{\Delta }}}(\check{t}) $$
(1.2)

holds.

This paper gives us the time scale versions of the results provided in [9]. These inequalities, proved here, extend some known results in the literature, and they are also unify the continuous and the discrete case.

Main results

In what follows, \(\mathbb{R} \) denotes the set of real numbers, \(\mathbb{R} _{+}= [ 0,+\infty )\), \(\breve{\mathbb{T}}_{1}\), \(\breve{\mathbb{T}}_{2}\) are two time scales and we put \(\varOmega = \breve{\mathbb{T}}_{1}\times \breve{\mathbb{T}}_{2}=\{(\breve{t}, \breve{s}):\breve{t}\in \breve{\mathbb{T}}_{1}, \breve{s}\in \breve{\mathbb{T}}_{2}\}\) which is a complete metric space with the metric ρ̆ defined by

$$ \breve{\rho }\bigl((\breve{t},\breve{s}),(\acute{t},\acute{s})\bigr)=\sqrt{( \breve{t}-\acute{t})^{2}+(\breve{s}-\acute{s})^{2}}, \quad \forall ( \breve{t},\breve{s}), (\acute{t},\acute{s})\in \breve{ \mathbb{T}}_{1} \times \breve{\mathbb{T}}_{2}. $$

\(C_{\mathrm{rd}}(\varOmega ,\mathbb{R} _{+})\) denotes the set of all right-dense continuous functions from Ω into \(\mathbb{R} _{+}\) and \(C^{1}_{\mathrm{rd}} ( \breve{\mathbb{T}}_{i},\breve{\mathbb{T}}_{i} ) \) denotes the set of all right-dense continuously delta-differentiable functions from \(\breve{\mathbb{T}}_{i}\) into \(\breve{\mathbb{T}}_{i}\), \(i=1,2\). The two-variables time scales calculus and multiple integration on time scales were introduced in [10, 11] (see also [12]).

Theorem 2.1

Suppose that\(a\in C_{\mathrm{rd}}(\varOmega ,\mathbb{R}_{+})\)is nondecreasing with respect to\((\breve{x},\breve{y}) \in \varOmega \), andg, u, p, \(f\in C_{\mathrm{rd}}(\varOmega ,\mathbb{R}_{+})\). Furthermore, suppose thatψ̄, \(\bar{\varphi } \in C(\mathbb{R}_{+},\mathbb{R}_{+})\)are nondecreasing functions with\(\{ \bar{\psi } ,\bar{\varphi } \} (u)>0\)for\(u>0\), and\(\lim_{u\rightarrow +\infty }\bar{\psi } (u)=+\infty \). If\(u(\breve{x},\breve{y}) \)satisfies

$$\begin{aligned} \bar{\psi } \bigl( u(\breve{x},\breve{y}) \bigr) \leq &a( \breve{x},\breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}} \bigl[ f(\breve{s},\breve{t}) \bar{\varphi } \bigl( u(\breve{s},\breve{t}) \bigr) +p(\breve{s},\breve{t}) \bigr] \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \biggl( \int _{0}^{\breve{s}}g(\breve{\tau },\breve{t})\bar{ \varphi } \bigl( u(\breve{\tau },\breve{t}) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s}, \end{aligned}$$

for\((\breve{x},\breve{y}) \in \varOmega \), then

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \biggl\{ \breve{G}^{-1} \biggl[ \breve{G} \bigl( q(\breve{x},\breve{y}) \bigr) + \int _{0} ^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \biggl( 1+ \int _{0}^{\breve{s}}g(\breve{\tau },\breve{t})\hat{ \Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \biggr] \biggr\} $$
(2.1)

for\(0\leq \breve{x}\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\), where

$$\begin{aligned}& q(\breve{x},\breve{y}) =a(\breve{x},\breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}p(\breve{s},\breve{t})\hat{\Delta }\breve{t} \hat{\Delta }\breve{s} , \end{aligned}$$
(2.2)
$$\begin{aligned}& \breve{G}(r)= \int _{r_{0}}^{r}\frac{\hat{\Delta }\breve{s}}{\bar{ \varphi } \circ \bar{\psi } ^{-1}(\breve{s})},\quad r\geq r_{0}>0,\qquad \breve{G}(+\infty )= \int _{r_{0}}^{+\infty }\frac{\hat{\Delta } \breve{s}}{\bar{\varphi } \circ \bar{\psi } ^{-1}(\breve{s})}=+\infty , \end{aligned}$$
(2.3)

and\(( \breve{x}_{1},\breve{y}_{1} ) \in \varOmega \)is chosen so that

$$ \biggl( \breve{G} \bigl( q(\breve{x},\breve{y}) \bigr) + \int _{0} ^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \biggl( 1+ \int _{0}^{\breve{s}}g(\breve{\tau },\breve{t})\hat{ \Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \biggr) \in \operatorname{Dom} \bigl( G^{-1} \bigr). $$

Proof

Assume that \(a ( \breve{x},\breve{y} ) >0\). Since \(q\geq 0\) and it is nondecreasing, fixing an arbitrary point \((\breve{\xi },\breve{\zeta }) \in \varOmega \) and defining \(z(\breve{x}, \breve{y}) \) by

$$\begin{aligned} z(\breve{x},\breve{y}) =&q(\breve{\xi },\breve{\zeta }) + \int _{0} ^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t})\bar{\varphi } \bigl( u(\breve{s},\breve{t}) \bigr) \hat{\Delta }\breve{t} \hat{\Delta } \breve{s} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \biggl( \int _{0}^{\breve{s}}g(\breve{\tau },\breve{t})\bar{ \varphi } \bigl( u(\breve{\tau },\breve{t}) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s}, \end{aligned}$$

which is a positive and nondecreasing function for \(0\leq \breve{x} \leq \breve{\xi }\leq \breve{x}_{1}\), \(0\leq \breve{x}\leq \breve{\zeta }\leq \breve{y}_{1}\), we have \(z(0,\breve{y}) =z( \breve{x},0) =q(\breve{\xi },\breve{\zeta }) \) and

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \bigl( z(\breve{x}, \breve{y}) \bigr) . $$
(2.4)

Differentiating \(z(\breve{x},\breve{y}) \), with respect to and using (2.4), we get

$$\begin{aligned} z^{\hat{\Delta } \breve{x}}(\breve{x},\breve{y}) =& \int _{0}^{ \breve{y}}f(\breve{x},\breve{t}) \biggl[ \bar{\varphi } \bigl( u( \breve{x},\breve{t}) \bigr) + \int _{0}^{\breve{x}}g(\breve{\tau }, \breve{t})\bar{ \varphi } \bigl( u(\breve{\tau },\breve{t}) \bigr) \hat{\Delta }\breve{\tau } \biggr] \hat{\Delta }\breve{t} \\ \leq & \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \biggl[ \bar{\varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{x},\breve{t}) \bigr) + \int _{0}^{\breve{x}}g(\breve{\tau } ,\breve{t})\bar{ \varphi } \circ \bar{ \psi } ^{-1} \bigl( z(\breve{\tau },\breve{t}) \bigr) \hat{\Delta } \breve{\tau } \biggr] \hat{\Delta }\breve{t}, \end{aligned}$$

since \(\bar{\varphi } \circ \bar{\psi } ^{-1}\) is nondecreasing with respect to \((\breve{x},\breve{y}) \in \mathbb{R} _{+}\times \mathbb{R} _{+}\), we have

$$\begin{aligned} z^{\hat{\Delta } \breve{x}}(\breve{x},\breve{y}) \leq & \int _{0}^{ \breve{y}}f(\breve{x},\breve{t}) \biggl[ \bar{\varphi } \circ \bar{ \psi } ^{-1} \bigl( z(\breve{x},\breve{t}) \bigr) +\bar{\varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{x}, \breve{t}) \bigr) \int _{0}^{\breve{x}}g(\breve{\tau },\breve{t})\hat{ \Delta }\breve{\tau } \biggr] \hat{\Delta }\breve{t} \\ \leq &\bar{\varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{x}, \breve{y}) \bigr) \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \biggl[ 1+ \int _{0}^{\breve{x}}g(\breve{\tau },\breve{t})\hat{ \Delta } \breve{\tau } \biggr] \hat{\Delta }\breve{t}, \end{aligned}$$
(2.5)

and from (2.5) we get

$$ \frac{z^{\hat{\Delta } \breve{x}}(\breve{x},\breve{y}) }{\bar{\varphi } \circ \bar{\psi } ^{-1} ( z(\breve{x},\breve{y}) ) } \leq \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \biggl( 1+ \int _{0} ^{\breve{x}}g(\breve{\tau },\breve{t})\hat{ \Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t}. $$
(2.6)

From (2.6) we get

$$ \breve{G} \bigl( z(\breve{x},\breve{y}) \bigr) \leq \breve{G} \bigl( q(\breve{ \xi },\breve{\zeta }) \bigr) + \int _{0}^{\breve{x}} \int _{0} ^{\breve{y}}f(\breve{s},\breve{t}) \biggl( 1+ \int _{0}^{\breve{s}}g( \breve{\tau },\breve{t})\hat{ \Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s}. $$

Since \((\breve{\xi },\breve{\zeta }) \in \varOmega \) is chosen arbitrarily,

$$ z(\breve{x},\breve{y}) \leq \breve{G}^{-1} \biggl[ \breve{G} \bigl( q( \breve{x},\breve{y}) \bigr) + \int _{0}^{\breve{x}} \int _{0}^{ \breve{y}}f(\breve{s},\breve{t}) \biggl( 1+ \int _{0}^{\breve{s}}g( \breve{\tau },\breve{t})\hat{ \Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \biggr] . $$
(2.7)

