On dynamic inequalities in two independent variables on time scales and their applications for boundary value problems

Abstract

Our work is based on the multiple inequalities illustrated by Boudeliou and Khalaf in 2015. With the help of the Leibniz integral rule on time scales, we generalize a number of those inequalities to a general time scale. Besides that, in order to obtain some new inequalities as special cases, we also extend our inequalities to discrete, quantum, and continuous calculus. These inequalities may be of use in the analysis of some kinds of partial dynamic equations on time scales and their applications in environmental phenomena, physical and engineering sciences described by partial differential equations.

1 Introduction

In 2015, Boudeliou and Khalaf [15] proved the following inequalities.

Theorem 1.1

Let u, f, $\phi \in C( {\Omega} ,\mathbb{R}_{+})$ and $a\in C( {\Omega} ,\mathbb{R}_{+})$ be nondecreasing with respect to $(x,y)\in I_{1}\times I_{2}$; let $\theta \in C^{1}(I_{1},I_{1})$, $\vartheta \in C^{1}(I_{2},I_{2})$ be nondecreasing with $\theta (x)\leq x$ on $I_{1}$, $\vartheta (y)\leq y$ on $I_{2}$. Let $\phi _{1}$, $\phi _{2}\in C( {\Omega} ,\mathbb{R}_{+})$. Further, let ψ, ω, $\eta \in C(\mathbb{R}_{+},\mathbb{R}_{+})$ be nondecreasing functions with $\{ \psi ,\omega ,\eta \} (u)>0$ for $u>0$, and $\lim_{u\rightarrow +\infty }\psi (u)=+\infty$.

$(A_{1})$ If u satisfies

$$\begin{aligned} \psi \bigl( u(x,y) \bigr) \leq &a(x,y)+ \int _{0}^{\theta (x)} \int _{0}^{ \vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t)\omega \bigl( u(s,t) \bigr) \\ &{} + \int _{0}^{s}\phi _{2}(\tau ,t) \omega \bigl( u(\tau ,t) \bigr) \,d\tau \biggr] \,dt\,ds \end{aligned}$$

for $(x,y)\in {\Omega}$, then

$$ u(x,y)\leq \psi ^{-1} \biggl\{ G^{-1} \biggl( p(x,y)+ \int _{0}^{\theta (x)} \int _{0}^{\vartheta (y)}\phi _{1}(s,t)f(s,t)\,dt\,ds \biggr) \biggr\} $$

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where G is defined by (2.3) and

$$ p(x,y)=G \bigl( a(x,y) \bigr) + \int _{0}^{\theta (x)} \int _{0}^{ \vartheta (y)}\phi _{1}(s,t) \biggl( \int _{0}^{s}\phi _{2}(\tau ,t)\,d \tau \biggr) \,dt\,ds $$

and $( x_{1},y_{1} ) \in {\Omega}$ is chosen so that $( p(x,y)+\int _{0}^{\theta (x)}\int _{0}^{\vartheta (y)}\phi _{1}(s,t)f(s,t)\,dt\,ds ) \in \operatorname{Dom} ( G^{-1} )$.

$(A_{2})$ If $u(x,y)$ satisfies

$$\begin{aligned} \psi \bigl( u(x,y) \bigr) \leq &a(x,y)+ \int _{0}^{\theta (x)} \int _{0}^{ \vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t)\omega \bigl( u(s,t) \bigr) \eta \bigl( u(s,t) \bigr) \\ &{} + \int _{0}^{s}\phi _{2}(\tau ,t) \omega \bigl( u(\tau ,t) \bigr) \,d\tau \biggr] \,dt\,ds \end{aligned}$$

for $(x,y)\in {\Omega}$, then

$$ u(x,y)\leq \psi ^{-1} \biggl\{ G^{-1} \biggl( F^{-1} \biggl[ F \bigl( p(x,y) \bigr) + \int _{0}^{\theta (x)} \int _{0}^{\vartheta (y)}\phi _{1}(s,t)f(s,t)\,dt\,ds \biggr] \biggr) \biggr\} $$

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where G and p are as in $(A_{1})$, and

$$ F(v)= \int _{v_{0}}^{v} \frac{ds}{\eta ( \psi ^{-1} ( G^{-1}(s) ) ) },\quad v\geq v_{0}>0,\qquad F(+\infty )=+\infty $$

and $( x_{1},y_{1} ) \in {\Omega}$ is chosen so that $[ F ( p(x,y) ) +\int _{0}^{\theta (x)}\int _{0}^{ \vartheta (y)}\phi _{1}(s,t)f(s,t)\,dt\,ds ] \in \operatorname{Dom} ( F^{-1} )$.

$(A_{3})$ If $u(x,y)$ satisfies

$$\begin{aligned} \psi \bigl( u(x,y) \bigr) \leq &a(x,y)+ \int _{0}^{\theta (x)} \int _{0}^{ \vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t)\omega \bigl( u(s,t) \bigr) \eta \bigl( u(s,t) \bigr) \\ &{} + \int _{0}^{s}\phi _{2}(\tau ,t) \omega \bigl( u( \tau ,t) \bigr) \eta \bigl( u(\tau ,t) \bigr) \,d\tau \biggr] \,dt\,ds \end{aligned}$$

for $(x,y)\in {\Omega}$, then

$$ u(x,y)\leq \psi ^{-1} \biggl\{ G^{-1} \biggl( F^{-1} \biggl[ p_{0}(x,y)+ \int _{0}^{\theta (x)} \int _{0}^{\vartheta (y)}\phi _{1}(s,t)f(s,t)\,dt\,ds \biggr] \biggr) \biggr\} $$

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where

$$ p_{0}(x,y)=F\bigl(G \bigl( a(x,y) \bigr) \bigr)+ \int _{0}^{\theta (x)} \int _{0}^{ \vartheta (y)}\phi _{1}(s,t) \biggl( \int _{0}^{s}\phi _{2}(\tau ,t)\,d \tau \biggr) \,dt\,ds $$

and $( x_{1},y_{1} ) \in {\Omega}$ is chosen so that $[ p_{0}(x,y)+\int _{0}^{\theta (x)}\int _{0}^{\vartheta (y)} \phi _{1}(s,t)f(s,t)\,dt\,ds ] \in \operatorname{Dom} ( F^{-1} )$.

Hilger in his PhD thesis [26] was the first to accomplish the unification and extension of differential equations, difference equations, q-difference equations, and so on to the encompassing theory of dynamic equations on time scales.

Throughout this work a knowledge and understanding of time scales and time-scale notation is assumed; for an excellent introduction to the calculus on time scales, see Bohner and Peterson [11, 13].

Over several decades Gronwall–Bellman-type inequalities, which have many applications in stability and oscillation theory, have attracted many researchers, and several refinements and extensions have been done to the previous results. For example, Yuzhen Mi [32] applied his results to study a boundary value problem of differential equations with impulsive terms. Also, we refer the reader to the works [1, 3, 4, 8, 18–20, 24, 34, 35, 40], see also [2, 5–7, 9, 10, 16, 17, 22, 27–30, 33, 36, 37].

Before we arrive at the main results in the next section, we need the following theorems and essential relations on some time scales such as $\mathbb{R}$, $\mathbb{Z}$, $h \mathbb{Z}$ and $\overline{q^{\mathbb{Z}}}$. Note that:

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

  $$ \sigma (t)=t,\quad \mu (t)=0,\quad \psi ^{\Delta}(t)= \psi ^{\prime }(t), \quad \int _{a}^{b}\psi (t)\Delta t= \int _{a}^{b}\psi (t)\,dt. $$

  (1.1)

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

  $$ \begin{aligned} &\sigma (t)=t+1,\qquad \mu (t)=1,\qquad \psi ^{\Delta}(t)= \psi (t+1)- \psi (t),\\ &\int _{a}^{b}\psi (t)\Delta t=\sum _{t=a}^{b-1}\psi (t). \end{aligned} $$

  (1.2)

• (iii) If $\mathbb{T}=h \mathbb{Z}$, then

  $$ \begin{aligned} &\sigma (t)=t+h,\qquad \mu (t)=h,\qquad \psi ^{\Delta}(t)= \frac{\psi (t+h)-\psi (t)}{h},\\ &\int _{a}^{b}\psi (t)\Delta t= \sum _{t=\frac{a}{h}}^{\frac{b}{h}-1}\psi (th)h. \end{aligned} $$

  (1.3)

• (iv) If $\mathbb{T}=\overline{q^{\mathbb{Z}}}$, then

  $$ \begin{aligned} &\sigma (t)=qt,\qquad \mu (t)=(q-1)t,\qquad \psi ^{\Delta}(t)= \frac{\psi (qt)-\psi (t)}{(q-1)t},\\ &\int _{a}^{b}\psi (t)\Delta t=(q-1) \sum _{t=(\log _{q}{a})}^{(\log _{q}{b})-1}\psi \bigl(q^{t} \bigr)q^{t}. \end{aligned} $$

  (1.4)

Theorem 1.2

If f is Δ-integrable on $[ {a}, {b}]$, then so is $\vert {f} \vert$, and

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

Theorem 1.3

(Chain rule on time scales [12])

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

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

(1.5)

Theorem 1.4

(see [14])

Let $t_{0}\in {\mathbb{T}}^{\kappa}$ and $k: {\mathbb{T}} \times {\mathbb{T}}^{\kappa}\rightarrow \mathbb{R}$ be continuous at $(t,t)$, where $t>t_{0}$ and $t\in {\mathbb{T}}^{\kappa}$. Assume that $k^{ {\Delta}}(t,\cdot )$ is rd-continuous on $[t_{0},\sigma (t)]$. If for any $\varepsilon > 0$ there exists a neighborhood U of t, independent of $\tau \in [t_{0},\sigma (t)]$, such that

$$ \bigl\vert \bigl[k\bigl(\sigma (t),\tau \bigr)-k(s,\tau )\bigr]-k^{ {\Delta}}(t, \tau )\bigl[\sigma (t)-s\bigr] \bigr\vert \leq \varepsilon \bigl\vert \sigma (t)-s \bigr\vert , \quad \forall s\in U. $$

If $k^{ {\Delta}}$ denotes the derivative of k with respect to the first variable, then

$$ f(t)= \int _{t_{0}}^{t}k(t,\tau ) {\Delta}\tau $$

yields

$$ f^{ {\Delta}}(t)= \int _{t_{0}}^{t}k^{ {\Delta}}(t,\tau ) { \Delta} \tau +k\bigl(\sigma (t),t\bigr). $$

Theorem 1.5

([21, Leibniz rule on time scales])

In the following, by $f^{\Delta }(t,s)$ we mean the delta derivative of $f(t,s)$ with respect to t. Similarly, $f^{\nabla }(t,s)$ is understood. If f, $f^{\Delta}$, and $f^{\nabla}$ are continuous and $u,h:\mathbb{T}\rightarrow \mathbb{T}$ are delta differentiable functions, then the following formulas hold $\forall t\in \mathbb{T^{\kappa}}$:

• (i) $[ \int ^{h(t)}_{u(t)}f(t,s)\Delta s ]^{ \Delta}= \int ^{h(t)}_{u(t)}f^{\Delta}(t,s)\Delta s + h^{ \Delta}(t)f(\sigma (t),h(t))- u^{\Delta}(t)f(\sigma (t),u(t))$;

• (ii) $[ \int ^{h(t)}_{u(t)}f(t,s)\Delta s ]^{ \nabla}= \int ^{h(t)}_{u(t)}f^{\nabla}(t,s)\Delta s + h^{ \nabla}(t)f(\rho (t),h(t))- u^{\nabla}(t)f(\rho (t),u(t))$;

• (iii) $[ \int ^{h(t)}_{u(t)}f(t,s)\nabla s ]^{ \Delta}= \int ^{h(t)}_{u(t)}f^{\Delta}(t,s)\nabla s + h^{ \Delta}(t)f(\sigma (t),h(t))- u^{\Delta}(t)f(\sigma (t),u(t))$;

• (iv) $[ \int ^{h(t)}_{u(t)}f(t,s)\nabla s ]^{ \nabla}= \int ^{h(t)}_{u(t)}f^{\nabla}(t,s)\nabla s + h^{ \nabla}(t)f(\rho (t),h(t))- u^{\nabla}(t)f(\rho (t),u(t))$.

