On some new double dynamic inequalities associated with Leibniz integral rule on time scales

Abstract

In 2020, El-Deeb et al. proved several dynamic inequalities. It is our aim in this paper to give the retarded time scales case of these inequalities. We also give a new proof and formula of Leibniz integral rule on 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 boundedness of some delay initial value problems by applying our results as application.

1 Introduction

In 2020, El-Deeb et al. [1] have proved the following inequalities:

$$\begin{aligned}& \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \\& \quad \leq a( \hat{ \varsigma },\hat{\varrho }) + \int _{0}^{\hat{\varsigma }} \int _{0}^{ \hat{\varrho }} \bigl[ f(\hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \\& \qquad {} + \int _{0}^{\hat{\varsigma }} \int _{0}^{\hat{\varrho }}b(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl[ h(\hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) + \int _{0}^{\hat{\xi }_{1}}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \end{aligned}$$

and

$$\begin{aligned}& \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \\& \quad \leq a( \hat{ \varsigma },\hat{\varrho }) + \int _{0}^{\hat{\varsigma }} \int _{0}^{ \hat{\varrho }}\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \bigl[ f(\hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1},\hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \\& \qquad {} + \int _{0}^{\hat{\varsigma }} \int _{0}^{\hat{\varrho }}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \biggl( \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2}) \tilde{\Psi } \bigl( u(\hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}. \end{aligned}$$

The objective of the theory of time scales, which was introduced by Stefan Hilger in his PhD thesis [2] in 1988, is to unify continuous and discrete calculus. Several foundational definitions and notations of basic calculus of time scales introduced in the excellent recent books [3, 4] by Bohner and Peterson will be employed in the sequel. For some Gronwall–Bellman-type integral, dynamic inequalities and other type of inequalities on time scales, see the papers [5–36].

We use the following notations:

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

  $$ \begin{gathered} \sigma (t)=t, \\ \mu (t)=0, \\ f^{\Delta }(t)=f'(t), \\ \int _{a}^{b}f(t)\Delta t= \int _{a}^{b}f(t)\,dt; \end{gathered} $$

  (1.1)

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

  $$ \begin{gathered} \sigma (t)=t+1, \\ \mu (t)=1, \\ f^{\Delta }(t)=\Delta f(t), \\ \int _{a}^{b}f(t)\Delta t=\sum _{t=a}^{b-1}f(t), \end{gathered} $$

  (1.2)

  where Δ is the forward difference operator.

Theorem 1.1

(Chain rule on time scales [3])

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

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

(1.3)

Theorem 1.2

(Chain rule on time scales [3])

Let $f : \mathbb{R}\rightarrow \mathbb{R}$ be continuously differentiable and suppose $g : \mathbb{T}\rightarrow \mathbb{R}$ is Δ-differentiable. Then $f \circ g : \mathbb{T}\rightarrow \mathbb{R}$ is Δ-differentiable and the formula

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

(1.4)

holds.

Theorem 1.3

([3])

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 $[t_{0},\sigma (t)]$. Suppose that 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

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

Other dynamic inequalities on time scales may be found in [37–40]. In this manuscript, we will discuss the retarded time scale case of the inequalities obtained in [1] using new techniques by replacing the upper limit ς̂ and ϱ̂ of the integral by the delay function $\hat{\alpha }(\hat{\varsigma })\leq \hat{\varsigma }$ and $\hat{\beta }(\hat{\varrho })\leq \hat{\varrho }$. Furthermore, these inequalities that we obtained here extend some known results in the literature, and they also unify the continuous and discrete cases.

2 Main results

Throughout the paper, we suppose that $\mathbb{T}_{1}$ and $\mathbb{T}_{2}$ are two time scales.

First, we prove the following result.