So from (2.7) and (2.4) we get the desired inequality in (2.1). For \(a(\breve{x},\breve{y}) =0\), we carry out the above procedure with \(\epsilon >0\) instead of \(a(\breve{x},\breve{y}) \) and subsequently let \(\epsilon \rightarrow 0\). This completes the proof. □

Corollary 2.2

If we take\(\breve{\mathbb{T}}=\mathbb{R}\)in Theorem2.1, then the inequality

$$\begin{aligned} \bar{\psi } \bigl( u(\breve{x},\breve{y}) \bigr) \leq &a( \breve{x},\breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}} \bigl[ f(\breve{s},\breve{t}) \bar{\varphi } \bigl( u(\breve{s},\breve{t}) \bigr) +p(\breve{s},\breve{t}) \bigr] \,d \breve{t}\,d \breve{s} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \biggl( \int _{0}^{\breve{s}} g(\breve{\tau },\breve{t})\bar{ \varphi } \bigl( u(\breve{\tau },\breve{t}) \bigr) \,d \breve{\tau } \biggr) \,d \breve{t}\,d \breve{s}, \end{aligned}$$

for\((\breve{x},\breve{y}) \in \varOmega \), implies

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \biggl\{ \breve{G}^{-1} \biggl[ \breve{G} \bigl( q(\breve{x},\breve{y}) \bigr) + \int _{0} ^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \biggl( 1+ \int _{0}^{\breve{s}}g(\breve{\tau },\breve{t})\,d \breve{ \tau } \biggr) \,d \breve{t}\,d \breve{s} \biggr] \biggr\} $$

for\(0\leq \breve{x}\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\), where

$$\begin{aligned}& q(\breve{x},\breve{y}) =a(\breve{x},\breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}p(\breve{s},\breve{t})\,d \breve{t}\,d \breve{s}, \\& \breve{G}(r)= \int _{r_{0}}^{r}\frac{d \breve{s}}{\bar{\varphi }\circ \bar{ \psi } ^{-1}(\breve{s})},\quad r\geq r_{0}>0,\qquad \breve{G}(+\infty )= \int _{r_{0}}^{+\infty }\frac{d \breve{s}}{\bar{\varphi }\circ \bar{ \psi } ^{-1}(\breve{s})}=+\infty , \end{aligned}$$
(2.8)

and\(( \breve{x}_{1},\breve{y}_{1} ) \in \varOmega \)is chosen so that

$$ \biggl( \breve{G} \bigl( q(\breve{x},\breve{y}) \bigr) + \int _{0} ^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \biggl( 1+ \int _{0}^{\breve{s}}g(\breve{\tau },\breve{t})\,d \breve{ \tau } \biggr) \,d \breve{t}\,d \breve{s} \biggr) \in \operatorname{Dom} \bigl( G^{-1} \bigr). $$

Corollary 2.3

The discrete form can be obtained by letting\(\breve{\mathbb{T}}= \mathbb{Z}\)in Theorem2.1:

$$\begin{aligned} \bar{\psi } \bigl( u(\breve{x},\breve{y}) \bigr) \leq &a( \breve{x},\breve{y}) +\sum_{\breve{s}=0}^{\breve{x}-1} \sum _{ \breve{t}=0}^{\breve{y}-1} \bigl[ f(\breve{s},\breve{t})\bar{ \varphi } \bigl( u(\breve{s},\breve{t}) \bigr) +p(\breve{s},\breve{t}) \bigr] \\ &{}+\sum_{\breve{s}=0}^{\breve{x}-1}\sum _{\breve{t}=0}^{\breve{y}-1}f( \breve{s},\breve{t}) \Biggl( \sum _{\breve{\tau }=0}^{\breve{s}-1} g( \breve{\tau },\breve{t}) \bar{\varphi } \bigl( u(\breve{\tau }, \breve{t}) \bigr) \Biggr) \end{aligned}$$

for\((\breve{x},\breve{y}) \in \varOmega \), which implies

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \Biggl\{ \breve{G}^{-1} \Biggl[ \breve{G} \bigl( q(\breve{x},\breve{y}) \bigr) + \sum_{\breve{s}=0}^{\breve{x}-1}\sum _{\breve{t}=0}^{\breve{y}-1}f( \breve{s},\breve{t}) \Biggl( 1+ \sum _{\breve{\tau }=0}^{\breve{s}-1}g( \breve{\tau },\breve{t}) \Biggr) \Biggr] \Biggr\} $$

for\(0\leq \breve{x}\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\), where

$$\begin{aligned}& q(\breve{x},\breve{y}) =a(\breve{x},\breve{y}) +\sum _{\breve{s}=0} ^{\breve{x}-1}\sum_{\breve{t}=0}^{\breve{y}-1}p( \breve{s},\breve{t}), \\& \breve{G}(r)=\sum_{\breve{s}=r_{0}}^{r-1} \frac{1}{\bar{\varphi } \circ \bar{\psi } ^{-1}(\breve{s})},\quad r\geq r_{0}>0,\qquad \breve{G}(+ \infty )=\sum_{\breve{s}=r_{0}}^{+\infty } \frac{1}{\bar{\varphi } \circ \bar{\psi } ^{-1}(\breve{s})}=+\infty , \end{aligned}$$
(2.9)

and\(( \breve{x}_{1},\breve{y}_{1} ) \in \varOmega \)is chosen so that

$$ \Biggl( \breve{G} \bigl( q(\breve{x},\breve{y}) \bigr) + \sum _{\breve{s}=0}^{\breve{x}-1}\sum_{\breve{t}=0}^{\breve{y}-1}f( \breve{s},\breve{t}) \Biggl( 1+\sum_{\breve{\tau }_{0}}^{\breve{s}}g( \breve{\tau },\breve{t}) \Biggr) \Biggr) \in \operatorname{Dom} \bigl( G^{-1} \bigr). $$

Theorem 2.4

Assume thath, \(b\in C_{\mathrm{rd}}(\varOmega ,\mathbb{R} _{+})\). Letg, f, p, a, u, ψ̄andφ̄be as in Theorem2.1, if\(u(\breve{x},\breve{y})\)satisfies

$$\begin{aligned} \bar{\psi } \bigl( u(\breve{x},\breve{y}) \bigr) \leq &a( \breve{x},\breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}} \bigl[ f(\breve{s},\breve{t}) \bar{\varphi } \bigl( u(\breve{s},\breve{t}) \bigr) +p(\breve{s},\breve{t}) \bigr] \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}b(\breve{s},\breve{t}) \biggl[ h( \breve{s},\breve{t})\bar{\varphi } \bigl( u(\breve{s}, \breve{t}) \bigr) + \int _{0}^{\breve{s}}g(\breve{\tau },\breve{t})\bar{ \varphi } \bigl( u(\breve{\tau },\breve{t}) \bigr) \hat{\Delta } \breve{\tau } \biggr] \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \end{aligned}$$

for\((\breve{x},\breve{y}) \in \varOmega \), then

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \biggl\{ \breve{G}^{-1} \biggl[ \breve{G} \bigl( q(\breve{x},\breve{y}) \bigr) +A(\breve{x}, \breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s}, \breve{t})\hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \biggr] \biggr\} $$
(2.10)

for\(0\leq \breve{x}\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\), whereq, Ğare defined by (2.2) and (2.3), respectively, and

$$ \breve{A}(\breve{x},\breve{y}) = \int _{0}^{\breve{x}} \int _{0}^{ \breve{y}}b(\breve{s},\breve{t}) \biggl[ h( \breve{s},\breve{t})+ \int _{0}^{\breve{s}}g(\breve{\tau },\breve{t})\hat{ \Delta }\breve{\tau } \biggr] \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} $$
(2.11)

and\(( \breve{x}_{1},\breve{y}_{1} ) \in \varOmega \)is chosen so that

$$ \biggl( \breve{G} \bigl( q(\breve{x},\breve{y}) \bigr) +\breve{A}( \breve{x}, \breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f( \breve{s},\breve{t})\hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \biggr) \in \operatorname{Dom} \bigl( \breve{G}^{-1} \bigr). $$

Proof

Assume that \(a(\breve{x},\breve{y}) >0\). Fixing an arbitrary \((\breve{\xi },\breve{\zeta }) \in \varOmega \), we define positive and nondecreasing function \(z(\breve{x},\breve{y}) \) by

$$\begin{aligned} z(\breve{x},\breve{y}) =&q(\breve{\xi },\breve{\zeta }) + \int _{0} ^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t})\bar{\varphi } \bigl( u(\breve{s},\breve{t}) \bigr) \hat{\Delta }\breve{t} \hat{\Delta } \breve{s} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}b(\breve{s},\breve{t}) \biggl[ h( \breve{s},\breve{t})\bar{\varphi } \bigl( u(\breve{s}, \breve{t}) \bigr) + \int _{0}^{\breve{s}}g(\breve{\tau },\breve{t})\bar{ \varphi } \bigl( u(\breve{\tau },\breve{t}) \bigr) \hat{\Delta } \breve{\tau } \biggr] \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \end{aligned}$$

for \(0\leq \breve{x}\leq \breve{\xi }\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{\zeta }\leq y_{1}\), then \(z(0,\breve{y}) =z( \breve{x},0) =q(\breve{\xi },\breve{\zeta }) \) and