In this manuscript, by applying Theorem 1.5, we discuss the retarded time scale case of the inequalities obtained in [15]. Furthermore, these inequalities that are proved here extend some known results in [23, 31, 38] and also unify the continuous, the discrete, and the quantum cases.

2 Main results

Lemma 2.1

Suppose that $\mathbb{T}_{1}$, $\mathbb{T}_{2}$ are two times scales and $a\in C (\Omega = \mathbb{T}_{1}\times \mathbb{T}_{2} ,\mathbb{R}_{+})$ is nondecreasing with respect to $(x,y) \in \Omega$. Assume that ϕ, u, $f\in C (\Omega ,\mathbb{R}_{+})$, $\theta \in C^{1} ( \mathbb{T}_{1},\mathbb{T}_{1} )$, and $\vartheta \in C^{1} ( \mathbb{T}_{2},\mathbb{T}_{2} )$ are nondecreasing functions with $\theta (x)\leq x$ on $\mathbb{T}_{1}$, $\vartheta (y)\leq y$ on $\mathbb{T}_{2}$. Furthermore, suppose that ψ, ${\omega} \in C(\mathbb{R}_{+},\mathbb{R}_{+})$ are nondecreasing functions with $\{ {\psi} , {\omega} \} (u)>0$ for $u>0$, and $\lim_{u\rightarrow +\infty } {\psi} (u)=+\infty$. If $u(x,y)$ satisfies

$$ {\psi} \bigl( u(x,y) \bigr) \leq a(x,y)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi (s,t)f(s,t) {\omega} \bigl( u(s,t) \bigr) \nabla t\Delta s $$

(2.1)

for $(x,y)\in \Omega$, then

$$ u(x,y)\leq {\psi} ^{-1} \biggl\{ G^{-1}G \bigl( a ( x,y ) \bigr) + \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)} \phi (s,t)f(s,t)\nabla t\Delta s \biggr\} $$

(2.2)

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where

$$ G(v)= \int _{v_{0}}^{v} \frac{\Delta s}{ {\omega} ( {\psi} ^{-1}(s) ) },\quad v\geq v_{0}>0, \qquad G(+\infty )= \int _{v_{0}}^{+\infty } \frac{\Delta s}{ {\omega} ( {\psi} ^{-1}(s) ) }=+\infty $$

(2.3)

and $( x_{1},y_{1} ) \in \Omega$ is chosen so that

$$ \biggl( G \bigl( a ( x,y ) \bigr) + \int _{x_{0}}^{ \theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t \Delta s \biggr) \in \operatorname {Dom}\bigl( G^{-1} \bigr) .$$

Proof

First we assume that $a ( x,y ) >0$. Fixing an arbitrary $(x_{0},y_{0})\in \Omega$, we define a positive and nondecreasing function $z(x,y)$ by

$$ z(x,y)=a(x_{0},y_{0})+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi (s,t)f(s,t) {\omega} \bigl( u(s,t) \bigr) \nabla t\Delta s $$

(2.4)

for $0\leq x\leq x_{0}\leq x_{1}$, $0\leq y\leq y_{0}\leq y_{1}$, then $z(x_{0},y)=z(x,y_{0})=a(x_{0},y_{0})$ and

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

(2.5)

Taking Δ-derivative for (2.4) with employing Theorem 1.5$(i)$, we have

$$\begin{aligned} z^{\Delta _{x}}(x,y) =&\theta ^{\Delta }(x) \int _{y_{0}}^{\vartheta (y)} \phi \bigl(\theta (x),t\bigr)f \bigl(\theta (x),t\bigr) {\omega} \bigl( u\bigl(\theta (x),t\bigr) \bigr) \nabla t \\ \leq &\theta ^{\Delta }(x) \int _{y_{0}}^{\vartheta (y)}\phi \bigl(\theta (x),t\bigr)f \bigl( \theta (x),t\bigr) {\omega} \bigl( {\psi} ^{-1} \bigl( z\bigl( \theta (x),t\bigr) \bigr) \bigr) \nabla t \\ \leq & {\omega} \bigl( {\psi} ^{-1} \bigl( z\bigl(\theta (x), \vartheta (y)\bigr) \bigr) \bigr) \theta ^{\Delta }(x) \int _{y_{0}}^{\vartheta (y)} \phi \bigl(\theta (x),t\bigr)f \bigl(\theta (x),t\bigr)\nabla t. \end{aligned}$$

(2.6)

Inequality (2.6) can be written in the form

$$ \frac{z^{\Delta _{x}}(x,y)}{ {\omega} ( {\psi} ^{-1} ( z(x,y) ) ) }\leq \theta ^{\Delta }(x) \int _{y_{0}}^{ \vartheta (y)}\phi \bigl(\theta (x),t\bigr)f \bigl(\theta (x),t\bigr)\nabla t. $$

(2.7)

Taking Δ-integral for inequality (2.7) leads to

$$\begin{aligned} G \bigl( z(x,y) \bigr) \leq &G \bigl( z(x_{0},y) \bigr) + \int _{x_{0}}^{ \theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi (s,t)f(s,t)\nabla t \Delta s \\ \leq &G \bigl( a(x_{0},y_{0}) \bigr) + \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi (s,t)f(s,t)\nabla t\Delta s. \end{aligned}$$

Since $(x_{0},y_{0})\in \Omega$ is chosen arbitrarily,

$$ z(x,y)\leq G^{-1} \biggl[ G \bigl( a(x,y) \bigr) + \int _{x_{0}}^{ \theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi (s,t)f(s,t)\nabla t \Delta s \biggr] . $$

(2.8)

From (2.8) and (2.5) we obtain the desired result (2.2). We carry out the above procedure with $\epsilon >0$ instead of $a(x,y)$ when $a(x,y)=0$ and subsequently let $\epsilon \rightarrow 0$. □

Now, as special cases of our results, we will give the continuous, discrete, and quantum inequalities. Namely, in the cases of time scales $\mathbb{T}=\mathbb{R}$, $\mathbb{T}=h\mathbb{Z}$, $\mathbb{T}=\mathbb{Z}$, and $\mathbb{T}=\overline{q^{\mathbb{Z}}}$.

Remark 2.2

If we take $\mathbb{T}=\mathbb{R}$, $x_{0}=0$, and $y_{0}=0$ in Lemma 2.1, then, by relation (1.1), inequality (2.1) becomes the inequality obtained in [15, Lemma 2.1].

Corollary 2.3

If we take $\mathbb{T}=h \mathbb{Z}$ in Lemma 2.1by relation (1.3), then the following inequality

$$ {\psi} \bigl( u(x,y) \bigr) \leq a(x,y)+h^{2}\sum _{s= \frac{x_{0}}{h}}^{\frac{\theta (x)}{h}-1}\sum_{t=\frac{y_{0}}{h}}^{ \frac{\vartheta (y)}{h}+1} \phi (sh,th)f(sh,th) {\omega} \bigl( u(sh,th) \bigr) $$

for $(x,y)\in \Omega$ implies

$$ u(x,y)\leq {\psi} ^{-1} \Biggl\{ G^{-1}G \bigl( a ( x,y ) \bigr) +h^{2}\sum_{s=\frac{x_{0}}{h}}^{\frac{\theta (x)}{h}-1} \sum _{t=\frac{y_{0}}{h}}^{\frac{\vartheta (y)}{h}+1}\phi (sh,th)f(sh,th) \Biggr\} $$

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where

$$ G(v)=\sum_{s=\frac{v_{0}}{h}}^{\frac{v}{h}-1} \frac{h}{ {\omega} ( {\psi} ^{-1}(sh) ) },\quad v\geq v_{0}>0, \qquad G(+\infty )= \sum_{s=\frac{v_{0}}{h}}^{+\infty } \frac{h}{ {\omega} ( {\psi} ^{-1}(sh) ) }=+ \infty $$

and $( x_{1},y_{1} ) \in \Omega$ is chosen so that

$$ \Biggl( G \bigl( a ( x,y ) \bigr) +h^{2}\sum _{s= \frac{x_{0}}{h}}^{\frac{\theta (x)}{h}-1}\sum_{t=\frac{y_{0}}{h}}^{ \frac{\vartheta (y)}{h}+1} \phi _{1}(sh,th)f(sh,th) \Biggr) \in \operatorname {Dom}\bigl( G^{-1} \bigr) .$$

Remark 2.4

In Corollary 2.3, if we take $h=1$, then the following inequality

$$ {\psi} \bigl( u(x,y) \bigr) \leq a(x,y)+\sum_{s=x_{0}}^{\theta (x)-1} \sum_{t=y_{0}}^{\vartheta (y)+1}\phi (s,t)f(s,t) {\omega} \bigl( u(s,t) \bigr) $$

for $(x,y)\in \Omega$ implies

$$ u(x,y)\leq {\psi} ^{-1} \Biggl\{ G^{-1}G \bigl( a ( x,y ) \bigr) +\sum_{s=x_{0}}^{\theta (x)-1}\sum _{t=y_{0}}^{\vartheta (y)+1} \phi (s,t)f(s,t) \Biggr\} $$

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where

$$ G(v)=\sum_{s=v_{0}}^{v-1} \frac{1}{ {\omega} ( {\psi} ^{-1}(s) ) },\quad v\geq v_{0}>0,\qquad G(+\infty )= \sum_{s=v_{0}}^{+\infty } \frac{1}{ {\omega} ( {\psi} ^{-1}(s) ) }=+ \infty $$

and $( x_{1},y_{1} ) \in \Omega$ is chosen so that

$$ \Biggl( G \bigl( a ( x,y ) \bigr) +\sum_{s=x_{0}}^{ \theta (x)-1} \sum_{t=y_{0}}^{\vartheta (y)+1}\phi _{1}(s,t)f(s,t) \Biggr) \in \operatorname {Dom}\bigl( G^{-1} \bigr) .$$

Corollary 2.5

If we take $\mathbb{T}=\overline{q^{\mathbb{Z}}}$ in Lemma 2.1by relation (1.4), then the following inequality

$$ {\psi} \bigl( u(x,y) \bigr) \leq a(x,y)+(q-1)^{2}\sum _{s=(\log _{q}{x_{0}})}^{( \log _{q}{\theta (x)})-1}\sum_{t=(\log _{q}{y_{0}})}^{(\log _{q}{ \vartheta (y)})+1} q^{(s+t)}\phi \bigl(q^{s},q^{t}\bigr)f \bigl(q^{s},q^{t}\bigr) {\omega} \bigl( u \bigl(q^{s},q^{t}\bigr) \bigr) $$

for $(x,y)\in \Omega$ implies

$$ u(x,y)\leq {\psi} ^{-1} \Biggl\{ G^{-1}G \bigl( a ( x,y ) \bigr) +(q-1)^{2}\sum_{s=(\log _{q}{x_{0}})}^{(\log _{q}{\theta (x)})-1} \sum_{t=(\log _{q}{y_{0}})}^{(\log _{q}{\vartheta (y)})+1}q^{(s+t)} \phi \bigl(q^{s},q^{t}\bigr)f\bigl(q^{s},q^{t} \bigr) \Biggr\} $$