Theorem 2.1

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

Proof

We will only prove part (i); the others may be proved similarly. Define a function g by

$$ g(t) = \int _{u(t)}^{h(t)}f(t,s)\Delta s,\quad \text{for } t\in \mathbb{T^{\kappa }}. $$

(2.1)

We notice that g is a continuous function. Indeed, we have two cases for t. In the first case, if t is right-scattered, from (2.1), we get

$$\begin{aligned} g^{\Delta }(t) =& \frac{g(\sigma (t))-g(t)}{\sigma (t)-t} \\ =& \frac{1}{\sigma (t)-t} \biggl[ \int ^{h(\sigma (t))}_{u(\sigma (t))}f \bigl( \sigma (t),s \bigr)\Delta s - \int ^{h(t)}_{u(t)}f(t,s)\Delta s \biggr] \\ =& \frac{1}{\sigma (t)-t} \biggl[- \int _{u(t)}^{u(\sigma (t))}f \bigl( \sigma (t),s \bigr)\Delta s + \int ^{h(t)}_{u(t)}f \bigl(\sigma (t),s \bigr)\Delta s \\ &{}+ \int ^{h(\sigma (t))}_{h(t)}f \bigl(\sigma (t),s \bigr)\Delta s- \int ^{h(t)}_{u(t)}f(t,s) \Delta s \biggr] \\ =& \int ^{h(t)}_{u(t)}\frac{f(\sigma (t),s) - f(t,s)}{\sigma (t)-t} \Delta s + \frac{1}{\sigma (t)-t} \int ^{h(\sigma (t))}_{h(t)}f \bigl( \sigma (t),s \bigr)\Delta s \\ &{}- \frac{1}{\sigma (t)-t} \int _{u(t)}^{u(\sigma (t))}f \bigl(\sigma (t),s \bigr) \Delta s \\ =& \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + \frac{h(\sigma (t))-h(t)}{\sigma (t)-t}f \bigl(\sigma (t),h(t) \bigr) \\ &{}- \frac{u(\sigma (t))-u(t)}{\sigma (t)-t}f \bigl(\sigma (t),u(t) \bigr) \\ =& \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + h^{\Delta }(t)f \bigl(\sigma (t),h(t) \bigr) -u^{\Delta }(t)f \bigl( \sigma (t),u(t) \bigr). \end{aligned}$$

(2.2)

From (2.2), we get the required result.

Now consider the second case when t is right-dense. Since f is continuous, it is rd-continuous, hence it has a delta partial anti-derivative with respect to the second variable s, say $F(t,s)$, that is, $f(t,s)=F^{\Delta _{s}}(t,s)$, and then we have

$$\begin{aligned} \biggl[ \int ^{h(t)}_{u(t)}f(t,s)\Delta s \biggr]^{\Delta } =& g^{\Delta }(t) \\ =& \lim_{r\to t}\frac{g(t)-g(r)}{t - r} \\ =& \lim_{r\to t}\frac{1}{t - r} \biggl[ \int ^{h(t)}_{u(t)}f(t,s) \Delta s- \int ^{h(r)}_{u(r)}f(r,s)\Delta s \biggr] \\ =& \lim_{r\to t}\frac{1}{t - r} \biggl[ \int ^{h(t)}_{u(t)}f(t,s) \Delta s - \int _{u(r)}^{u(t)}f(r,s)\Delta s \\ &{}- \int _{u(t)}^{h(t)}f(r,s)\Delta s - \int _{h(t)}^{h(r)}f(r,s) \Delta s \biggr] \\ =& \lim_{r\to t} \int ^{h(t)}_{u(t)}\frac{f(t,s)-f(r,s)}{t - r} \Delta s + \lim_{r\to t}\frac{1}{t - r} \int _{h(r)}^{h(t)}F^{\Delta _{s}}(r,s) \Delta s \\ &{}- \lim_{r\to t}\frac{1}{t - r} \int _{u(r)}^{u(t)}F^{\Delta _{s}}(r,s) \Delta s. \end{aligned}$$

(2.3)