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \bigl( z(\breve{x}, \breve{y}) \bigr); $$

then we have

$$\begin{aligned} z^{\hat{\Delta } \breve{x}}(\breve{x},\breve{y}) =& \int _{0}^{ \breve{y}}f(\breve{x},\breve{t}) \bar{ \varphi } \bigl( u(\breve{x}, \breve{t}) \bigr) \hat{\Delta }\breve{t} \\ &{}+ \int _{0}^{\breve{y}}b(\breve{x},\breve{t}) \biggl( h( \breve{x}, \breve{t}) \bar{\varphi } \bigl( u(\breve{x},\breve{t}) \bigr) + \int _{0}^{\breve{x} }g(\breve{\tau },\breve{t})\bar{ \varphi } \bigl( u(\breve{\tau },\breve{t}) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t} \\ \leq & \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \bar{ \varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{x},\breve{t}) \bigr) \hat{\Delta }\breve{t}+ \int _{0}^{\breve{y}}b(\breve{x},\breve{t}) \\ &{}\times \biggl( h(\breve{x},\breve{t}) \bar{\varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{x},\breve{t}) \bigr) + \int _{0}^{\breve{x} }g( \breve{\tau },\breve{t})\bar{ \varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{\tau },\breve{t}) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t} \\ \leq &\bar{\varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{x}, \breve{y}) \bigr) \biggl[ \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \hat{\Delta }\breve{t} \\ &{} + \int _{0}^{\breve{y}}b(\breve{x},\breve{t}) \biggl( h( \breve{x},\breve{t}) + \int _{0}^{\breve{x} }g(\breve{\tau },\breve{t}) \hat{ \Delta }\breve{\tau } \biggr) \biggr] \hat{\Delta }\breve{t}, \end{aligned}$$

then

$$ \frac{z^{\hat{\Delta } \breve{x}}(\breve{x},\breve{y}) }{\bar{\varphi } \circ \bar{\psi } ^{-1} ( z(\breve{x},\breve{y}) ) } \leq \biggl[ \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \hat{\Delta } \breve{t}+ \int _{0}^{\breve{y}}b(\breve{x},\breve{t}) \biggl( h( \breve{x},\breve{t}) + \int _{0}^{\breve{x}}g(\breve{\tau },\breve{t}) \hat{ \Delta }\breve{\tau } \biggr) \biggr] \hat{\Delta }\breve{t}. $$
(2.12)

Integrating (2.12) and using (2.3) and (2.11), we get

$$ \breve{G} \bigl( z(\breve{x},\breve{y}) \bigr) \leq \breve{G} \bigl( q(\breve{ \xi },\breve{\zeta }) \bigr) +\breve{A}(\breve{x}, \breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s}, \breve{t})\hat{\Delta }\breve{t}\hat{\Delta }\breve{s}. $$

Since \((\breve{\xi },\breve{\zeta }) \in \varOmega \) is chosen arbitrarily,

$$ z(\breve{x},\breve{y}) \leq \breve{G}^{-1} \biggl[ \breve{G} \bigl( q( \breve{x},\breve{y}) \bigr) +\breve{A}(\breve{x},\breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \biggr] . $$
(2.13)

From (2.13) and \(u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} ( z(\breve{x},\breve{y}) ) \), we get the required inequality in (2.10). For \(a(\breve{x},\breve{y}) =0\), we carry out the above procedure with \(\epsilon >0\) instead of \(a(\breve{x}, \breve{y}) \) and subsequently let \(\epsilon \rightarrow 0\). This completes the proof. □

Corollary 2.5

If we take\(\breve{\mathbb{T}}=\mathbb{R}\)in Theorem2.4, then the inequality

$$\begin{aligned} \bar{\psi } \bigl( u(\breve{x},\breve{y}) \bigr) \leq &a( \breve{x},\breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}} \bigl[ f(\breve{s},\breve{t}) \bar{\varphi } \bigl( u(\breve{s},\breve{t}) \bigr) +p(\breve{s},\breve{t}) \bigr] \,d \breve{t}\,d \breve{s} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}b(\breve{s},\breve{t}) \biggl[ h( \breve{s},\breve{t})\bar{\varphi } \bigl( u(\breve{s}, \breve{t}) \bigr) + \int _{0}^{\breve{s}}g(\breve{\tau },\breve{t}) \bar{ \varphi } \bigl( u(\breve{\tau },\breve{t}) \bigr) \,d \breve{\tau } \biggr] \,d \breve{t}\,d \breve{s}, \end{aligned}$$

for\((\breve{x},\breve{y}) \in \varOmega \), implies

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \biggl\{ \breve{G}^{-1} \biggl[ \breve{G} \bigl( q(\breve{x},\breve{y}) \bigr) +A(\breve{x}, \breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s}, \breve{t})\,d \breve{t}\,d \breve{s} \biggr] \biggr\} $$

for\(0\leq \breve{x}\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\), whereĞis defined by (2.8) and

$$ \breve{A}(\breve{x},\breve{y}) = \int _{0}^{\breve{x}} \int _{0}^{ \breve{y}}b(\breve{s},\breve{t}) \biggl[ h( \breve{s},\breve{t})+ \int _{0}^{\breve{s}}g(\breve{\tau },\breve{t})\,d \breve{ \tau } \biggr] \,d \breve{t}\,d \breve{s} $$

and\(( \breve{x}_{1},\breve{y}_{1} ) \in \varOmega \)is chosen so that

$$ \biggl( \breve{G} \bigl( q(\breve{x},\breve{y}) \bigr) +\breve{A}( \breve{x}, \breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f( \breve{s},\breve{t})\,d \breve{t}\,d \breve{s} \biggr) \in \operatorname{Dom} \bigl( \breve{G}^{-1} \bigr). $$

Corollary 2.6

The discrete form can be obtained by letting\(\breve{\mathbb{T}}= \mathbb{Z}\)in Theorem2.4:

$$\begin{aligned} \bar{\psi } \bigl( u(\breve{x},\breve{y}) \bigr) \leq &a( \breve{x},\breve{y}) +\sum_{\breve{s}=0}^{\breve{x}-1} \sum _{ \breve{t}=0}^{\breve{y}-1} \bigl[ f(\breve{s},\breve{t})\bar{ \varphi } \bigl( u(\breve{s},\breve{t}) \bigr) +p(\breve{s},\breve{t}) \bigr] \\ &{}+\sum_{\breve{s}=0}^{\breve{x}-1}\sum _{\breve{t}=0}^{\breve{y}-1}b( \breve{s},\breve{t}) \Biggl[ h( \breve{s},\breve{t})\bar{\varphi } \bigl( u(\breve{s},\breve{t}) \bigr) +\sum _{\breve{\tau }=0}^{ \breve{s}-1}g(\breve{\tau },\breve{t}) \bar{\varphi } \bigl( u( \breve{\tau },\breve{t}) \bigr) \Biggr], \end{aligned}$$

for\((\breve{x},\breve{y}) \in \varOmega \), implies

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \Biggl\{ G^{-1} \Biggl[ G \bigl( q(\breve{x},\breve{y}) \bigr) +A(\breve{x}, \breve{y}) + \sum_{\breve{s}=0}^{\breve{x}-1}\sum _{\breve{t}=0}^{\breve{y}-1}f( \breve{s},\breve{t}) \Biggr] \Biggr\} $$

for\(0\leq \breve{x}\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\), whereĞis defined by (2.9) and

$$ \breve{A}(\breve{x},\breve{y}) =\sum_{\breve{s}=0}^{\breve{x}-1} \sum_{\breve{t}=0}^{\breve{y}-1}b(\breve{s},\breve{t}) \Biggl[ h( \breve{s},\breve{t})+\sum_{\breve{\tau }=0}^{\breve{s}-1}g( \breve{\tau },\breve{t}) \Biggr] $$

and\(( \breve{x}_{1},\breve{y}_{1} ) \in \varOmega \)is chosen so that

$$ \Biggl( \breve{G} \bigl( q(\breve{x},\breve{y}) \bigr) +\breve{A}( \breve{x}, \breve{y}) +\sum_{\breve{s}=0}^{\breve{x}-1}\sum _{\breve{t}=0} ^{\breve{y}-1}f(\breve{s},\breve{t}) \Biggr) \in \operatorname{Dom} \bigl( \breve{G}^{-1} \bigr). $$