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where

$$ G(v)=\sum_{s=(\log _{q}{v_{0}})}^{(\log _{q}{v})-1} \frac{(q-1)q^{s}}{ {\omega} ( {\psi} ^{-1}(q^{s}) ) },\quad v \geq v_{0}>0,\qquad G(+\infty )= \sum_{s=(\log _{q}{v_{0}})}^{+ \infty } \frac{(q-1)q^{s}}{ {\omega} ( {\psi} ^{-1}(q^{s}) ) }=+ \infty $$

and $( x_{1},y_{1} ) \in \Omega$ is chosen so that

$$ \Biggl( G \bigl( a ( x,y ) \bigr) +(q-1)^{2}\sum _{s=( \log _{q}{x_{0}})}^{(\log _{q}{\theta (x)})-1}\sum_{t=(\log _{q}{y_{0}})}^{( \log _{q}{\vartheta (y)})+1}q^{(s+t)} \phi _{1}\bigl(q^{s},q^{t}\bigr)f \bigl(q^{s},q^{t}\bigr) \Biggr) \in \operatorname {Dom}\bigl( G^{-1} \bigr) .$$

Theorem 2.6

Let u, a, f, θ, and ϑ be as in Lemma 2.1. Let $\phi _{1},\phi _{2}\in C (\Omega ,\mathbb{R}_{+})$. If $u(x,y)$ satisfies

$$\begin{aligned} {\psi} \bigl( u(x,y) \bigr) \leq &a(x,y)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t) {\omega} \bigl( u(s,t) \bigr) \\ &{} + \int _{x_{0}}^{s}\phi _{2}(\tau ,t) { \omega} \bigl( u( \tau ,t) \bigr) \Delta \tau \biggr] \nabla t\Delta s \end{aligned}$$

(2.9)

for $(x,y)\in \Omega$, then

$$ u(x,y)\leq {\psi} ^{-1} \biggl\{ G^{-1} \biggl( p(x,y)+ \int _{x_{0}}^{ \theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t \Delta s \biggr) \biggr\} $$

(2.10)

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where G is defined by (2.3) and

$$ p(x,y)=G \bigl( a(x,y) \bigr) + \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi _{1}(s,t) \biggl( \int _{x_{0}}^{s}\phi _{2}(\tau ,t) \Delta \tau \biggr) \nabla t\Delta s $$

(2.11)

and $( x_{1},y_{1} ) \in \Omega$ is chosen so that

$$ \biggl( p(x,y)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)} \phi _{1}(s,t)f(s,t) \nabla t\Delta s \biggr) \in \operatorname {Dom}\bigl( G^{-1} \bigr) .$$

Proof

By the same steps of the proof of Lemma 2.1, we can obtain (2.10) with suitable changes. □

Remark 2.7

If we take $\phi _{2}(x,y)=0$, then Theorem 2.6 reduces to Lemma 2.1.

Corollary 2.8

Let the functions u, f, $\phi _{1}$, $\phi _{2}$, a, θ, and ϑ be as in Theorem 2.6. Further, suppose that $q>p>0$ are constants. If $u(x,y)$ satisfies

$$\begin{aligned} u^{q}(x,y) \leq &a(x,y)+\frac{q}{q-p} \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t)u^{p}(s,t) \\ &{} + \int _{x_{0}}^{s}\phi _{2}(\tau ,t)u^{p}( \tau ,t)\Delta \tau \biggr] \nabla t\Delta s \end{aligned}$$

(2.12)

for $(x,y)\in \Omega$, then

$$ u(x,y)\leq \biggl\{ p(x,y)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t\Delta s \biggr\} ^{ \frac{1}{q-p}}, $$

(2.13)

where

$$ p(x,y)= \bigl( a(x,y) \bigr) ^{\frac{q-p}{q}}+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t) \biggl( \int _{x_{0}}^{s} \phi _{2}(\tau ,t) \Delta \tau \biggr) \nabla t\Delta s. $$

Proof

In Theorem 2.6, by letting ${\psi} (u)=u^{q}$, ${\omega} (u)=u^{p}$, we have

$$ G(v)= \int _{v_{0}}^{v} \frac{\Delta s}{ {\omega} ( {\psi} ^{-1}(s) ) }= \int _{v_{0}}^{v}\frac{\Delta s}{s^{\frac{p}{q}}}\geq \frac{q}{q-p} \bigl( v^{\frac{q-p}{q}}-v_{0}^{\frac{q-p}{q}} \bigr) ,\quad v\geq v_{0}>0 $$

and

$$ G^{-1}(v)\geq \biggl\{ v_{0}^{\frac{q-p}{q}}+ \frac{q-p}{q}v \biggr\} ^{ \frac{1}{q-p}}. $$

We obtain inequality (2.13). □

Theorem 2.9

Under the hypotheses of Theorem 2.6, further, let ψ, ω, $\eta \in C(\mathbb{R}_{+},\mathbb{R}_{+})$ be nondecreasing functions with $\{ \psi ,\omega ,\eta \} (u)>0$ for $u>0$, and $\lim_{u\rightarrow +\infty } \psi (u)=+\infty$. If $u(x,y)$ satisfies

$$\begin{aligned} {\psi} \bigl( u(x,y) \bigr) \leq &a(x,y)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t) {\omega} \bigl( u(s,t) \bigr) \eta \bigl( u(s,t) \bigr) \\ &{} + \int _{x_{0}}^{s}\phi _{2}(\tau ,t) { \omega} \bigl( u( \tau ,t) \bigr) \Delta \tau \biggr] \nabla t\Delta s \end{aligned}$$

(2.14)

for $(x,y)\in \Omega$, then

$$ u(x,y)\leq {\psi} ^{-1} \biggl\{ G^{-1} \biggl( F^{-1} \biggl[ F \bigl( p(x,y) \bigr) + \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)} \phi _{1}(s,t)f(s,t) \nabla t\Delta s \biggr] \biggr) \biggr\} $$

(2.15)

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where G and p are as in (2.3), (2.11) respectively and

$$ F(v)= \int _{v_{0}}^{v} \frac{\Delta s}{\eta ( {\psi} ^{-1} ( G^{-1}(s) ) ) },\quad v\geq v_{0}>0,\qquad F(+\infty )=+\infty , $$

(2.16)

and $( x_{1},y_{1} ) \in \Omega$ is chosen so that

$$ \biggl[ F \bigl( p(x,y) \bigr) + \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t\Delta s \biggr] \in \operatorname {Dom}\bigl( F^{-1} \bigr) .$$

Proof

Assume that $a(x,y)>0$. Fixing arbitrary $(x_{0},y_{0})\in \Omega$, we define a positive and nondecreasing function $z(x,y)$ by

$$\begin{aligned} z(x,y) =&a(x_{0},y_{0})+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t) {\omega} \bigl( u(s,t) \bigr) \eta \bigl( u(s,t) \bigr) \end{aligned}$$

(2.17)

$$\begin{aligned} & {}+ \int _{x_{0}}^{s}\phi _{2}(\tau ,t) { \omega} \bigl( u( \tau ,t) \bigr) \Delta \tau \biggr] \nabla t\Delta s \end{aligned}$$

(2.18)

for $0\leq x\leq x_{0}\leq x_{1}$, $0\leq y\leq y_{0}\leq y_{1}$, then $z(x_{0},y)=z(x,y_{0})=a(x_{0},y_{0})$ and

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

(2.19)

Taking Δ-derivative for (2.17) with employing Theorem 1.5(i) gives

$$\begin{aligned} z^{\Delta _{x}}(x,y) =&\theta ^{\Delta }(x) \int _{y_{0}}^{\vartheta (y)} \phi _{1}\bigl(\theta (x),t\bigr) \biggl[ f\bigl(\theta (x),t\bigr) {\omega} \bigl( u\bigl( \theta (x),t\bigr) \bigr) \eta \bigl( u\bigl(\theta (x),t\bigr) \bigr) \end{aligned}$$

(2.20)

$$\begin{aligned} &{} + \int _{x_{0}}^{\theta (x)}\phi _{2}( \tau ,t) { \omega} \bigl( u(\tau ,t) \bigr) \Delta \tau \biggr] \nabla t \end{aligned}$$

(2.21)

$$\begin{aligned} \leq &\theta ^{\Delta }(x) \int _{y_{0}}^{\vartheta (y)}\phi _{1}\bigl( \theta (x),t\bigr) \biggl[ f\bigl(\theta (x),t\bigr) {\omega} \bigl( {\psi} ^{-1} \bigl( z\bigl(\theta (x),t\bigr) \bigr) \bigr) \end{aligned}$$

(2.22)

$$\begin{aligned} & {}\times \eta\bigl( { \psi} ^{-1} \bigl( z\bigl(\theta (x),t\bigr) \bigr) \bigr)+ \int _{x_{0}}^{\theta (x)}\phi _{2}(\tau ,t) { \omega} \bigl( {\psi} ^{-1} \bigl( z(\tau ,t) \bigr) \bigr) \Delta \tau \biggr] \nabla t \end{aligned}$$

(2.23)

$$\begin{aligned} \leq &\theta ^{\Delta }(x). {\omega} \bigl( {\psi} ^{-1} \bigl( z\bigl( \theta (x),\vartheta (y)\bigr) \bigr) \bigr) \end{aligned}$$

(2.24)

$$\begin{aligned} &{}\times \int _{y_{0}}^{\vartheta (y)}\phi _{1}\bigl(\theta (x),t\bigr) \biggl[ f\bigl( \theta (x),t\bigr)\eta \bigl( {\psi} ^{-1} \bigl( z\bigl(\theta (x),t\bigr) \bigr) \bigr) \end{aligned}$$

(2.25)

$$\begin{aligned} &{} + \int _{x_{0}}^{\theta (x)}\phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t. \end{aligned}$$

(2.26)

From (2.20) we have

$$\begin{aligned} \frac{z^{\Delta _{x}}(x,y)}{ {\omega} ( {\psi} ^{-1} ( z(x,y) ) ) } \leq &\theta ^{\Delta }(x) \int _{y_{0}}^{ \vartheta (y)}\phi _{1}\bigl(\theta (x),t\bigr) \biggl[ f\bigl(\theta (x),t\bigr)\eta \bigl( {\psi} ^{-1} \bigl( z\bigl(\theta (x),t\bigr) \bigr) \bigr) \end{aligned}$$

(2.27)

$$\begin{aligned} &{} + \int _{x_{0}}^{\theta (x)}\phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t. \end{aligned}$$

(2.28)

Taking Δ-integral for (2.27) gives

$$\begin{aligned} G \bigl( z(x,y) \bigr) \leq &G \bigl( z(x_{0},y) \bigr) + \int _{x_{0}}^{ \theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t) \eta \bigl( {\psi} ^{-1} \bigl( z(s,t) \bigr) \bigr) \\ & {}+ \int _{x_{0}}^{s}\phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t\Delta s \\ \leq &G \bigl( a(x_{0},y_{0}) \bigr) + \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t)\eta \bigl( { \psi} ^{-1} \bigl( z(s,t) \bigr) \bigr) \\ & {}+ \int _{x_{0}}^{s}\phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t\Delta s. \end{aligned}$$

Since $(x_{0},y_{0})\in \Omega$ is chosen arbitrarily, the last inequality can be rewritten as

$$ G \bigl( z(x,y) \bigr) \leq p(x,y)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi _{1}(s,t)f(s,t) \eta \bigl( {\psi} ^{-1} \bigl( z(s,t) \bigr) \bigr) \nabla t\Delta s. $$

(2.29)

Since $p(x,y)$ is a nondecreasing function, an application of Lemma 2.1 to (2.29) gives us

$$ z(x,y)\leq G^{-1} \biggl( F^{-1} \biggl[ F \bigl( p(x,y) \bigr) + \int _{x_{0}}^{ \theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t \Delta s \biggr] \biggr) . $$

(2.30)

From (2.19) and (2.30) we obtain the desired inequality (2.15).