Thus, from (2.3), we get

$$\begin{aligned} \biggl[ \int ^{h(t)}_{u(t)}f(t,s)\Delta s \biggr]^{\Delta } =& \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + \lim_{r\to t}\frac{1}{t - r} \bigl[F \bigl(r,h(t) \bigr)-F \bigl(r,h(r) \bigr) \bigr] \\ &{}- \lim_{r\to t}\frac{1}{t - r} \bigl[F \bigl(r,u(t) \bigr)-F \bigl(r,u(r) \bigr) \bigr] \\ =& \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + \lim_{r\to t} \frac{h(t)-h(r)}{t - r}\frac{F(r,h(t))-F(r,h(r))}{h(t)-h(r)} \\ &{}- \lim_{r\to t}\frac{u(t)-u(r)}{t - r} \frac{F(r,u(t))-F(r,u(r))}{u(t)-u(r)} \\ =& \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + \lim_{r\to t} \frac{h(t)-h(r)}{t - r}\lim_{r\to t} \frac{F(r,h(t))-F(r,h(r))}{h(t)-h(r)} \\ &{}- \lim_{r\to t}\frac{u(t)-u(r)}{t - r}\lim_{r\to t} \frac{F(r,u(t))-F(r,u(r))}{u(t)-u(r)} \\ =& \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + h^{\Delta }(t)F^{ \Delta _{s}} \bigl(t,h(t) \bigr) - u^{\Delta }(t)F^{\Delta _{s}} \bigl(t,u(t) \bigr) \\ =& \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + h^{\Delta }(t)f \bigl(t,h(t) \bigr)- u^{\Delta }(t)f \bigl(t,u(t) \bigr). \end{aligned}$$

This completes the proof. □

Remark 2.2

If we take $h(t)=t$ and $u(t)=a$ (where a is constant), then Theorem 2.1 reduces to [4, Theorem 5.37, p. 139].

Now, by using the result of Theorem 2.1, we state and prove the rest of our main results:

Theorem 2.3

Suppose $a\in C_{\mathrm{rd}}(\Omega ,\mathbb{R}_{+})$ is nondecreasing with respect to $(\hat{\varsigma },\hat{\varrho }) \in \Omega$, and g, u, p, $f\in C_{\mathrm{rd}}(\Omega ,\mathbb{R}_{+})$. Also let $\hat{\alpha }\in C^{1}_{\mathrm{rd}} ( \mathbb{T}_{1},\mathbb{T}_{1} )$ and $\hat{\beta }\in C^{1}_{\mathrm{rd}} ( \mathbb{T}_{2},\mathbb{T}_{2} )$ be nondecreasing functions with $\hat{\alpha }(\hat{\varsigma })\leq \hat{\varsigma }$ on $\mathbb{T}_{1}$, $\hat{\beta }(\hat{\varrho })\leq \hat{\varrho }$ on $\mathbb{T}_{2}$. Furthermore, suppose Φ̃, $\tilde{\Psi } \in C(\mathbb{R}_{+},\mathbb{R}_{+})$ are nondecreasing functions with $\{ \tilde{\Phi } ,\tilde{\Psi } \} (u)>0$ for $u>0$, and $\underset{u\rightarrow +\infty }{\lim }\tilde{\Phi } (u)=+\infty$. If $u(\hat{\varsigma },\hat{\varrho })$ satisfies

$$\begin{aligned} \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &a( \hat{ \varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })} \bigl[ f( \hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \end{aligned}$$

(2.4)

for $(\hat{\varsigma },\hat{\varrho }) \in \Omega$, then

$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \biggl\{ \tilde{\Lambda }^{-1} \biggl[ \tilde{\Lambda } \bigl( q(\hat{ \varsigma }, \hat{\varrho }) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2}) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr] \biggr\} $$

(2.5)

for $0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho } \leq \hat{\varrho }_{1}$, where

$$\begin{aligned}& q(\hat{\varsigma },\hat{\varrho }) =a(\hat{\varsigma },\hat{ \varrho }) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}p(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} , \end{aligned}$$