Theorem 2.7

Assume thatg, a, u, f, p, ψ̄andφ̄are as in Theorem2.1. If\(u(\breve{x},\breve{y}) \)satisfies

$$\begin{aligned} \bar{\psi } \bigl( u(\breve{x},\breve{y}) \bigr) \leq &a( \breve{x},\breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}\bar{ \varphi } \bigl( u( \breve{s},\breve{t}) \bigr) \bigl[ f(\breve{s}, \breve{t})\bar{\varphi } \bigl( u( \breve{s},\breve{t}) \bigr) +p( \breve{s},\breve{t}) \bigr] \hat{\Delta }\breve{t} \hat{\Delta } \breve{s} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t})\bar{ \varphi } \bigl( u(\breve{s},\breve{t}) \bigr) \biggl( \int _{0}^{ \breve{s}}g(\breve{\tau },\breve{t})\bar{ \varphi } \bigl( u( \breve{\tau },\breve{t}) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s}, \end{aligned}$$

for\((\breve{x},\breve{y}) \in \varOmega \), then

$$\begin{aligned} u(\breve{x},\breve{y}) \leq& \bar{\psi } ^{-1} \biggl\{ \breve{G}^{-1} \biggl( \breve{F}^{-1} \biggl[ \breve{F} \bigl( q_{1} ( \breve{x}, \breve{y} ) \bigr) \\ &{} + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f( \breve{s},\breve{t}) \biggl( 1+ \int _{0}^{\breve{s}}g(\breve{\tau }, \breve{t})\hat{ \Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t} \hat{\Delta }\breve{s} \biggr] \biggr) \biggr\} , \end{aligned}$$
(2.14)

for\(0\leq \breve{x}\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\), whereĞis defined in (2.3) and

$$\begin{aligned}& q_{1} ( \breve{x},\breve{y} ) =\breve{G} \bigl( a( \breve{x},\breve{y}) \bigr) + \int _{0}^{\breve{x}} \int _{0}^{ \breve{y}}p(\breve{s},\breve{t})\hat{\Delta }\breve{t}\hat{\Delta } \breve{s}, \end{aligned}$$
(2.15)
$$\begin{aligned}& \begin{gathered} \breve{F}(r)= \int _{r_{0}}^{r}\frac{\hat{\Delta }\breve{s}}{ ( ( \bar{\varphi } \circ \bar{\psi } ^{-1} ) \circ \breve{G}^{-1} ) (\breve{s} ) },\quad r\geq r_{0}>0,\\ \breve{F}(+\infty )= \int _{r_{0}}^{+\infty }\frac{\hat{\Delta } \breve{s}}{ ( \bar{\varphi } \circ \bar{\psi } ^{-1} ) \circ \breve{G}^{-1}(\breve{s})}=+\infty , \end{gathered} \end{aligned}$$
(2.16)

and\(( \breve{x}_{1},\breve{y}_{1} ) \in \varOmega \)is chosen so that

$$ \biggl( \breve{F} \bigl( q_{1} ( \breve{x},\breve{y} ) \bigr) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s}, \breve{t}) \biggl( 1+ \int _{0}^{\breve{s}}g(\breve{\tau },\breve{t}) \hat{ \Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t} \hat{\Delta }\breve{s} \biggr) \in \operatorname{Dom} \bigl( \breve{F}^{-1} \bigr). $$

Proof

Suppose that \(a(\breve{\xi },\breve{\zeta }) >0\). Fixing an arbitrary \((\breve{\xi },\breve{\zeta }) \in \varOmega \), we define a positive and nondecreasing function \(z(\breve{x},\breve{y})\) by

$$\begin{aligned} z(\breve{x},\breve{y}) =&a(\breve{\xi },\breve{\zeta })+ \int _{0}^{ \breve{x}} \int _{0}^{\breve{y}}\bar{\varphi } \bigl( u(\breve{s}, \breve{t}) \bigr) \bigl[ f(\breve{s},\breve{t})\bar{\varphi } \bigl( u(\breve{s}, \breve{t}) \bigr) +p(\breve{s},\breve{t}) \bigr] \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t})\bar{ \varphi } \bigl( u(\breve{s},\breve{t}) \bigr) \biggl( \int _{0}^{ \breve{s}}g(\breve{\tau },\breve{t})\bar{ \varphi } \bigl( u( \breve{\tau },\breve{t}) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s}, \end{aligned}$$

for \(0\leq \breve{x}\leq \breve{\xi }\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{\zeta }\leq \breve{y}_{1}\), then \(z(0,\breve{y}) =z(\breve{x},0) =a(\breve{\xi },\breve{\zeta })\) and

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \bigl( z(\breve{x}, \breve{y}) \bigr) , $$

then we have

$$\begin{aligned} z^{\hat{\Delta } \breve{x}}(\breve{x},\breve{y}) =& \int _{0}^{ \breve{y}}\bar{\varphi } \bigl( u(\breve{x}, \breve{t}) \bigr) \bigl[ f(\breve{x},\breve{t}) \bar{\varphi } \bigl( u(\breve{x}, \breve{t}) \bigr) +p(\breve{x},\breve{t}) \bigr] \hat{\Delta } \breve{t} \\ &{}+ \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \bar{ \varphi } \bigl( u( \breve{x},\breve{t}) \bigr) \biggl( \int _{0}^{\breve{x} }g( \breve{\tau } ,\breve{t})\bar{ \varphi } \bigl( u(\breve{\tau }, \breve{t}) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{\Delta } \breve{t} \\ \leq & \int _{0}^{\breve{y}}\bar{\varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{x},\breve{t}) \bigr) \bigl[ f(\breve{x}, \breve{t}) \bar{\varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{x}, \breve{t}) \bigr) +p(\breve{x},\breve{t}) \bigr] \hat{\Delta }\breve{t} \\ &{}+ \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \bar{ \varphi } \circ \bar{ \psi } ^{-1} \bigl( z(\breve{x},\breve{t}) \bigr) \biggl( \int _{0} ^{\breve{x} }g(\breve{\tau },\breve{t})\bar{ \varphi } \circ \bar{ \psi } ^{-1} \bigl( z(\breve{\tau },\breve{t}) \bigr) \hat{\Delta } \breve{\tau } \biggr) \hat{\Delta }\breve{t} \\ \leq &\bar{\varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{x}, \breve{y}) \bigr) \int _{0}^{\breve{y}} \bigl[ f(\breve{x},\breve{t}) \bar{\varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{x},\breve{t}) \bigr) +p(\breve{x},\breve{t}) \bigr] \hat{\Delta }\breve{t} \\ &{}+\bar{\varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{x}, \breve{y}) \bigr) \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \biggl( \int _{0}^{\breve{x} }g(\breve{\tau },\breve{t})\bar{ \varphi } \circ \bar{ \psi } ^{-1} \bigl( z(\breve{\tau },\breve{t}) \bigr) \hat{\Delta } \breve{\tau } \biggr) \hat{\Delta }\breve{t}, \end{aligned}$$

or

$$\begin{aligned} \frac{z^{\hat{\Delta } \breve{x}}(\breve{x},\breve{y}) }{\bar{\varphi } \circ \bar{\psi } ^{-1} ( z(\breve{x},\breve{y}) ) } \leq & \int _{0}^{\breve{y}} \bigl[ f(\breve{x},\breve{t}) \bar{\varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{x},\breve{t}) \bigr) +p( \breve{x},\breve{t}) \bigr] \hat{\Delta }\breve{t} \\ &{}+ \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \biggl( \int _{0}^{ \breve{x} }g(\breve{\tau },\breve{t})\bar{ \varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{\tau },\breve{t}) \bigr) \hat{\Delta } \breve{\tau } \biggr) \hat{\Delta }\breve{t}. \end{aligned}$$
(2.17)

Integrating (2.17) and using (2.3), we get

$$\begin{aligned} \breve{G} \bigl( z(\breve{x},\breve{y}) \bigr) \leq &\breve{G} \bigl( a(\breve{ \xi },\breve{\zeta }) \bigr) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}} \bigl[ f(\breve{s},\breve{t}) \bar{\varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{s},\breve{t}) \bigr) +p( \breve{s},\breve{t}) \bigr] \hat{\Delta }\breve{t}\hat{\Delta } \breve{s} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \biggl( \int _{0}^{\breve{s}}g(\breve{\tau } ,\breve{t})\bar{ \varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{\tau },\breve{t}) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t} \hat{\Delta } \breve{s}. \end{aligned}$$

\((\breve{\xi },\breve{\zeta }) \in \varOmega \) is chosen arbitrarily, then from (2.15) we have

$$\begin{aligned} \breve{G} \bigl( z(\breve{x},\breve{y}) \bigr) \leq &q_{1} ( \breve{x}, \breve{y} ) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f( \breve{s},\breve{t})\bar{ \varphi } \circ \bar{\psi } ^{-1} \bigl( z( \breve{s},\breve{t}) \bigr) \hat{\Delta }\breve{t}\hat{\Delta } \breve{s} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \biggl( \int _{0}^{\breve{s}}g(\breve{\tau } ,\breve{t})\bar{ \varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{\tau },\breve{t}) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t} \hat{\Delta } \breve{s}. \end{aligned}$$