Now, we take the case $a(x,y)=0$ for some $(x,y)\in \Omega$. Let $a_{\epsilon }(x,y)=a(x,y)+\epsilon$ for all $(x,y)\in \Omega$, where $\epsilon >0$ is arbitrary, then $a_{\epsilon }(x,y)>0$ and $a_{\epsilon }(x,y)\in C(\Omega ,\mathbb{R}_{+})$ are nondecreasing with respect to $(x,y)\in \Omega$. We carry out the above procedure with $a_{\epsilon }(x,y)>0$ instead of $a(x,y)$, and we get

$$ u(x,y)\leq {\psi} ^{-1} \biggl\{ G^{-1} \biggl( F^{-1} \biggl[ F \bigl( p_{ \epsilon }(x,y) \bigr) + \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t\Delta s \biggr] \biggr) \biggr\} , $$

where

$$ p_{\epsilon }(x,y)=G \bigl( a_{\epsilon }(x,y) \bigr) + \int _{x_{0}}^{ \theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t) \biggl( \int _{x_{0}}^{s} \phi _{2}(\tau ,t) \Delta \tau \biggr) \nabla t\Delta s. $$

Letting $\epsilon \rightarrow 0^{+}$, we obtain (2.15). The proof is complete. □

Now, as special cases of our results, we will give the continuous, discrete, and quantum inequalities. Namely, in the cases of time scales $\mathbb{T}=\mathbb{R}$, $\mathbb{T}=h\mathbb{Z}$, $\mathbb{T}=\mathbb{Z}$, and $\mathbb{T}=\overline{q^{\mathbb{Z}}}$.

Remark 2.10

If we take $\mathbb{T}=\mathbb{R}$, $x_{0}=0$, and $y_{0}=0$ in Theorem 2.9, then, by relation (1.1), inequality (2.14) becomes the inequality obtained in [15, Theorem 2.2(A_2)].

Corollary 2.11

If we take $\mathbb{T}=h \mathbb{Z}$ in Theorem 2.9by relation (1.3), then the following inequality

$$\begin{aligned} {\psi} \bigl( u(x,y) \bigr) \leq &a(x,y)+h^{2}\sum _{s= \frac{x_{0}}{h}}^{\frac{\theta (x)}{h}-1}\sum_{t=\frac{y_{0}}{h}}^{ \frac{\vartheta (y)}{h}+1} \phi _{1}(sh,th) \Biggl[ f(sh,th) {\omega} \bigl( u(sh,th) \bigr) \eta \bigl( u(sh,th) \bigr) \\ & {}+h\sum_{t=\frac{x_{0}}{h}}^{\frac{s}{h}-1}\phi _{2}(\tau ,th) {\omega} \bigl( u(\tau ,th) \bigr) \Biggr] \end{aligned}$$

for $(x,y)\in \Omega$ implies

$$ u(x,y)\leq {\psi} ^{-1} \Biggl\{ G^{-1} \Biggl( F^{-1} \Biggl[ F \bigl( p(x,y) \bigr) +h^{2}\sum _{s=\frac{x_{0}}{h}}^{\frac{\theta (x)}{h}-1} \sum_{t=\frac{y_{0}}{h}}^{\frac{\vartheta (y)}{h}+1} \phi _{1}(sh,th)f(sh,th) \Biggr] \Biggr) \Biggr\} $$

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where G and p are as in (2.3) and (2.11), respectively, and

$$ F(v)=\sum_{s=\frac{v_{0}}{h}}^{\frac{v}{h}} \frac{h}{\eta ( {\psi} ^{-1} ( G^{-1}(sh) ) ) },\quad v\geq v_{0}>0,\qquad F(+\infty )=+\infty $$

and $( x_{1},y_{1} ) \in \Omega$ is chosen so that

$$ \Biggl[ F \bigl( p(x,y) \bigr) +h^{2}\sum _{s=\frac{x_{0}}{h}}^{ \frac{\theta (x)}{h}-1}\sum_{t=\frac{y_{0}}{h}}^{ \frac{\vartheta (y)}{h}+1} \phi _{1}(sh,th)f(sh,th) \Biggr] \in \operatorname {Dom}\bigl( F^{-1} \bigr) .$$

Remark 2.12

In Corollary 2.11, if we take $h=1$, then the following inequality

$$\begin{aligned} {\psi} \bigl( u(x,y) \bigr) \leq &a(x,y)+\sum_{s=x_{0}}^{\theta (x)-1} \sum_{t=y_{0}}^{\vartheta (y)+1}\phi _{1}(s,t) \Biggl[ f(s,t) {\omega} \bigl( u(s,t) \bigr) \eta \bigl( u(s,t) \bigr) \\ & {}+\sum_{t=x_{0}}^{s-1}\phi _{2}(\tau ,t) {\omega} \bigl( u( \tau ,t) \bigr) \Biggr] \end{aligned}$$

for $(x,y)\in \Omega$ implies

$$ u(x,y)\leq {\psi} ^{-1} \Biggl\{ G^{-1} \Biggl( F^{-1} \Biggl[ F \bigl( p(x,y) \bigr) +\sum _{s=x_{0}}^{\theta (x)-1}\sum_{t=y_{0}}^{\vartheta (y)+1} \phi _{1}(s,t)f(s,t) s \Biggr] \Biggr) \Biggr\} $$

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where G and p are as in (2.3), and

$$ F(v)=\sum_{s=v_{0}}^{v-1} \frac{1}{\eta ( {\psi} ^{-1} ( G^{-1}(s) ) ) },\quad v\geq v_{0}>0,\qquad F(+\infty )=+\infty , $$

and $( x_{1},y_{1} ) \in \Omega$ is chosen so that

$$ \Biggl[ F \bigl( p(x,y) \bigr) +\sum_{s=x_{0}}^{\theta (x)-1} \sum_{t=y_{0}}^{ \vartheta (y)+1}\phi _{1}(s,t)f(s,t) \Biggr] \in \operatorname {Dom}\bigl( F^{-1} \bigr) .$$

Corollary 2.13

If we take $\mathbb{T}=\overline{q^{\mathbb{Z}}}$ in Theorem 2.9by relation (1.4), then the following inequality

$$\begin{aligned} {\psi} \bigl( u(x,y) \bigr) \leq &a(x,y)+(q-1)^{2}\sum _{s=(\log _{q}{x_{0}})}^{( \log _{q}{\theta (x)})-1}\sum_{t=(\log _{q}{y_{0}})}^{(\log _{q}{ \vartheta (y)})+1}q^{(s+t)} \phi _{1}\bigl(q^{s},q^{t}\bigr) \\ & {}\times\Biggl[ f \bigl(q^{s},q^{t}\bigr) {\omega} \bigl( u \bigl(q^{s},q^{t}\bigr) \bigr) \eta \bigl( u \bigl(q^{s},q^{t}\bigr) \bigr) \\ & {}+(q-1)\sum_{t=(\log _{q}{x_{0}})}^{(\log _{q}{s})-1}q^{t} \phi _{2}\bigl(\tau ,q^{t}\bigr) {\omega} \bigl( u(\tau ,t) \bigr) \Biggr] \end{aligned}$$

for $(x,y)\in \Omega$, then

$$\begin{aligned} u(x,y) \leq& {\psi} ^{-1} \Biggl\{ G^{-1} \Biggl( F^{-1} \Biggl[ F \bigl( p(x,y) \bigr)\\ &{} +(q-1)^{2}\sum _{s=(\log _{q}{x_{0}})}^{(\log _{q}{\theta (x)})-1} \sum _{t=(\log _{q}{y_{0}})}^{(\log _{q}{\vartheta (y)})+1}q^{(s+t)} \phi _{1}\bigl(q^{s},q^{t}\bigr)f \bigl(q^{s},q^{t}\bigr) s \Biggr] \Biggr) \Biggr\} \end{aligned}$$

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where G and p are as in (2.3), and

$$ F(v)=\sum_{s=(\log _{q}{v_{0}})}^{(\log _{q}{v})-1} \frac{(q-1)q^{s}}{\eta ( {\psi} ^{-1} ( G^{-1}(q^{s}) ) ) },\quad v\geq v_{0}>0,\qquad F(+\infty )=+\infty , $$

and $( x_{1},y_{1} ) \in \Omega$ is chosen so that

$$ \Biggl[ F \bigl( p(x,y) \bigr) +(q-1)^{2}\sum _{s=(\log _{q}{x_{0}})}^{( \log _{q}{\theta (x)})-1}\sum_{t=(\log _{q}{y_{0}})}^{(\log _{q}{ \vartheta (y)})+1}q^{(s+t)} \phi _{1}\bigl(q^{s},q^{t}\bigr)f \bigl(q^{s},q^{t}\bigr) \Biggr] \in \operatorname {Dom}\bigl( F^{-1} \bigr) .$$

Corollary 2.14

Let the functions u, a, f, $\phi _{1}$, $\phi _{2}$, θ, and ϑ be as in Theorem 2.6. Further, suppose that q, p, and r are constants with $p>0$, $r>0$, and $q>p+r$. If $u(x,y)$ satisfies

$$\begin{aligned} u^{q}(x,y) \leq &a(x,y)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t)u^{p}(s,t)u^{r}(s,t) \\ & {}+ \int _{x_{0}}^{s}\phi _{2}(\tau ,t)u^{p}(\tau ,t)\Delta \tau \biggr] \nabla t\Delta s \end{aligned}$$

(2.31)

for $(x,y)\in \Omega$, then

$$ u(x,y)\leq \biggl\{ \bigl[ p(x,y) \bigr] ^{\frac{q-p-r}{q-p}}+ \frac{q-p-r}{q}\int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t\Delta s \biggr\} ^{\frac{1}{q-p-r}}, $$

(2.32)

where

$$ p(x,y)= \bigl( a(x,y) \bigr) ^{\frac{q-p}{q}}+\frac{q-p}{q} \int _{x_{0}}^{ \theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t) \biggl( \int _{x_{0}}^{s} \phi _{2}(\tau ,t) \Delta \tau \biggr) \nabla t\Delta s. $$

Proof

An application of Theorem 2.9 with ${\psi} ( u ) =u^{q}$, ${\omega} ( u ) =u^{p}$, and $\eta ( u ) =u^{r}$ yields the desired inequality (2.32). □