(2.6)

$$\begin{aligned}& \tilde{\Lambda }(r)= \int _{r_{0}}^{r} \frac{\Delta \hat{\xi }_{1}}{\omega \circ \tilde{\Phi } ^{-1}(\hat{\xi }_{1})},\quad r \geq r_{0}>0,\qquad \tilde{\Lambda }(+\infty )= \int _{r_{0}}^{+\infty } \frac{\Delta \hat{\xi }_{1}}{\omega \circ \tilde{\Phi } ^{-1}(\hat{\xi }_{1})}=+\infty , \end{aligned}$$

(2.7)

and $( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega$ is chosen so that

$$ \biggl( \tilde{\Lambda } \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \biggr) \in \operatorname{Dom}\bigl( G^{-1} \bigr) . $$

Proof

Assume that $a ( \hat{\varsigma },\hat{\varrho } ) >0$. Since $q\geq 0$ and it is nondecreasing, fixing an arbitrary point $(\breve{\xi },\breve{\zeta }) \in \Omega$ and defining $z(\hat{\varsigma },\hat{\varrho })$ by

$$\begin{aligned} z(\hat{\varsigma },\hat{\varrho }) =&q(\breve{\xi },\breve{\zeta }) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2}\Delta \hat{ \xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}, \end{aligned}$$

which is a positive and nondecreasing function for $0\leq \hat{\varsigma }\leq \breve{\xi }\leq \hat{\varsigma }_{1}$, $0\leq \hat{\varsigma }\leq \breve{\zeta }\leq \hat{\varrho }_{1}$, we then get $z(0,\hat{\varrho }) =z(\hat{\varsigma },0) =q(\breve{\xi },\breve{\zeta })$ and

$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \bigl( z( \hat{\varsigma },\hat{\varrho }) \bigr) . $$

(2.8)

By applying Theorem 2.1, differentiating $z(\hat{\varsigma },\hat{\varrho })$ with respect to ς̂, and using (2.8), we get

$$\begin{aligned}& z^{\Delta }_{ \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) \\& \quad = \hat{ \alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl[ \tilde{\Psi } \bigl( u \bigl(\hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2}) \tilde{\Psi } \bigl( u(\hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2} \\& \quad \leq \hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl[ \tilde{\Psi } \circ \tilde{ \Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }( \hat{\varsigma }), \hat{\xi }_{2} \bigr) \bigr) \\& \qquad {}+ \int _{0}^{\hat{\alpha }( \hat{\varsigma })}g(\hat{\zeta } ,\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2}. \end{aligned}$$

Since $\tilde{\Psi } \circ \tilde{\Phi } ^{-1}$ is nondecreasing with respect to $(\hat{\varsigma },\hat{\varrho }) \in \mathbb{R} _{+}\times \mathbb{R} _{+}$, we then have

$$\begin{aligned}& z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) \\& \quad \leq \hat{\alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl[ \tilde{\Psi } \circ \tilde{ \Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }(\hat{\varsigma }), \hat{\xi }_{2} \bigr) \bigr) \\& \qquad {}+\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl( \hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \int _{0}^{ \hat{\alpha }(\hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2} \\& \quad \leq \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{ \alpha }( \hat{\varsigma }),\hat{\beta }(\hat{\varrho }) \bigr) \bigr)\hat{\alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl[ 1+ \int _{0}^{ \hat{\alpha }(\hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2}, \end{aligned}$$

(2.9)

from which $\tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\alpha }( \hat{\varsigma }),\hat{\beta }(\hat{\varrho })) )\leq \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\varsigma },\hat{\varrho }) )$, so from (2.9), we get

$$ \frac{z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) }{\tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\varsigma },\hat{\varrho }) ) }\leq \hat{\alpha }^{\Delta }( \hat{ \varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( 1+ \int _{0}^{\hat{\alpha }( \hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}. $$

(2.10)

Now from (2.10), we get

$$ \tilde{\Lambda } \bigl( z(\hat{\varsigma },\hat{\varrho }) \bigr) \leq \tilde{ \Lambda } \bigl( q(\breve{\xi },\breve{\zeta }) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}. $$