Since \(q_{1} (\breve{x},\breve{y} )>0 \) is a nondecreasing function, fixing an arbitrary point \(( \breve{\xi }, \breve{\zeta } ) \in \varOmega \) and defining \(v(\breve{x}, \breve{y}) >0\) to be a nondecreasing function by

$$\begin{aligned} v(\breve{x},\breve{y}) =&q_{1} ( \breve{\xi },\breve{\zeta } ) + \int _{0}^{\breve{x}} \int _{0}^{ \breve{y}}f(\breve{s},\breve{t})\bar{\varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{s},\breve{t}) \bigr) \hat{ \Delta }\breve{t} \hat{\Delta }\breve{s} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \biggl( \int _{0}^{\breve{s}}g(\breve{\tau } ,\breve{t})\bar{ \varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{\tau },\breve{t}) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t} \hat{\Delta } \breve{s}, \end{aligned}$$

for \(0\leq \breve{x}\leq \breve{\xi }\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{\zeta }\leq y_{1}\), we have \(v(0,\breve{y}) =v( \breve{x},0) =q_{1}(\breve{\xi },\breve{\zeta })\) and

$$ z(\breve{x},\breve{y}) \leq \breve{G}^{-1} \bigl( v(\breve{x}, \breve{y}) \bigr) ; $$
(2.18)

then we have

$$\begin{aligned} v^{\hat{\Delta } \breve{x}}(\breve{x},\breve{y}) =& \int _{0}^{ \breve{y}}f(\breve{x},\breve{t}) \bar{ \varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{x},\breve{t}) \bigr) \hat{\Delta }\breve{t} \\ &{}+ \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \biggl( \int _{0}^{ \breve{x} }g(\breve{\tau },\breve{t})\bar{ \varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{\tau },\breve{t}) \bigr) \hat{\Delta } \breve{\tau } \biggr) \hat{\Delta }\breve{t} \\ \leq & \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \bar{ \varphi } \circ \bar{\psi } ^{-1} \bigl( G^{-1} \bigl( v( \breve{x},\breve{t}) \bigr) \bigr) \hat{\Delta }\breve{t} \\ &{}+ \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \biggl( \int _{0}^{ \breve{x} }g(\breve{\tau },\breve{t})\bar{ \varphi } \circ \bar{\psi } ^{-1} \bigl( G^{-1} \bigl( v( \breve{\tau },\breve{t}) \bigr) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{ \Delta }\breve{t} \\ \leq & \bigl( \bar{\varphi } \circ \bar{\psi } ^{-1} \bigr) \circ \breve{G}^{-1}\bigl(v ( \breve{x},\breve{y} ) \bigr) \biggl[ \int _{0} ^{\breve{y}}f(\breve{x},\breve{t}) \hat{ \Delta }\breve{t}+ \int _{0} ^{\breve{y}}f(\breve{x},\breve{t}) \biggl( \int _{0}^{\breve{x} }g( \breve{\tau },\breve{t})\hat{ \Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t} \biggr] , \end{aligned}$$

or

$$ \frac{v^{\hat{\Delta } \breve{x}}(\breve{x},\breve{y}) }{ ( \bar{ \varphi } \circ \bar{\psi } ^{-1} ) \circ \breve{G}^{-1}(v ( \breve{x},\breve{y} ) )}\leq \biggl[ \int _{0}^{\breve{y}}f( \breve{x},\breve{t}) \hat{ \Delta }\breve{t}+ \int _{0}^{\breve{y}}f( \breve{x},\breve{t}) \biggl( \int _{0}^{\breve{x} }g(\breve{\tau }, \breve{t})\hat{ \Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t} \biggr] . $$
(2.19)

Integrating (2.19) and using (2.16), we get

$$ \breve{F} \bigl( v ( \breve{x},\breve{y} ) \bigr) \leq \breve{F} \bigl( q_{1}(\breve{\xi },\breve{\zeta }) \bigr) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \biggl[ 1+ \int _{0}^{\breve{s}}g(\breve{\tau } ,\breve{t})\hat{ \Delta } \breve{\tau } \biggr] \hat{\Delta }\breve{t}\hat{\Delta }\breve{s}. $$

Since we can choose \((\breve{\xi },\breve{\zeta }) \in \varOmega \) arbitrarily, we have

$$ v ( \breve{x},\breve{y} ) \leq \breve{F}^{-1} \biggl[ \breve{F} \bigl( q_{1}(\breve{x},\breve{y}) \bigr) + \int _{0}^{ \breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \biggl[ 1+ \int _{0}^{\breve{s}}g(\breve{\tau } ,\breve{t})\hat{ \Delta }\breve{\tau } \biggr] \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \biggr] . $$
(2.20)

From (2.20), (2.18) and \(u(\breve{x},\breve{y}) \leq \bar{ \psi } ^{-1} ( z(\breve{x},\breve{y}) ) \) we get the desired inequality in (2.14). For \(a(\breve{x},\breve{y}) =0\), we carry out the above procedure with \(\epsilon >0\) instead of \(a(\breve{x}, \breve{y}) \) and subsequently let \(\epsilon \rightarrow 0\). This completes the proof. □

Corollary 2.8

If we take\(\breve{\mathbb{T}}=\mathbb{R}\)in Theorem2.7, then the inequality

$$\begin{aligned} \bar{\psi } \bigl( u(\breve{x},\breve{y}) \bigr) \leq &a( \breve{x},\breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}\bar{ \varphi } \bigl( u( \breve{s},\breve{t}) \bigr) \bigl[ f(\breve{s}, \breve{t})\bar{\varphi } \bigl( u( \breve{s},\breve{t}) \bigr) +p( \breve{s},\breve{t}) \bigr] \,d \breve{t}\,d \breve{s} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t})\bar{ \varphi } \bigl( u(\breve{s},\breve{t}) \bigr) \biggl( \int _{0}^{ \breve{s}}g(\breve{\tau },\breve{t})\bar{ \varphi } \bigl( u( \breve{\tau },\breve{t}) \bigr) \hat{\Delta }\breve{\tau } \biggr) \,d \breve{t}\,d \breve{s}, \end{aligned}$$

for\((\breve{x},\breve{y}) \in \varOmega \), implies

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \biggl\{ \breve{G}^{-1} \biggl( \breve{F}^{-1} \biggl[ \breve{F} \bigl( q_{2} ( \breve{x}, \breve{y} ) \bigr) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f( \breve{s},\breve{t}) \biggl( 1+ \int _{0}^{\breve{s}}g(\breve{\tau }, \breve{t})\,d \breve{ \tau } \biggr) \,d \breve{t}\,d \breve{s} \biggr] \biggr) \biggr\} , $$

for\(0\leq \breve{x}\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\), whereĞis as defined in (2.8) and

$$\begin{aligned}& q_{2} ( \breve{x},\breve{y} ) =\breve{G} \bigl( a( \breve{x}, \breve{y}) \bigr) + \int _{0}^{\breve{x}} \int _{0}^{ \breve{y}}p(\breve{s},\breve{t})\,d \breve{t}\,d \breve{s}, \\& \begin{gathered} \breve{F}(r)= \int _{r_{0}}^{r}\frac{d \breve{s}}{ ( ( \bar{ \varphi } \circ \bar{\psi } ^{-1} ) \circ \breve{G}^{-1} ) (\breve{s} ) },\quad r\geq r_{0}>0,\\ \breve{F}(+\infty )= \int _{r_{0}}^{+\infty }\frac{d \breve{s}}{ ( \bar{\varphi } \circ \bar{\psi } ^{-1} ) \circ \breve{G}^{-1}(\breve{s})}=+ \infty , \end{gathered} \end{aligned}$$

and\(( \breve{x}_{1},\breve{y}_{1} ) \in \varOmega \)is chosen so that

$$ \biggl( \breve{F} \bigl( q_{2} ( \breve{x},\breve{y} ) \bigr) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}} f(\breve{s}, \breve{t}) \biggl( 1+ \int _{0}^{\breve{s}}g(\breve{\tau },\breve{t})\,d \breve{ \tau } \biggr) \,d \breve{t}\,d \breve{s} \biggr) \in \operatorname{Dom} \bigl( \breve{F}^{-1} \bigr). $$

Corollary 2.9

The discrete form of Theorem2.7can be obtained by letting\(\breve{\mathbb{T}}=\mathbb{Z}\):

$$\begin{aligned} \bar{\psi } \bigl( u(\breve{x},\breve{y}) \bigr) \leq &a( \breve{x},\breve{y}) +\sum_{\breve{s}=0}^{\breve{x}-1}\sum _{\breve{t}=0} ^{\breve{y}-1}\bar{\varphi } \bigl( u(\breve{s}, \breve{t}) \bigr) \bigl[ f(\breve{s},\breve{t})\bar{\varphi } \bigl( u(\breve{s}, \breve{t}) \bigr) +p(\breve{s},\breve{t}) \bigr] \\ &{}+\sum_{\breve{s}=0}^{\breve{x}-1}\sum _{\breve{t}=0}^{\breve{y}-1}f( \breve{s},\breve{t})\bar{\varphi } \bigl( u(\breve{s},\breve{t}) \bigr) \Biggl( \sum_{\breve{\tau }=0}^{ \breve{s}-1}g( \breve{\tau },\breve{t})\bar{\varphi } \bigl( u( \breve{\tau },\breve{t}) \bigr) \Biggr), \end{aligned}$$

for\((\breve{x},\breve{y}) \in \varOmega \), implies

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \Biggl\{ \bar{G}^{-1} \Biggl( \bar{F}^{-1} \Biggl[ \bar{F} \bigl( \bar{q}_{2} ( \breve{x},\breve{y} ) \bigr) +\sum _{\breve{s}=0}^{\breve{x}-1} \sum_{\breve{t}=0}^{\breve{y}-1}f( \breve{s},\breve{t}) \Biggl( 1+ \sum_{\breve{\tau }=0}^{\breve{s}-1}g( \breve{\tau },\breve{t}) \Biggr) \Biggr] \Biggr) \Biggr\} , $$

for\(0\leq \breve{x}\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\), whereĞis as defined in (2.9) and

$$\begin{aligned}& \bar{q}_{2} ( \breve{x},\breve{y} ) =\breve{G} \bigl( a( \breve{x}, \breve{y}) \bigr) +\sum_{\breve{s}=0}^{\breve{x}-1} \sum _{\breve{t}=0}^{\breve{y}-1}p(\breve{s},\breve{t}), \\& \begin{gathered} \bar{F}(r)=\sum_{\breve{s}=r_{0}}^{r-1} \frac{1}{ ( ( \bar{ \varphi } \circ \bar{\psi } ^{-1} ) \circ \bar{G}^{-1} ) (\breve{s} ) },\quad r\geq r_{0}>0,\\ \bar{F}(+\infty )= \sum_{\breve{s}=r_{0}}^{+\infty }\frac{1}{ ( \bar{\varphi } \circ \bar{\psi } ^{-1} ) \circ \bar{G}^{-1}(\breve{s})}=+ \infty , \end{gathered} \end{aligned}$$

and\(( \breve{x}_{1},\breve{y}_{1} ) \in \varOmega \)is chosen so that

$$ \Biggl( \bar{F} \bigl( \bar{q}_{2} ( \breve{x},\breve{y} ) \bigr) + \sum_{\breve{s}=0}^{\breve{x}-1}\sum _{\breve{t}=0}^{ \breve{y}-1} f(\breve{s},\breve{t}) \Biggl( 1+\sum _{\breve{\tau }=0} ^{\breve{s}-1}g(\breve{\tau },\breve{t}) \Biggr) \Biggr) \in \operatorname{Dom} \bigl( \bar{F}^{-1} \bigr). $$