Theorem 2.15

Under the hypotheses of Theorem 2.9. If $u(x,y)$ satisfies

$$\begin{aligned} {\psi} \bigl( u(x,y) \bigr) \leq &a(x,y)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t) {\omega} \bigl( u(s,t) \bigr) \eta \bigl( u(s,t) \bigr) \\ &{} + \int _{x_{0}}^{s}\phi _{2}(\tau ,t) { \omega} \bigl( u(\tau ,t) \bigr) \eta \bigl( u(\tau ,t) \bigr) \Delta \tau \biggr] \nabla t\Delta s \end{aligned}$$

(2.33)

for $(x,y)\in \Omega$, then

$$ u(x,y)\leq {\psi} ^{-1} \biggl\{ G^{-1} \biggl( F^{-1} \biggl[ p_{0}(x,y)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t \Delta s \biggr] \biggr) \biggr\} $$

(2.34)

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where

$$ p_{0}(x,y)=F\bigl(G \bigl( a(x,y) \bigr) \bigr)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t) \biggl( \int _{x_{0}}^{s} \phi _{2}(\tau ,t) \Delta \tau \biggr) \nabla t\Delta s, $$

and $( x_{1},y_{1} ) \in \Omega$ is chosen so that

$$ \biggl[ p_{0}(x,y)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t\Delta s \biggr] \in \operatorname {Dom}\bigl( F^{-1} \bigr) .$$

Proof

Assume that $a(x,y)>0$. Fixing arbitrary $(x_{0},y_{0})\in \Omega$, we define a positive and nondecreasing function $z(x,y)$ by

$$\begin{aligned} z(x,y) =&a(x_{0},y_{0})+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t) {\omega} \bigl( u(s,t) \bigr) \eta \bigl( u(s,t) \bigr) \\ & {}+ \int _{x_{0}}^{s}\phi _{2}(\tau ,t) { \omega} \bigl( u( \tau ,t) \bigr) \eta \bigl( u(\tau ,t) \bigr) \Delta \tau \biggr] \nabla t\Delta s \end{aligned}$$

for $0\leq x\leq x_{0}\leq x_{1}$, $0\leq y\leq y_{0}\leq y_{1}$, then $z(x_{0},y)=z(x,y_{0})=a(x_{0},y_{0})$, and

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

(2.35)

By the same steps as the proof of Theorem 2.9, we obtain

$$\begin{aligned} z(x,y) \leq &G^{-1} \biggl\{ G \bigl( a(x_{0},y_{0}) \bigr) + \int _{x_{0}}^{ \theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t) \eta \bigl( {\psi} ^{-1} \bigl( z(s,t) \bigr) \bigr) \\ & {}+ \int _{x_{0}}^{s}\phi _{2}(\tau ,t)\eta \bigl( { \psi} ^{-1} \bigl( z(\tau ,t) \bigr) \bigr) \Delta \tau \biggr] \nabla t\Delta s \biggr\} . \end{aligned}$$

We define a nonnegative and nondecreasing function $v(x,y)$ by

$$\begin{aligned} v(x,y) =&G \bigl( a(x_{0},y_{0}) \bigr) + \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t) \biggl[ \bigl[ f(s,t) \eta \bigl( {\psi} ^{-1} \bigl( z(s,t) \bigr) \bigr) \bigr] \\ & {}+ \int _{x_{0}}^{s}\phi _{2}(\tau ,t)\eta \bigl( {\psi} ^{-1} \bigl( z(\tau ,t) \bigr) \bigr) \Delta \tau \biggr] \nabla t \Delta s, \end{aligned}$$

then $v(x_{0},y)=v(x,y_{0})=G ( a(x_{0},y_{0}) )$,

$$ z(x,y)\leq G^{-1} \bigl[ v(x,y) \bigr], $$

(2.36)

and then

$$\begin{aligned} v^{\Delta x}(x,y) \leq &\theta ^{\Delta }(x) \int _{y_{0}}^{ \vartheta (y)}\phi _{1}\bigl(\theta (x),t\bigr) \biggl[ f\bigl(\theta (x),t\bigr)\eta \bigl( {\psi} ^{-1} \bigl( G^{-1} \bigl( v\bigl(\theta (x),y\bigr) \bigr) \bigr) \bigr) \\ &{} + \int _{x_{0}}^{\theta (x)}\phi _{2}(\tau ,t) \eta \bigl( {\psi} ^{-1} \bigl( G^{-1} \bigl( v(\tau ,y) \bigr) \bigr) \bigr) \Delta \tau \biggr] \nabla t \\ \leq &\theta ^{\Delta }(x)\eta \bigl( {\psi} ^{-1} \bigl( G^{-1} \bigl( v\bigl(\theta (x),\vartheta (y)\bigr) \bigr) \bigr) \bigr) \int _{y_{0}}^{ \vartheta (y)}\phi _{1}\bigl(\theta (x),t\bigr) \biggl[ f\bigl(\theta (x),t\bigr) \\ & {}+ \int _{x_{0}}^{\theta (x)}\phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t \end{aligned}$$

or

$$\begin{aligned} \frac{v^{\Delta x}(x,y)}{\eta ( {\psi} ^{-1} ( G^{-1} ( v(x,y) ) ) ) } \leq &\theta ^{ \Delta }(x) \int _{y_{0}}^{\vartheta (y)}\phi _{1}\bigl(\theta (x),t\bigr) \biggl[ f\bigl(\theta (x),t\bigr) + \int _{x_{0}}^{\theta (x)}\phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t. \end{aligned}$$

Taking Δ-integral for the above inequality gives

$$ F\bigl(v(x,y)\bigr)\leq F\bigl(v(x_{0},y)\bigr)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t)+ \int _{x_{0}}^{s}\phi _{2}( \tau ,t) \Delta \tau \biggr] \nabla t\Delta s $$

or

$$\begin{aligned} v(x,y) \leq &F^{-1} \biggl\{ F\bigl(G \bigl( a(x_{0},y_{0}) \bigr) \bigr)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t) \\ & {}+ \int _{x_{0}}^{s}\phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t\Delta s \biggr\} . \end{aligned}$$

(2.37)

From (2.35)–(2.37), and since $(x_{0},y_{0})\in \Omega$ is chosen arbitrarily, we obtain the desired inequality (2.34). If $a(x,y)=0$, we carry out the above procedure with $\epsilon >0$ instead of $a(x,y)$ and subsequently let $\epsilon \rightarrow 0$. The proof is complete. □

Now, as special cases of our results, we will give the continuous, discrete, and quantum inequalities. Namely, in the cases of time scales $\mathbb{T}=\mathbb{R}$, $\mathbb{T}=h\mathbb{Z}$, $\mathbb{T}=\mathbb{Z}$, and $\mathbb{T}=\overline{q^{\mathbb{Z}}}$.

Remark 2.16

If we take $\mathbb{T}=\mathbb{R}$ and $x_{0}=0$ and $y_{0}=0$ in Theorem 2.15, then, by relation (1.1), inequality (2.33) becomes the inequality obtained in [15, Theorem 2.2(A₃)].

Corollary 2.17

If we take $\mathbb{T}=h \mathbb{Z}$ in Theorem 2.15by relation (1.3), then the following inequality

$$\begin{aligned} {\psi} \bigl( u(x,y) \bigr) \leq &a(x,y)+h^{2}\sum _{s= \frac{x_{0}}{h}}^{\frac{\theta (x)}{h}-1}\sum_{t=\frac{y_{0}}{h}}^{ \frac{\vartheta (y)}{h}+1} \phi _{1}(sh,th) \Biggl[ f(sh,th) {\omega} \bigl( u(sh,th) \bigr) \eta \bigl( u(sh,th) \bigr) \\ &{} +h\sum_{t=x_{0}}^{\frac{s}{h}-1}\phi _{2}(\tau ,th) { \omega} \bigl( u(\tau ,th) \bigr) \eta \bigl( u( \tau ,th) \bigr) \Biggr] \end{aligned}$$

for $(x,y)\in \Omega$ implies

$$ u(x,y)\leq {\psi} ^{-1} \Biggl\{ G^{-1} \Biggl( F^{-1} \Biggl[ p_{0}(x,y)+h^{2} \sum _{s=\frac{x_{0}}{h}}^{\frac{\theta (x)}{h}-1}\sum _{t= \frac{y_{0}}{h}}^{\frac{\vartheta (y)}{h}+1}\phi _{1}(sh,th)f(sh,th) \Biggr] \Biggr) \Biggr\} $$

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where

$$ p_{0}(x,y)=F\bigl(G \bigl( a(x,y) \bigr) \bigr)+h^{2} \sum_{s=\frac{x_{0}}{h}}^{ \frac{\theta (x)}{h}-1}\sum _{t=\frac{y_{0}}{h}}^{ \frac{\vartheta (y)}{h}+1}\phi _{1}(sh,th) \Biggl( \sum_{t= \frac{x_{0}}{h}}^{\frac{s}{h}}\phi _{2}( \tau ,th) \Biggr), $$

and $( x_{1},y_{1} ) \in \Omega$ is chosen so that

$$ \Biggl[ p_{0}(x,y)+h^{2}\sum _{s=\frac{x_{0}}{h}}^{ \frac{\theta (x)}{h}-1}\sum_{t=\frac{y_{0}}{h}}^{ \frac{\vartheta (y)}{h}+1} \phi _{1}(sh,th)f(sh,th) \Biggr] \in \operatorname {Dom}\bigl( F^{-1} \bigr) .$$

Remark 2.18

In Corollary 2.17, if we take $h=1$, then the following inequality

$$\begin{aligned} {\psi} \bigl( u(x,y) \bigr) \leq &a(x,y)+\sum_{s=x_{0}}^{\theta (x)-1} \sum_{t=y_{0}}^{\vartheta (y)+1}\phi _{1}(s,t) \Biggl[ f(s,t) {\omega} \bigl( u(s,t) \bigr) \eta \bigl( u(s,t) \bigr) \\ &{} +\sum_{s=x_{0}}^{s-1}\phi _{2}(\tau ,t) {\omega} \bigl( u(\tau ,t) \bigr) \eta \bigl( u(\tau ,t) \bigr) \Biggr] \end{aligned}$$

for $(x,y)\in \Omega$ implies

$$ u(x,y)\leq {\psi} ^{-1} \Biggl\{ G^{-1} \Biggl( F^{-1} \Biggl[ p_{0}(x,y)+ \sum _{s=x_{0}}^{\theta (x)-1}\sum_{t=y_{0}}^{\vartheta (y)+1} \phi _{1}(s,t)f(s,t) \Biggr] \Biggr) \Biggr\} $$

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where

$$ p_{0}(x,y)=F\bigl(G \bigl( a(x,y) \bigr) \bigr)+\sum _{s=x_{0}}^{\theta (x)-1} \sum_{t=y_{0}}^{\vartheta (y)+1} \phi _{1}(s,t) \Biggl( \sum_{t=x_{0}}^{s-1} \phi _{2}(\tau ,t) \Biggr), $$

and $( x_{1},y_{1} ) \in \Omega$ is chosen so that

$$ \Biggl[ p_{0}(x,y)+\sum_{s=x_{0}}^{\theta (x)-1} \sum_{t=y_{0}}^{ \vartheta (y)+1}\phi _{1}(s,t)f(s,t) \Biggr] \in \operatorname {Dom}\bigl( F^{-1} \bigr) .$$