Theorem 2.10

Assume thatg, a, f, u, ,ψ̄andφ̄be as in Theorem2.1. If\(u(\breve{x},\breve{y}) \)satisfies

$$\begin{aligned} \bar{\psi } \bigl( u(\breve{x},\breve{y}) \bigr) \leq &a( \breve{x},\breve{y}) + \biggl( \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f( \breve{s},\breve{t})\bar{ \varphi } \bigl( u(\breve{s},\breve{t}) \bigr) \hat{\Delta }\breve{t}\hat{\Delta } \breve{s} \biggr) ^{2} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t})\bar{ \varphi } \bigl( u(\breve{s},\breve{t}) \bigr) \biggl( \int _{0}^{ \breve{s}}g(\breve{\tau },\breve{t})\bar{ \varphi } \bigl( u( \breve{\tau },\breve{t}) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s}, \end{aligned}$$

for\((\breve{x},\breve{y}) \in \varOmega \), then

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \biggl\{ \breve{H}^{-1} \biggl[ \breve{H} \bigl( a ( \breve{x},\breve{y} ) \bigr) +\breve{B}(\breve{x},\breve{y}) + \biggl( \int _{0}^{\breve{x}} \int _{0} ^{\breve{y}}f(\breve{s},\breve{t})\hat{\Delta }\breve{t}\hat{\Delta } \breve{s} \biggr) ^{2} \biggr] \biggr\} , $$
(2.21)

for\(0\leq \breve{x} \leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\), where

$$\begin{aligned}& \breve{B}(\breve{x},\breve{y}) = \int _{0}^{\breve{x}} \int _{0}^{ \breve{y}}f(\breve{s},\breve{t}) \biggl( \int _{0}^{\breve{s}}g( \breve{\tau },\breve{t})\hat{ \Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s}, \end{aligned}$$
(2.22)
$$\begin{aligned}& \begin{gathered} \breve{H}(r)= \int _{r_{0}}^{r}\frac{\hat{\Delta }\breve{s}}{ ( \bar{ \varphi } \circ \bar{\psi } ^{-1} ) ^{2} ( \breve{s} ) },\quad r\geq r_{0}>0,\\ \breve{H}(+\infty )= \int _{r_{0}}^{+\infty }\frac{ \hat{\Delta }\breve{s}}{ ( \bar{\varphi } \circ \bar{\psi } ^{-1} ) ^{2} ( \breve{s} ) }=+\infty , \end{gathered} \end{aligned}$$
(2.23)

and\(( \breve{x}_{1},\breve{y}_{1} ) \in \varOmega \)is chosen so that

$$ \biggl( \breve{H} \bigl( a ( \breve{x},\breve{y} ) \bigr) +B(\breve{x}, \breve{y}) + \biggl( \int _{0}^{\breve{x}} \int _{0}^{ \breve{y}}f(\breve{s},\breve{t})\hat{\Delta }\breve{t}\hat{\Delta } \breve{s} \biggr) ^{2} \biggr) \in \operatorname{Dom} \bigl( \breve{H}^{-1} \bigr). $$

Proof

Assume that \(a(\breve{x},\breve{y}) >0\). Taking \((\breve{\xi }, \breve{\zeta })\in \varOmega \) as a fixed arbitrary point, we define \(z(\breve{x},\breve{y}) >0\) to be a nondecreasing function by

$$\begin{aligned} z(\breve{x},\breve{y}) =&a(\breve{\xi },\breve{\zeta })+ \biggl( \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t})\bar{\varphi } \bigl( u(\breve{s},\breve{t}) \bigr) \hat{\Delta }\breve{t} \hat{\Delta } \breve{s} \biggr) ^{2} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t})\bar{ \varphi } \bigl( u(\breve{s},\breve{t}) \bigr) \biggl( \int _{0}^{ \breve{s}}g(\breve{\tau },\breve{t})\bar{ \varphi } \bigl( u( \breve{\tau },\breve{t}) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s}, \end{aligned}$$
(2.24)

for \(0\leq \breve{x}\leq \breve{\xi }\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{\zeta }\leq \breve{y}_{1}\), hence \(z(0, \breve{y}) =z(\breve{x},0) =a(\breve{\xi },\breve{\zeta })\) and

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \bigl( z(\breve{x}, \breve{y}) \bigr). $$

From (2.24), and applying the chain rule on time scales, Theorem 1.4, we get

$$\begin{aligned} z^{\hat{\Delta } \breve{x}}(\breve{x},\breve{y}) =&2 \biggl( \int _{0} ^{c} \int _{0}^{\breve{y}}f(\breve{s},\breve{t})\bar{\varphi } \bigl( u( \breve{s},\breve{t}) \bigr) \hat{\Delta }\breve{t}\hat{\Delta } \breve{s} \biggr) \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \bar{ \varphi } \bigl( u(\breve{x},\breve{t}) \bigr) \hat{\Delta } \breve{t} \\ &{}+ \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \bar{ \varphi } \bigl( u( \breve{x},\breve{t}) \bigr) \biggl( \int _{0}^{\breve{x}}g( \breve{\tau } ,\breve{t})\bar{ \varphi } \bigl( u(\breve{\tau }, \breve{t}) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{\Delta } \breve{t} \\ \leq &2 \biggl( \int _{0}^{c} \int _{0}^{\breve{y}}f(\breve{s},\breve{t})\bar{ \varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{s},\breve{t}) \bigr) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \biggr) \int _{0}^{ \breve{y}}f(\breve{x},\breve{t}) \bar{ \varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{x},\breve{t}) \bigr) \hat{\Delta }\breve{t} \\ &{}+ \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \bar{ \varphi } \circ \bar{ \psi } ^{-1} \bigl( z(\breve{x},\breve{t}) \bigr) \biggl( \int _{0} ^{\breve{x}}g(\breve{\tau },\breve{t})\bar{ \varphi } \circ \bar{ \psi } ^{-1} \bigl( z(\breve{\tau },\breve{t}) \bigr) \hat{\Delta } \breve{\tau } \biggr) \hat{\Delta }\breve{t} \\ \leq &2 \bigl( \bar{\varphi } \circ \bar{\psi } ^{-1} \bigl( z( \breve{x},\breve{y}) \bigr) \bigr) ^{2} \biggl( \int _{0}^{c} \int _{0}^{\breve{y}}f(\breve{s},\breve{t})\hat{\Delta }\breve{t} \hat{\Delta }\breve{s} \biggr) \int _{0}^{\breve{y}}f(\breve{x}, \breve{t}) \hat{ \Delta }\breve{t} \\ &{}+ \bigl( \bar{\varphi } \circ \bar{\psi } ^{-1} \bigl( z(\breve{x}, \breve{y}) \bigr) \bigr) ^{2} \int _{0}^{\breve{y}}f(\breve{x}, \breve{t}) \biggl( \int _{0}^{\breve{x} }g(\breve{\tau },\breve{t}) \hat{ \Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t}, \end{aligned}$$

thus, we have

$$\begin{aligned} \frac{z^{\hat{\Delta } \breve{x}}(\breve{x},\breve{y}) }{ ( \bar{ \varphi } \circ \bar{\psi } ^{-1} ( z(\breve{x},\breve{y}) ) ) ^{2}} \leq &2 \biggl( \int _{0}^{c} \int _{0}^{\breve{y}}f( \breve{s},\breve{t})\hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \biggr) \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \hat{\Delta }\breve{t} \\ &{}+ \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \biggl( \int _{0}^{ \breve{x}}g(\breve{\tau },\breve{t})\hat{ \Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t} \\ =& \biggl[ \biggl( \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s}, \breve{t})\hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \biggr)^{2} \biggr] ^{\hat{\Delta }_{\breve{x}}} \\ &{}+ \int _{0}^{\breve{y}}f(\breve{x},\breve{t}) \biggl( \int _{0}^{ \breve{x}}g(\breve{\tau },\breve{t})\hat{ \Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t}. \end{aligned}$$
(2.25)

Integrating (2.25) and using (2.23), we get

$$\begin{aligned} \breve{H} \bigl( z(\breve{x},\breve{y}) \bigr) \leq &\breve{H} \bigl( a(\breve{ \xi },\breve{\zeta }) \bigr) + \biggl( \int _{0}^{ \breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t})\hat{\Delta } \breve{t}\hat{\Delta }\breve{s} \biggr) ^{2} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \biggl( \int _{0}^{\breve{s}}g(\breve{\tau } ,\breve{t})\hat{ \Delta } \breve{\tau } \biggr) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s}. \end{aligned}$$

Since \((\breve{\xi },\breve{\zeta })\in \varOmega \) is chosen arbitrarily,

$$ z(\breve{x},\breve{y}) \leq \breve{H}^{-1} \biggl[ \breve{H} \bigl( a( \breve{x},\breve{y}) \bigr) +\breve{B}(\breve{x},\breve{y}) + \biggl( \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \biggr) ^{2} \biggr] . $$
(2.26)

From (2.26) and \(u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} ( z(\breve{x},\breve{y}) ) \), we get the desired inequality (2.21). For \(a(\breve{x},\breve{y}) =0\), we carry out the above procedure with \(\epsilon >0\) instead of \(a(\breve{x},\breve{y}) \) and subsequently let \(\epsilon \rightarrow 0\). This completes the proof. □

Corollary 2.11

If we take\(\breve{\mathbb{T}}=\mathbb{R}\)in Theorem2.10, then the inequality

$$\begin{aligned} \bar{\psi } \bigl( u(\breve{x},\breve{y}) \bigr) \leq &a( \breve{x},\breve{y}) + \biggl( \int _{0}^{\breve{x}} \int _{0}^{\breve{y}} f(\breve{s},\breve{t})\bar{ \varphi } \bigl( u(\breve{s},\breve{t}) \bigr) \,d \breve{t}\,d \breve{s} \biggr) ^{2} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t})\bar{ \varphi } \bigl( u(\breve{s},\breve{t}) \bigr) \biggl( \int _{0}^{ \breve{s}}g(\breve{\tau },\breve{t})\bar{ \varphi } \bigl( u( \breve{\tau },\breve{t}) \bigr) \,d \breve{\tau } \biggr) \,d \breve{t}\,d \breve{s}, \end{aligned}$$

for\((\breve{x},\breve{y}) \in \varOmega \), implies

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \biggl\{ \breve{H}^{-1} \biggl[ \breve{H} \bigl( a ( \breve{x},\breve{y} ) \bigr) +\breve{B}(\breve{x},\breve{y}) + \biggl( \int _{0}^{\breve{x}} \int _{0} ^{\breve{y}}f(\breve{s},\breve{t})\,d \breve{t}\,d \breve{s} \biggr) ^{2} \biggr] \biggr\} , $$

for\(0\leq \breve{x} \leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\), where

$$\begin{aligned}& \breve{B}(\breve{x},\breve{y}) = \int _{0}^{\breve{x}} \int _{0}^{ \breve{y}}f(\breve{s},\breve{t}) \biggl( \int _{0}^{\breve{s}}g( \breve{\tau },\breve{t})\,d \breve{ \tau } \biggr) \,d \breve{t}\,d \breve{s}, \\& \breve{H}(r)= \int _{r_{0}}^{r}\frac{d \breve{s}}{ ( \bar{\varphi } \circ \bar{\psi } ^{-1} ) ^{2} ( \breve{s} ) },\quad r \geq r_{0}>0,\qquad \breve{H}(+\infty )= \int _{r_{0}}^{+\infty }\frac{d \breve{s}}{ ( \bar{\varphi } \circ \bar{\psi } ^{-1} ) ^{2} ( \breve{s} ) }=+\infty , \end{aligned}$$

and\(( \breve{x}_{1},\breve{y}_{1} ) \in \varOmega \)is chosen so that

$$ \biggl( \breve{H} \bigl( a ( \breve{x},\breve{y} ) \bigr) +B(\breve{x}, \breve{y}) + \biggl( \int _{0}^{\breve{x}} \int _{0}^{ \breve{y}} f(\breve{s},\breve{t})\,d \breve{t}\,d \breve{s} \biggr) ^{2} \biggr) \in \operatorname{Dom} \bigl( \breve{H}^{-1} \bigr). $$

Corollary 2.12

The discrete form can be obtained by letting\(\breve{\mathbb{T}}= \mathbb{Z}\)in Theorem2.10:

$$\begin{aligned} \bar{\psi } \bigl( u(\breve{x},\breve{y}) \bigr) \leq &a( \breve{x},\breve{y}) + \Biggl( \sum_{\breve{s}=0}^{\breve{x}-1} \sum _{\breve{t}=0}^{\breve{y}-1} f(\breve{s},\breve{t})\bar{\varphi } \bigl( u(\breve{s},\breve{t}) \bigr) \Biggr) ^{2} \\ &{}+\sum_{\breve{s}=0}^{\breve{x}-1}\sum _{\breve{t}=0}^{\breve{y}-1}f( \breve{s},\breve{t})\bar{\varphi } \bigl( u(\breve{s},\breve{t}) \bigr) \Biggl( \sum_{\breve{\tau }=0}^{ \breve{s}-1}g( \breve{\tau },\breve{t})\bar{\varphi } \bigl( u( \breve{\tau },\breve{t}) \bigr) \Biggr), \end{aligned}$$

for\((\breve{x},\breve{y}) \in \varOmega \), implies

$$ u(\breve{x},\breve{y}) \leq \bar{\psi } ^{-1} \Biggl\{ \breve{H}^{-1} \Biggl[ \breve{H} \bigl( a ( \breve{x},\breve{y} ) \bigr) +\breve{B}(\breve{x},\breve{y}) + \Biggl( \sum _{\breve{s}=0}^{ \breve{x}-1}\sum_{\breve{t}=0}^{\breve{y}-1}f( \breve{s},\breve{t}) \Biggr) ^{2} \Biggr] \Biggr\} , $$

for\(0\leq \breve{x} \leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\), where

$$\begin{aligned}& \breve{B}(\breve{x},\breve{y}) =\sum_{\breve{s}=0}^{\breve{x}-1} \sum_{\breve{t}=0}^{\breve{y}}f(\breve{s},\breve{t}) \Biggl( \sum_{\breve{\tau }=0}^{\breve{s}-1}g(\breve{\tau }, \breve{t}) \Biggr), \\& \breve{H}(r)=\sum_{\breve{s}=r_{0}}^{r-1} \frac{1}{ ( \bar{\varphi } \circ \bar{\psi } ^{-1} ) ^{2} ( \breve{s} ) },\quad r \geq r_{0}>0,\qquad \breve{H}(+ \infty )=\sum_{\breve{s}=r_{0}}^{+ \infty } \frac{1}{ ( \bar{\varphi } \circ \bar{\psi } ^{-1} ) ^{2} ( \breve{s} ) }=+\infty , \end{aligned}$$

and\(( \breve{x}_{1},\breve{y}_{1} ) \in \varOmega \)is chosen so that

$$ \Biggl( \breve{H} \bigl( a ( \breve{x},\breve{y} ) \bigr) +B(\breve{x}, \breve{y}) + \Biggl( \sum_{\breve{s}=0}^{\breve{x}-1} \sum _{\breve{t}=0}^{\breve{y}-1} f(\breve{s},\breve{t}) \Biggr) ^{2} \Biggr) \in \operatorname{Dom} \bigl( \breve{H}^{-1} \bigr). $$

Applications

The present section illustrates how Theorems 2.7 and 2.1 can be used to study the boundedness of the solutions of some initial boundary value problem for partial dynamic equations in two independent variables.

Let us consider the problem

$$\begin{aligned}& u^{\hat{\Delta } \breve{x}\hat{\Delta } \breve{y}}(\breve{x}, \breve{y}) =\breve{F} \biggl( \breve{x}, \breve{y},u ( \breve{x}, \breve{y} ) , \int _{0}^{\breve{x}}\breve{k} \bigl( \breve{s}, \breve{y},u ( s,\breve{y} ) \bigr) \hat{\Delta } \breve{s} \biggr), \end{aligned}$$
(3.1)
$$\begin{aligned}& u ( \breve{x},0 ) =a_{1}(\breve{x}),\qquad u(0,\breve{y}) =a_{2}( \breve{y}),\qquad a_{1}(0)=a_{2}(0)=0, \end{aligned}$$
(3.2)

for any \((\breve{x},\breve{y}) \in \varOmega \), where \(\breve{k}\in C _{\mathrm{rd}} ( \varOmega \times \mathbb{R} ,\mathbb{R} )\), \(\breve{F} \in C_{\mathrm{rd}} ( \varOmega \times \mathbb{R} \times \mathbb{R} , \mathbb{R} ) \), \(a_{1}\in C_{\mathrm{rd}} ( \breve{\mathbb{T}}_{1}, \mathbb{R} ) \) and \(a_{2}\in C_{\mathrm{rd}} ( \breve{\mathbb{T}} _{2},\mathbb{R} ) \).