Corollary 2.19

If we take $\mathbb{T}=\overline{q^{\mathbb{Z}}}$ in Theorem 2.15by relation (1.4), then the following inequality

$$\begin{aligned} {\psi} \bigl( u(x,y) \bigr) \leq &a(x,y)+(q-1)^{2}\sum _{s=(\log _{q}{x_{0}})}^{( \log _{q}{\theta (x)})-1}\sum_{t=(\log _{q}{y_{0}})}^{(\log _{q}{ \vartheta (y)})+1} q^{(s+t)} \\ &{}\times \phi _{1}\bigl(q^{s},q^{t} \bigr) \Biggl[ f\bigl(q^{s},q^{t}\bigr){\omega} \bigl( u \bigl(q^{s},q^{t}\bigr) \bigr) \eta \bigl( u \bigl(q^{s},q^{t}\bigr) \bigr) \\ &{} +(q-1)\sum_{s=(\log _{q}{x_{0}})}^{(\log _{q}{s})-1}q^{t} \phi _{2}\bigl(\tau ,q^{t}\bigr) {\omega} \bigl( u \bigl(\tau ,q^{t}\bigr) \bigr) \eta \bigl( u\bigl(\tau ,q^{t}\bigr) \bigr) \Biggr] \end{aligned}$$

for $(x,y)\in \Omega$ implies

$$\begin{aligned} u(x,y) \leq& {\psi} ^{-1} \Biggl\{ G^{-1} \Biggl( F^{-1} \Biggl[ p_{0}(x,y)\\ &{}+(q-1)^{2} \sum _{s=(\log _{q}{x_{0}})}^{(\log _{q}{\theta (x)})-1}\sum _{t=( \log _{q}{y_{0}})}^{(\log _{q}{\vartheta (y)})+1} q^{(s+t)}\phi _{1}\bigl(q^{s},q^{t}\bigr)f \bigl(q^{s},q^{t}\bigr) \Biggr] \Biggr) \Biggr\} \end{aligned}$$

for $0\leq x\leq x_{1}$, $0\leq y\leq y_{1}$, where

$$ p_{0}(x,y)=F\bigl(G \bigl( a(x,y) \bigr) \bigr)+(q-1)^{2} \sum_{s=(\log _{q}{x_{0}})}^{( \log _{q}{\theta (x)})-1}\sum _{t=(\log _{q}{y_{0}})}^{(\log _{q}{ \vartheta (y)})+1} q^{(s+t)}\phi _{1}\bigl(q^{s},q^{t}\bigr) \Biggl( \sum _{t=( \log _{q}{x_{0}})}^{(\log _{q}{s})-1}\phi _{2}\bigl( \tau ,q^{t}\bigr) \Biggr) $$

and $( x_{1},y_{1} ) \in \Omega$ is chosen so that

$$ \Biggl[ p_{0}(x,y)+(q-1)^{2}\sum _{s=(\log _{q}{x_{0}})}^{(\log _{q}{ \theta (x)})-1}\sum_{t=(\log _{q}{y_{0}})}^{(\log _{q}{\vartheta (y)})+1} q^{(s+t)}\phi _{1}\bigl(q^{s},q^{t} \bigr)f\bigl(q^{s},q^{t}\bigr) \Biggr] \in \operatorname {Dom}\bigl( F^{-1} \bigr) .$$

Corollary 2.20

Under the hypotheses of Corollary 2.14. If $u(x,y)$ satisfies

$$\begin{aligned} u^{q}(x,y) \leq &a(x,y)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t)u^{p}(s,t)u^{r}(s,t) \\ & {}+ \int _{x_{0}}^{s}\phi _{2}(\tau ,t)u^{p}(\tau ,t)u^{r}( \tau ,t)\Delta \tau \biggr] \nabla t\Delta s \end{aligned}$$

(2.38)

for $(x,y)\in \Omega$, then

$$ u(x,y)\leq \biggl\{ p_{0}(x,y)+\frac{q-p-r}{q} \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t\Delta s \biggr\} ^{\frac{1}{q-p-r}}, $$

(2.39)

where

$$ p_{0}(x,y)= \bigl( a ( x,y ) \bigr) ^{\frac{q-p-r}{q}}+ \frac{q-p-r}{q} \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t) \biggl( \int _{x_{0}}^{s}\phi _{2}(\tau ,t) \Delta \tau \biggr) \nabla t\Delta s. $$

Proof

An application of Theorem 2.15 with ${\psi} ( u ) =u^{q}$, ${\omega} ( u ) =u^{p}$, and $\eta ( u ) =u^{r}$ yields the desired inequality (2.39). □

Theorem 2.21

Under the hypotheses of Theorem 2.9. If $u(x,y)$ satisfies

$$\begin{aligned} {\psi} \bigl( u(x,y) \bigr) \leq &a(x,y)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t)\eta \bigl( u ( s,t ) \bigr) \\ & {}\times\biggl[ f(s,t) {\omega} \bigl( u(s,t) \bigr) + \int _{x_{0}}^{s} \phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t\Delta s \end{aligned}$$

(2.40)

for $(x,y)\in \Omega$, then

$$ u ( x,y ) \leq {\psi} ^{-1} \biggl\{ G_{1}^{-1} \biggl( F_{1}^{-1} \biggl[ F_{1} \bigl( p_{1} ( x,y ) \bigr) + \int _{x_{0}}^{ \theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t \Delta s \biggr] \biggr) \biggr\} $$

(2.41)

for $0\leq x\leq x_{2}$, $0\leq y\leq y_{2}$, where

$$\begin{aligned}& G_{1}(v)= \int _{v_{0}}^{v} \frac{\Delta s}{\eta ( {\psi} ^{-1}(s) ) },\quad v\geq v_{0}>0,\qquad G_{1}(+ \infty )= \int _{v_{0}}^{+\infty } \frac{\Delta s}{\eta ( {\psi} ^{-1}(s) ) }=+ \infty \\& F_{1}(v)= \int _{v_{0}}^{v} \frac{\Delta s}{ {\omega} [ {\psi} ^{-1} ( G_{1}^{-1} ( s ) ) ] },\quad v\geq v_{0}>0,\qquad F_{1}(+ \infty )=+\infty \\& p_{1} ( x,y ) =G_{1}\bigl(a(x,y)\bigr)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t) \biggl( \int _{x_{0}}^{s} \phi _{2}(\tau ,t) \Delta \tau \biggr) \nabla t\Delta s, \end{aligned}$$

and $( x_{2},y_{2} ) \in \Omega$ is chosen so that

$$ \biggl[ F_{1} \bigl( p_{1} ( x,y ) \bigr) + \int _{x_{0}}^{ \theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t \Delta s \biggr] \in \operatorname {Dom}\bigl( F_{1}^{-1} \bigr). $$

Proof

Suppose that $a(x,y)>0$. Fixing an arbitrary $(x_{0},y_{0})\in \Omega$, we define a positive and nondecreasing function $z(x,y)$ by

$$\begin{aligned} z ( x,y ) =&a(x_{0},y_{0})+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t)\eta \bigl( u ( s,t ) \bigr) \biggl[ f(s,t) {\omega} \bigl( u(s,t) \bigr) \\ & {}+ \int _{x_{0}}^{s}\phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t\Delta s \end{aligned}$$

for $0\leq x\leq x_{0}\leq x_{2}$, $0\leq y\leq y_{0}\leq y_{2}$, then $z ( x_{0},y ) =z(x,y_{0})=a(x_{0},y_{0})$,

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

(2.42)

and

$$\begin{aligned} z^{\Delta _{x}}(x,y) \leq &\theta ^{\Delta }(x) \int _{y_{0}}^{ \vartheta (y)}\phi _{1}\bigl(\theta (x),t\bigr)\eta \bigl[ {\psi} ^{-1} \bigl( z\bigl( \theta (x),t\bigr) \bigr) \bigr] \biggl[ f\bigl(\theta (x),t\bigr) {\omega} \bigl( {\psi} ^{-1} \bigl( z\bigl(\theta (x),t\bigr) \bigr) \bigr) \\ & {}+ \int _{x_{0}}^{\theta (x)}\phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t \\ \leq &\theta ^{\Delta }(x)\eta \bigl[ {\psi} ^{-1} \bigl( z \bigl( \theta (x),\vartheta (y) \bigr) \bigr) \bigr] \int _{y_{0}}^{ \vartheta (y)}\phi _{1}\bigl(\theta (x),t\bigr) \biggl[ f\bigl(\theta (x),t\bigr) {\omega} \bigl( {\psi} ^{-1} \bigl( z\bigl(\theta (x),t\bigr) \bigr) \bigr) \\ & {}+ \int _{x_{0}}^{\theta (x)}\phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t, \end{aligned}$$

then

$$\begin{aligned} \frac{z^{\Delta _{x}}(x,y)}{\eta [ {\psi} ^{-1} ( z ( x,y ) ) ] } \leq &\theta ^{\Delta }(x) \int _{y_{0}}^{\vartheta (y)}\phi _{1}\bigl(\theta (x),t\bigr) \biggl[ f\bigl(\theta (x),t\bigr) {\omega} \bigl( {\psi} ^{-1} \bigl( z\bigl(\theta (x),t\bigr) \bigr) \bigr) \\ & {}+ \int _{x_{0}}^{\theta (x)}\phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t. \end{aligned}$$

Taking Δ-integral for the above inequality gives

$$\begin{aligned} G_{1} \bigl( z ( x,y ) \bigr) \leq &G_{1} \bigl( z ( 0,y ) \bigr) + \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi _{1}(s,t) \biggl[ f(s,t) {\omega} \bigl( {\psi} ^{-1} \bigl( z(s,t) \bigr) \bigr) \\ & {}+ \int _{x_{0}}^{s}\phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t\Delta s, \end{aligned}$$

then

$$\begin{aligned} G_{1} \bigl( z ( x,y ) \bigr) \leq &G_{1} \bigl( a(x_{0},y_{0}) \bigr) + \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)} \phi _{1}(s,t) \biggl[ f(s,t) {\omega} \bigl( {\psi} ^{-1} \bigl( z(s,t) \bigr) \bigr) \\ & {}+ \int _{x_{0}}^{s}\phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t\Delta s. \end{aligned}$$

Since $(x_{0},y_{0})\in \Omega$ is chosen arbitrarily, the last inequality can be restated as

$$ G_{1} \bigl( z ( x,y ) \bigr) \leq p_{1} ( x,y ) + \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)} \phi _{1}(s,t)f(s,t) { \omega} \bigl( {\psi} ^{-1} \bigl( z(s,t) \bigr) \bigr) \nabla t\Delta s. $$

(2.43)

It is easy to observe that $p_{1} ( x,y )$ is a positive and nondecreasing function for all $(x,y)\in \Omega$, then an application of Lemma 2.1 to (2.43) yields the inequality

$$ z ( x,y ) \leq G_{1}^{-1} \biggl( F_{1}^{-1} \biggl[ F_{1} \bigl( p_{1} ( x,y ) \bigr) + \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t\Delta s \biggr] \biggr) . $$

(2.44)

From (2.44) and (2.42) we get the desired inequality (2.41).

If $a(x,y)=0$, we carry out the above procedure with $\epsilon >0$ instead of $a(x,y)$ and subsequently let $\epsilon \rightarrow 0$. The proof is complete. □

Now, as special cases of our results, we will give the continuous, discrete, and quantum inequalities. Namely, in the cases of time scales $\mathbb{T}=\mathbb{R}$, $\mathbb{T}=h\mathbb{Z}$, $\mathbb{T}=\mathbb{Z}$, and $\mathbb{T}=\overline{q^{\mathbb{Z}}}$.