Theorem 3.1

Suppose that the functions, , \(a_{2}\), \(a_{1}\)in (3.1) and (3.2) satisfy the conditions

$$\begin{aligned}& \begin{aligned}[b] \bigl\vert \breve{F} ( \breve{x},\breve{y},u(\breve{x}, \breve{y},v ) \bigr\vert \leq {}&\bar{\varphi } \bigl( \bigl\vert u ( \breve{x},\breve{y} ) \bigr\vert \bigr) \bigl[ f ( \breve{x},\breve{y} ) \bar{\varphi } \bigl( \bigl\vert u (\breve{x},\breve{y} ) \bigr\vert \bigr) +p ( \breve{x},\breve{y} ) \bigr] \\ &{}+f ( \breve{x},\breve{y} ) \bar{\varphi } \bigl( \bigl\vert u ( \breve{x}, \breve{y} ) \bigr\vert \bigr) v, \end{aligned} \end{aligned}$$
(3.3)
$$\begin{aligned}& \bigl\vert \breve{k} \bigl( \breve{x},\breve{y},u ( \breve{x}, \breve{y} ) \bigr) \bigr\vert \leq g ( \breve{x}, \breve{y} ) \bar{ \varphi } \bigl( \bigl\vert u ( \breve{x},\breve{y} ) \bigr\vert \bigr), \end{aligned}$$
(3.4)
$$\begin{aligned}& \bigl\vert a_{1}(\breve{x})+a_{2}( \breve{y}) \bigr\vert \leq a( \breve{x},\breve{y}), \end{aligned}$$
(3.5)

where the functionsp, g, a, f, andφ̄are defined as in Theorem2.7with\(a(\breve{x},\breve{y}) >0\), for all\((\breve{x},\breve{y}) \in \varOmega \), then

$$ \bigl\vert u ( \breve{x},\breve{y} ) \bigr\vert \leq \breve{G}^{-1} \biggl( \breve{F}^{-1} \biggl[ \breve{F} \bigl( q_{2}( \breve{x},\breve{y}) \bigr) + \int _{0}^{\breve{x}} \int _{0}^{ \breve{y}}f(\breve{s},\breve{t}) \biggl[ 1+ \int _{0}^{\breve{s}}g ( \breve{\tau },\breve{t} ) \hat{ \Delta }\breve{\tau } \biggr] \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \biggr] \biggr), $$
(3.6)

for\(0\leq \breve{x}\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\), whereF, \(q_{2}\)andGare defined as in Theorem2.7.

Proof

If the problem (3.1) and (3.2) has a solution \(u( \breve{x},\breve{y}) \), it can be written as

$$ u(\breve{x},\breve{y}) =a_{1}(\breve{x})+a_{2}( \breve{y})+ \int _{0} ^{\breve{x}} \int _{0}^{\breve{y}}\breve{F} \biggl( \breve{s}, \breve{t},u (\breve{s},\breve{t} ) , \int _{0}^{\breve{s}}\breve{k} \bigl( \breve{\tau }, \breve{t},u ( \breve{\tau } ,t ) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{ \Delta }\breve{t} \hat{\Delta }\breve{s}, $$
(3.7)

for any \((\breve{x},\breve{y}) \in \varOmega \). Using the conditions (3.3), (3.4) and (3.5) in (3.7), we get

$$\begin{aligned} \bigl\vert u ( \breve{x},\breve{y} ) \bigr\vert \leq &a( \breve{x}, \breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}\bar{ \varphi } \bigl( \bigl\vert u ( \breve{s},\breve{t} ) \bigr\vert \bigr) \bigl[ f ( \breve{s}, \breve{t} ) \bar{ \varphi } \bigl( \bigl\vert u ( \breve{s},\breve{t} ) \bigr\vert \bigr) +p ( \breve{s},\breve{t} ) \bigr] \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f ( s,t ) \bar{ \varphi } \bigl( \bigl\vert u ( s,t ) \bigr\vert \bigr) \biggl( \int _{0}^{\breve{s}}g ( \breve{\tau },\breve{t} ) \bar{ \varphi } \bigl( \bigl\vert u ( \breve{\tau },\breve{t} ) \bigr\vert \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s}, \end{aligned}$$
(3.8)

for any \((\breve{x},\breve{y}) \in \varOmega \). Now, an application of Theorem 2.7 to (3.8) yields the required inequality in (3.6) where \(\bar{\psi }(u)=u\). □

Let us consider the initial boundary value problem of the form

$$\begin{aligned}& \bigl(z^{q} \bigr)^{\hat{\Delta } \breve{x} \hat{\Delta } \breve{y}}( \breve{x},\breve{y}) = \breve{A} \biggl( \breve{x},\breve{y},z ( \breve{x},\breve{y} ) , \int _{0}^{\breve{x}}h \bigl( \breve{s}, \breve{y},z ( \breve{s},\breve{y} ) \bigr) \hat{\Delta } \breve{s} \biggr) \end{aligned}$$
(3.9)
$$\begin{aligned}& z ( \breve{x},0 ) =a_{1}(\breve{x}),\qquad z(0,\breve{y}) =a_{2}( \breve{y}),\qquad a_{1}(0)=a_{2}(0)=0, \end{aligned}$$
(3.10)

for any \((\breve{x},\breve{y}) \in \varOmega \).

Theorem 3.2

Assume that the functionsh, A, \(a_{2}\), \(a_{1}\)in (3.9) and (3.10) satisfy the conditions

$$\begin{aligned}& \bigl\vert A ( \breve{x},\breve{y},z(\breve{x},\breve{y},v ) \bigr\vert \leq f ( \breve{x},\breve{y} ) \bigl\vert z ^{r} ( \breve{x},\breve{y} ) \bigr\vert +f ( \breve{x},\breve{y} ) v, \end{aligned}$$
(3.11)
$$\begin{aligned}& \bigl\vert h \bigl( \breve{x},\breve{y},z ( \breve{x},\breve{y} ) \bigr) \bigr\vert \leq g ( \breve{x},\breve{y} ) \bigl\vert z^{r} ( \breve{x}, \breve{y} ) \bigr\vert , \end{aligned}$$
(3.12)
$$\begin{aligned}& \bigl\vert a_{1}(\breve{x})+a_{2}(\breve{y}) \bigr\vert \leq a( \breve{x},\breve{y}) , \end{aligned}$$
(3.13)

where\(r\geq q>0\), then

$$ \bigl\vert z(\breve{x},\breve{y}) \bigr\vert \leq \biggl[ \bigl( a( \breve{x},\breve{y}) \bigr) ^{\frac{q-r}{q}}+\frac{q-r}{q} \int _{0} ^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \biggl( 1+ \int _{0}^{\breve{s}}g ( \breve{\tau },\breve{t} ) \hat{ \Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t} \hat{\Delta }\breve{s} \biggr] ^{\frac{1}{q-r}}, $$
(3.14)

for\(0\leq \breve{x}\leq \breve{x}_{1}\), \(0\leq \breve{y}\leq \breve{y} _{1}\).

Proof

If the problem (3.9) and (3.10), has a solution \(z(\breve{x},\breve{y}) \) it can be written as

$$ z^{q}(\breve{x},\breve{y}) =a_{1}(x)+a_{2}(y)+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}\breve{F} \biggl( \breve{s}, \breve{st},u ( \breve{s},\breve{t} ) , \int _{0}^{\breve{s}}\breve{k} \bigl( \breve{\tau }, \breve{t},u ( \breve{\tau },\breve{t} ) \bigr) \hat{\Delta }\breve{\tau } \biggr) \hat{\Delta }\breve{t} \hat{\Delta }\breve{s}, $$
(3.15)

for any \((\breve{x},\breve{y}) \in \varOmega \). Using the conditions (3.11), (3.12) and (3.13) in (3.15), we get

$$\begin{aligned} \bigl\vert z^{q} ( \breve{x},\breve{y} ) \bigr\vert \leq &a( \breve{x},\breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f ( \breve{s},\breve{t} ) \bigl\vert z^{r} ( s,t ) \bigr\vert \hat{\Delta }\breve{t}\hat{\Delta }\breve{s} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f ( \breve{s}, \breve{t} ) \biggl( \int _{0}^{\breve{s}}g ( \breve{\tau }, \breve{t} ) \bigl\vert z^{r} ( \breve{\tau },\breve{t} ) \bigr\vert \hat{ \Delta } \breve{\tau } \biggr) \hat{\Delta }\breve{t}\hat{\Delta }\breve{s}, \end{aligned}$$
(3.16)

from (3.16), we get

$$\begin{aligned} \bigl\vert z^{q} ( \breve{x},\breve{y} ) \bigr\vert \leq &a( \breve{x},\breve{y}) + \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f( \breve{s},\breve{t}) \bigl\vert z^{r} ( \breve{t},\breve{t} ) \bigr\vert \hat{\Delta } \breve{t}\hat{\Delta }\breve{s} \\ &{}+ \int _{0}^{\breve{x}} \int _{0}^{\breve{y}}f(\breve{s},\breve{t}) \biggl( \int _{0}^{\breve{s}}g ( \breve{\tau },\breve{t} ) \bigl\vert z ^{r} ( \breve{\tau },\breve{t} ) \bigr\vert \hat{\Delta } \breve{\tau } \biggr) \hat{\Delta }\breve{t} \hat{\Delta }\breve{s}, \end{aligned}$$
(3.17)

for any \((\breve{x},\breve{y}) \in \varOmega \). A suitable application of Theorem 2.1 to (3.17) with \(\bar{\psi } (u)=u^{q}\), \(\bar{ \varphi } ( u ) =u^{r}\) and \(p(\breve{x},\breve{y}) =0\) gives the required inequality in (3.14). □

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Acknowledgements

We are immensely thankful to the editor and anonymous referees for their valuable remarks, which helped to improve the paper.

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This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.

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El-Deeb, A.A., Khan, Z.A. Certain new dynamic nonlinear inequalities in two independent variables and applications. Bound Value Probl 2020, 31 (2020). https://doi.org/10.1186/s13661-020-01338-z

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Keywords

  • Gronwall-type inequality
  • Boundedness
  • Time scales