Remark 2.22

If we take $\mathbb{T}=\mathbb{R}$ and $x_{0}=0$ and $y_{0}=0$ in Theorem 2.21, then, by relation (1.1), inequality (2.41) becomes the inequality obtained in [15, Theorem 2.7].

Corollary 2.23

If we take $\mathbb{T}=h \mathbb{Z}$ in Theorem 2.15by relation (1.3), then the following inequality

$$\begin{aligned} {\psi} \bigl( u(x,y) \bigr) \leq &a(x,y)+h^{2}\sum _{s= \frac{x_{0}}{h}}^{\frac{\theta (x)}{h}-1}\sum_{t=\frac{y_{0}}{h}}^{ \frac{\vartheta (y)}{h}+1} \phi _{1}(sh,th)\eta \bigl( u ( sh,th ) \bigr) \\ & {}\times\Biggl[ f(sh,th) {\omega} \bigl( u(sh,th) \bigr) +\sum _{t= \frac{x_{0}}{h}}^{\frac{s}{h}-1}\phi _{2}(\tau ,th) \Biggr] \end{aligned}$$

for $(x,y)\in \Omega$, then

$$ u ( x,y ) \leq {\psi} ^{-1} \Biggl\{ G_{1}^{-1} \Biggl( F_{1}^{-1} \Biggl[ F_{1} \bigl( p_{1} ( x,y ) \bigr) +h^{2}\sum _{s= \frac{x_{0}}{h}}^{\frac{\theta (x)}{h}-1}\sum_{t=\frac{y_{0}}{h}}^{ \frac{\vartheta (y)}{h}+1} \phi _{1}(sh,th)f(sh,th) \Biggr] \Biggr) \Biggr\} $$

for $0\leq x\leq x_{2}$, $0\leq y\leq y_{2}$, where

$$\begin{aligned}& G_{1}(v)=\sum_{s=\frac{v_{0}}{h}}^{\frac{v}{h}-1} \frac{h}{\eta ( {\psi} ^{-1}(sh) ) },\quad v\geq v_{0}>0,\qquad G_{1}(+ \infty )=\sum _{s=\frac{v_{0}}{h}}^{+\infty } \frac{h}{\eta ( {\psi} ^{-1}(sh) ) }=+\infty \\& F_{1}(v)=\sum_{s=\frac{v_{0}}{h}}^{\frac{v}{h}-1} \frac{h}{ {\omega} [ {\psi} ^{-1} ( G_{1}^{-1} ( sh ) ) ] },\quad v\geq v_{0}>0,\qquad F_{1}(+ \infty )=+ \infty \\& p_{1} ( x,y ) =G_{1}\bigl(a(x,y)\bigr)+h^{2} \sum_{s=\frac{x_{0}}{h}}^{ \frac{\theta (x)}{h}-1}\sum _{t=\frac{y_{0}}{h}}^{ \frac{\vartheta (y)}{h}+1}\phi _{1}(sh,th) \Biggl( h \sum_{t= \frac{x_{0}}{h}}^{\frac{s}{h}-1}\phi _{2}( \tau ,th) \Biggr), \end{aligned}$$

and $( x_{2},y_{2} ) \in \Omega$ is chosen so that

$$ \Biggl[ F_{1} \bigl( p_{1} ( x,y ) \bigr) +h^{2}\sum_{s= \frac{x_{0}}{h}}^{\frac{\theta (x)}{h}-1}\sum _{t=\frac{y_{0}}{h}}^{ \frac{\vartheta (y)}{h}+1}\phi _{1}(sh,th)f(sh,th) \Biggr] \in \operatorname {Dom}\bigl( F_{1}^{-1} \bigr). $$

Corollary 2.24

In Corollary 2.23, if we take $h=1$, then the following inequality

$$\begin{aligned} {\psi} \bigl( u(x,y) \bigr) \leq &a(x,y)+\sum_{s=x_{0}}^{\theta (x)-1} \sum_{t=y_{0}}^{\vartheta (y)+1}\phi _{1}(s,t) \eta \bigl( u ( s,t ) \bigr) \\ & {}\times\Biggl[ f(s,t) {\omega} \bigl( u(s,t) \bigr) +\sum _{t=x_{0}}^{s-1} \phi _{2}(\tau ,t) \Biggr] \end{aligned}$$

for $(x,y)\in \Omega$, then

$$ u ( x,y ) \leq {\psi} ^{-1} \Biggl\{ G_{1}^{-1} \Biggl( F_{1}^{-1} \Biggl[ F_{1} \bigl( p_{1} ( x,y ) \bigr) +\sum_{s=x_{0}}^{ \theta (x)-1} \sum_{t=y_{0}}^{\vartheta (y)+1}\phi _{1}(s,t)f(s,t) \Biggr] \Biggr) \Biggr\} $$

for $0\leq x\leq x_{2}$, $0\leq y\leq y_{2}$, where

$$\begin{aligned}& G_{1}(v)=\sum_{s=v_{0}}^{v-1} \frac{1}{\eta ( {\psi} ^{-1}(s) ) },\quad v\geq v_{0}>0,\qquad G_{1}(+ \infty )=\sum _{s=v_{0}}^{+\infty } \frac{1}{\eta ( {\psi} ^{-1}(s) ) }=+ \infty, \\& F_{1}(v)=\sum_{s=v_{0}}^{v-1} \frac{1}{ {\omega} [ {\psi} ^{-1} ( G_{1}^{-1} ( s ) ) ] },\quad v\geq v_{0}>0,\qquad F_{1}(+ \infty )=+ \infty,\\& p_{1} ( x,y ) =G_{1}\bigl(a(x,y)\bigr)+\sum _{s=x_{0}}^{\theta (x)-1} \sum_{t=y_{0}}^{\vartheta (y)+1} \phi _{1}(s,t) \Biggl( \sum_{t=x_{0}}^{s-1} \phi _{2}(\tau ,t) \Biggr) \end{aligned}$$

and $( x_{2},y_{2} ) \in \Omega$ is chosen so that

$$ \Biggl[ F_{1} \bigl( p_{1} ( x,y ) \bigr) +\sum _{s=x_{0}}^{ \theta (x)-1}\sum_{t=y_{0}}^{\vartheta (y)+1} \phi _{1}(s,t)f(s,t) \Biggr] \in \operatorname {Dom}\bigl( F_{1}^{-1} \bigr). $$

Corollary 2.25

If we take $\mathbb{T}=\overline{q^{\mathbb{Z}}}$ in Theorem 2.21by relation (1.4), then the following inequality

$$\begin{aligned} {\psi} \bigl( u(x,y) \bigr) \leq &a(x,y)+(q-1)^{2}\sum _{s=(\log _{q}{x_{0}})}^{( \log _{q}{\theta (x)})-1}\sum_{t=(\log _{q}{y_{0}})}^{(\log _{q}{ \vartheta (y)})+1} q^{(s+t)}\phi _{1}\bigl(q^{s},q^{t} \bigr)\eta \bigl( u \bigl( q^{s},q^{t} \bigr) \bigr) \\ &{}\times \Biggl[ f\bigl(q^{s},q^{t}\bigr) {\omega} \bigl( u\bigl(q^{s},q^{t}\bigr) \bigr) +(q-1) \sum _{t=(\log _{q}{x_{0}})}^{(\log _{q}{s})-1}q^{t}\phi _{2}\bigl(\tau ,q^{t}\bigr) \Biggr] \end{aligned}$$

for $(x,y)\in \Omega$, then

$$\begin{aligned} u ( x,y ) \leq &{\psi} ^{-1} \Biggl\{ G_{1}^{-1} \Biggl( F_{1}^{-1} \Biggl[ F_{1} \bigl( p_{1} ( x,y ) \bigr) \\ &{}+(q-1)^{2} \sum _{s=(\log _{q}{x_{0}})}^{(\log _{q}{\theta (x)})-1}\sum_{t=( \log _{q}{y_{0}})}^{(\log _{q}{\vartheta (y)})+1} q^{(s+t)}\phi _{1}\bigl(q^{s},q^{t} \bigr)f\bigl(q^{s},q^{t}\bigr) \Biggr] \Biggr) \Biggr\} \end{aligned}$$

for $0\leq x\leq x_{2}$, $0\leq y\leq y_{2}$, where

$$\begin{aligned}& G_{1}(v)=\sum_{s=(\log _{q}{v_{0}})}^{(\log _{q}{v})-1} \frac{(q-1)q^{s}}{\eta ( {\psi} ^{-1}(q^{s}) ) },\quad v\geq v_{0}>0,\\& G_{(q-1)q^{s}}(+ \infty )= \sum_{s=(\log _{q}{v_{0}})}^{+\infty } \frac{(q-1)q^{s}}{\eta ( {\psi} ^{-1}(q^{s}) ) }=+ \infty ,\\& F_{1}(v)=\sum_{s=(\log _{q}{v_{0}})}^{(\log _{q}{v})-1} \frac{(q-1)q^{s}}{ {\omega} [ {\psi} ^{-1} ( G_{1}^{-1} ( q^{s} ) ) ] },\quad v\geq v_{0}>0,\qquad F_{1}(+ \infty )=+ \infty,\\& p_{1} ( x,y ) =G_{1}\bigl(a(x,y)\bigr)+(q-1)^{2} \sum_{s=(\log _{q}{x_{0}})}^{( \log _{q}{\theta (x)})-1}\sum _{t=(\log _{q}{y_{0}})}^{(\log _{q}{ \vartheta (y)})+1} q^{(s+t)}\phi _{1}\bigl(q^{s},q^{t}\bigr) \Biggl( \sum _{t=x_{0}}^{s-1} \phi _{2}\bigl( \tau ,q^{t}\bigr) \Biggr) \end{aligned}$$

and $( x_{2},y_{2} ) \in \Omega$ is chosen so that

$$ \Biggl[ F_{1} \bigl( p_{1} ( x,y ) \bigr) +(q-1)^{2} \sum_{s=(\log _{q}{x_{0}})}^{(\log _{q}{\theta (x)})-1} \sum_{t=( \log _{q}{y_{0}})}^{(\log _{q}{\vartheta (y)})+1} q^{(s+t)}\phi _{1}\bigl(q^{s},q^{t}\bigr)f \bigl(q^{s},q^{t}\bigr) \Biggr] \in \operatorname {Dom}\bigl( F_{1}^{-1} \bigr). $$

Theorem 2.26

Under the hypotheses of Theorem 2.9, and let p be a nonnegative constant. If $u(x,y)$ satisfies

$$\begin{aligned} {\psi} \bigl( u(x,y) \bigr) \leq &a(x,y)+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t)u^{p} ( s,t ) \\ &{}\times \biggl[ f(s,t) {\omega} \bigl( u(s,t) \bigr) + \int _{x_{0}}^{s} \phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t\Delta s \end{aligned}$$

(2.45)

for $(x,y)\in \Omega$, then

$$ u ( x,y ) \leq {\psi} ^{-1} \biggl\{ G_{1}^{-1} \biggl( F_{1}^{-1} \biggl[ F_{1} \bigl( p_{1} ( x,y ) \bigr) + \int _{x_{0}}^{ \theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t \Delta s \biggr] \biggr) \biggr\} $$

(2.46)

for $0\leq x\leq x_{2}$, $0\leq y\leq y_{2}$, where

$$ G_{1}(v)= \int _{v_{0}}^{v} \frac{\Delta s}{ [ {\psi} ^{-1}(s) ] ^{p}},\quad v\geq v_{0}>0,\qquad G_{1}(+ \infty )= \int _{v_{0}}^{+\infty } \frac{\Delta s}{ [ {\psi} ^{-1}(s) ] ^{p}}=+\infty , $$

(2.47)

and $F_{1}$, $p_{1}$ are as in Theorem 2.21and $( x_{2},y_{2} ) \in \Omega$ is chosen so that

$$ \biggl[ F_{1} \bigl( p_{1} ( x,y ) \bigr) + \int _{x_{0}}^{ \theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t \Delta s \biggr] \in \operatorname {Dom}\bigl( F_{1}^{-1} \bigr) . $$

Proof

An application of Theorem 2.21 with $\eta ( u ) =u^{p}$ yields the desired inequality (2.46). □

Remark 2.27

Taking $\mathbb{T}=\mathbb{R}$. The inequality established in Theorem 2.26 generalizes [38, Theorem 1] (with $p=1$, $a(x,y)=b(x)+c(y)$, $x_{0}=0$, $y_{0}=0$, $\phi _{1}(s,t)f(s,t)=h(s,t)$, and $\phi _{1}(s,t) ( \int _{x_{0}}^{s}\phi _{2}(\tau ,t)\Delta \tau ) =g(s,t)$).

Corollary 2.28

Under the hypotheses of Theorem 2.26, and let $q>p>0$ be constants. If $u(x,y)$ satisfies

$$\begin{aligned} u^{q}(x,y) \leq &a(x,y)+\frac{p}{p-q} \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi _{1}(s,t)u^{p} ( s,t ) \\ &{} \times\biggl[ f(s,t) {\omega} \bigl( u(s,t) \bigr) + \int _{x_{0}}^{s} \phi _{2}(\tau ,t) \Delta \tau \biggr] \nabla t\Delta s \end{aligned}$$

(2.48)

for $(x,y)\in \Omega$, then

$$ u(x,y)\leq \biggl\{ F_{1}^{-1} \biggl[ F_{1} \bigl( p_{1} ( x,y ) \bigr) + \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{ \vartheta (y)}\phi _{1}(s,t)f(s,t) \nabla t\Delta s \biggr] \biggr\} ^{\frac{1}{q-p}} $$

(2.49)

for $0\leq x\leq x_{2}$, $0\leq y\leq y_{2}$, where

$$ p_{1} ( x,y ) = \bigl[ a(x,y) \bigr] ^{\frac{q-p}{q}}+ \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)}\phi _{1}(s,t) \biggl( \int _{x_{0}}^{s}\phi _{2}(\tau ,t) \Delta \tau \biggr) \nabla t\Delta s $$

and $F_{1}$ is defined in Theorem 2.21.

Proof

An application of Theorem 2.26 with ${\psi} ( u(x,y) ) =u^{p}$ to (2.48) yields inequality (2.49); to save space, we omit the details. □

Remark 2.29

Taking $\mathbb{T}=\mathbb{R}$, $x_{0}=0$, $y_{0}=0$, $a(x,y)=b(x)+c(y)$, $\phi _{1}(s,t)f(s,t)=h(s,t)$, and $\phi _{1}(s,t) ( \int _{x_{0}}^{s}\phi _{2}(\tau ,t)\Delta \tau ) =g(s,t)$ in Corollary 2.28, we obtain [39, Theorem 1].

Remark 2.30

Taking $\mathbb{T}=\mathbb{R}$, $x_{0}=0$, $y_{0}=0$, $a(x,y)=c^{\frac{p}{p-q}}$, $\phi _{1}(s,t)f(s,t)=h(t)$, and $\phi _{1}(s, t) ( \int _{x_{0}}^{s}\phi _{2}(\tau ,t)\Delta \tau ) =g(t)$ and keeping y fixed in Corollary 2.28, we obtain [25, Theorem 2.1].

3 Application

In what follows, we discus the boundedness of the solutions of the initial boundary value problem for partial delay dynamic equation of the form

$$\begin{aligned}& \begin{gathered} \bigl(z^{q}\bigr)^{\Delta _{x}\nabla _{y}}(x,y)=A \biggl( x,y,z \bigl( x-h_{1}(x),y-h_{2}(y) \bigr) , \int _{x_{0}}^{x}B \bigl( s,y,z \bigl(s-h_{1}(s),y\bigr) \bigr) \Delta s \biggr), \\ z(x,y_{0})=a_{1}(x),\qquad z(x_{0},y)=a_{2}(y),\qquad a_{1}(x_{0})=a_{y_{0}}(0)=0 \end{gathered} \end{aligned}$$

(3.1)

for $(x,y)\in \Omega$, where $z,b\in C ( \Omega ,\mathbb{R}_{+} )$, $A\in C(\Omega \times R^{2},R)$, $B\in C ( \Omega \times R,R )$, and $h_{1}\in C^{1} ( \mathbb{T}_{1},\mathbb{R}_{+} )$, $h_{2} \in C^{1} (\mathbb{T}_{2},\mathbb{R}_{+} )$ are nondecreasing functions such that $h_{1}(x)\leq x$ on $\mathbb{T}_{1}$, $h_{2}(y)\leq y$ on $\mathbb{T}_{2}$, and $h_{1}^{\Delta }(x)<1$, $h_{2}^{\Delta }(y)<1$.

Theorem 3.1

Assume that the functions $a_{1}$, $a_{2}$, A, B in (3.1) satisfy the conditions

$$\begin{aligned}& \bigl\vert a_{1}(x)+a_{2}(y) \bigr\vert \leq a(x,y) \end{aligned}$$

(3.2)

$$\begin{aligned}& \bigl\vert A ( s,t,z,u ) \bigr\vert \leq \frac{q}{q-p} \phi _{1}(s,t) \bigl[ f(s,t) \vert z \vert ^{p}+ \vert u \vert \bigr] \end{aligned}$$

(3.3)

$$\begin{aligned}& \bigl\vert B ( \tau ,t,z ) \bigr\vert \leq \phi _{2}( \tau ,t) \vert z \vert ^{p}, \end{aligned}$$

(3.4)

where $a(x,y)$, $\phi _{1}(s,t)$, $f(s,t)$, and $\phi _{2}(\tau ,t)$ are as in Theorem 2.6, $q>p>0$ are constants. If $z(x,y)$ satisfies (3.1), then

$$ \bigl\vert z(x,y) \bigr\vert \leq \biggl\{ p(x,y)+M_{1}M_{2} \int _{x_{0}}^{ \theta (x)} \int _{y_{0}}^{\vartheta (y)}\overset{-}{\phi _{1}}(s,t) \overset{-}{f}(s,t)\nabla t\Delta s \biggr\} ^{\frac{1}{q-p}}, $$

(3.5)

where

$$\begin{aligned} p(x,y) =& \bigl( a(x,y) \bigr) ^{\frac{q-p}{q}} \\ &{}+M_{1}M_{2} \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)} \overset{-}{\phi _{1}}(s,t) \biggl( M_{1} \int _{x_{0}}^{s}\overset{-}{\phi _{2}}(\tau ,t) \Delta \tau \biggr) \nabla t\Delta s \end{aligned}$$

and

$$ M_{1}=\underset{x\in I_{1}}{\operatorname{Max}} \frac{1}{1-h_{1}^{\Delta }(x)}, \qquad M_{2}=\underset{y\in I_{2}}{\operatorname{Max}}\frac{1}{1-h_{2}^{\Delta }(y)} $$

and $\overset{-}{\phi _{1}}(\gamma ,\xi )=\phi _{1} ( \gamma +h_{1}(s), \xi +h_{2}(t) )$, $\overset{-}{\phi _{2}} ( \mu ,\xi ) =\phi _{2} ( \mu ,\xi +h_{2}(t) )$, $\overset{-}{f}(\gamma ,\xi )=f ( \gamma +h_{1}(s),\xi +h_{2}(t) )$.

Proof

If $z(x,y)$ is any solution of (3.1), then

$$\begin{aligned} z^{q}(x,y) =&a_{1}(x)+a_{2}(y) \\ &{}+ \int _{x_{0}}^{x} \int _{y_{0}}^{y}A \biggl( s,t,z \bigl( s-h_{1}(s),t-h_{2}(t) \bigr) , \\ &{} \int _{x_{0}}^{s}B \bigl( \tau ,t,z\bigl(\tau -h_{1}(\tau ),t\bigr) \bigr) \Delta \tau \biggr) \nabla t\Delta s. \end{aligned}$$

(3.6)

Using conditions (3.2)–(3.4) in (3.6), we obtain

$$\begin{aligned} \bigl\vert z(x,y) \bigr\vert ^{q} \leq &a(x,y)+ \frac{q-p}{q}\int _{x_{0}}^{x} \int _{y_{0}}^{y}\phi _{1}(s,t) \biggl[ f(s,t) \bigl\vert z \bigl( s-h_{1}(s),t-h_{2}(t) \bigr) \bigr\vert ^{p} \\ &{} + \int _{x_{0}}^{s}\phi _{2}(\tau ,t) \bigl\vert z(\tau ,t) \bigr\vert ^{p}\Delta \tau \biggr] \nabla t \Delta s. \end{aligned}$$

(3.7)

Now, making a change of variables on the right-hand side of (3.7), $s-h_{1}(s)=\gamma$, $t-h_{2}(t)=\xi$, $x-h_{1}(x)=\theta (x)$ for $x\in \mathbb{T}_{1}$, $y-h_{2}(y)=\vartheta (y)$ for $y\in \mathbb{T}_{2}$, we obtain the inequality

$$\begin{aligned} \bigl\vert z(x,y) \bigr\vert ^{q} \leq &a(x,y)+ \frac{q-p}{q}M_{1}M_{2} \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)} \overset{-}{\phi _{1}}(\gamma ,\xi ) \biggl[ \overset{-}{f}(\gamma ,\xi ) \bigl\vert z ( \gamma ,\xi ) \bigr\vert ^{p} \\ & {}+M_{1} \int _{x_{0}}^{\gamma }\overset{-}{\phi _{2}} ( \mu ,\xi ) \bigl\vert z(\mu ,t) \bigr\vert ^{p}\Delta \mu \biggr] \nabla \xi \Delta \gamma . \end{aligned}$$

(3.8)

We can rewrite inequality (3.8) as follows:

$$\begin{aligned} \bigl\vert z(x,y) \bigr\vert ^{q} \leq &a(x,y)+ \frac{q-p}{q}M_{1}M_{2} \int _{x_{0}}^{\theta (x)} \int _{y_{0}}^{\vartheta (y)} \overset{-}{\phi _{1}}(s,t) \biggl[ \overset{-}{f}(s,t) \bigl\vert z ( s,t ) \bigr\vert ^{p} \\ & {}+M_{1} \int _{x_{0}}^{s}\overset{-}{\phi _{2}} ( \tau ,t ) \bigl\vert z(\tau ,t) \bigr\vert ^{p}\Delta \tau \biggr] \nabla t\Delta s. \end{aligned}$$

(3.9)

As an application of Corollary 2.8 to (3.9) with $u(x,y)= \vert z(x,y) \vert$, we obtain the desired inequality (3.5). The proof is complete. □

4 Conclusion

In this article, we explored new generalizations of the integral retarded inequality given in [15] by the utilization of the integral rule on time scales. We generalized a number of those inequalities to a general time scale. Besides that, in order to obtain some new inequalities as special cases, we also extended our inequalities to discrete, quantum, and continuous calculus. Also, we studied the qualitative properties of solutions of some types of dynamic equations on time scales.