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

$$ z(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Lambda }^{-1} \biggl[ \tilde{\Lambda } \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \biggr] . $$

(2.11)

So from (2.11) and (2.8), we get the desired inequality in (2.5). For $a(\hat{\varsigma },\hat{\varrho }) =0$, we carry out the above procedure with $\epsilon >0$ instead of $a(\hat{\varsigma },\hat{\varrho })$ and subsequently let $\epsilon \rightarrow 0$. This completes the proof. □

Remark 2.4

If we take $\hat{\alpha }(\hat{\varsigma })= \hat{\varsigma }$ and $\hat{\alpha }(\hat{\varrho })= \hat{\varrho }$, then Theorem 2.3 reduces to [1, Theorem 2.1].

Corollary 2.5

The discrete form can be obtained by letting $\mathbb{T}=\mathbb{Z}$, with the help of relations (1.2), and $\hat{\alpha }(\hat{\varsigma })=\hat{\varsigma }$, $\hat{\beta }(\hat{\varrho })=\hat{\varrho }$ in Theorem 2.3. If

$$\begin{aligned} \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &a( \hat{ \varsigma },\hat{\varrho }) +\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1} \bigl[ f( \hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \\ &{}+\sum_{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1}\sum _{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1}f(\hat{\xi }_{1},\hat{\xi }_{2}) \Biggl( \sum_{ \hat{\zeta }=0}^{\hat{\xi }_{1}-1} g(\hat{\zeta },\hat{\xi }_{2}) \tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Biggr) \end{aligned}$$

holds for $(\hat{\varsigma },\hat{\varrho }) \in \Omega$, then

$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \Biggl\{ \tilde{\Lambda }^{-1} \Biggl[ \tilde{\Lambda } \bigl( q(\hat{ \varsigma }, \hat{\varrho }) \bigr) +\sum_{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1}f(\hat{\xi }_{1},\hat{\xi }_{2}) \Biggl( 1+ \sum _{\hat{\zeta }=0}^{\hat{\xi }_{1}-1}g(\hat{\zeta }, \hat{\xi }_{2}) \Biggr) \Biggr] \Biggr\} $$

for $0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho } \leq \hat{\varrho }_{1}$, where

$$\begin{aligned}& q(\hat{\varsigma },\hat{\varrho }) =a(\hat{\varsigma },\hat{\varrho }) + \sum _{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1}\sum _{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1}p(\hat{\xi }_{1},\hat{\xi }_{2}), \\& \tilde{\Lambda }(r)=\sum_{\hat{\xi }_{1}=r_{0}}^{r-1} \frac{1}{\omega \circ \tilde{\Phi } ^{-1}(\hat{\xi }_{1})},\quad r\geq r_{0}>0,\qquad \tilde{\Lambda }(+\infty )=\sum_{\hat{\xi }_{1}=r_{0}}^{+ \infty } \frac{1}{\omega \circ \tilde{\Phi } ^{-1}(\hat{\xi }_{1})}=+\infty , \end{aligned}$$

and $( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega$ is chosen so that

$$ \Biggl( \tilde{\Lambda } \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) +\sum _{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1}\sum _{ \breve{st}=0}^{\hat{\varrho }-1}f(\hat{\xi }_{1},\hat{\xi }_{2}) \Biggl( 1+ \sum_{\hat{\zeta }0}^{\hat{\xi }_{1}}g( \hat{\zeta },\hat{\xi }_{2}) \Biggr) \Biggr) \in \operatorname{Dom}\bigl( G^{-1} \bigr) . $$

Theorem 2.6

Assume that h, $b\in C_{\mathrm{rd}}(\Omega ,\mathbb{R} _{+})$. Let g, f, p, a, u, Φ̃, and Ψ̃ be as in Theorem 2.3. If $u(\hat{\varsigma },\hat{\varrho })$ satisfies

$$\begin{aligned} \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &a( \hat{ \varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })} \bigl[ f( \hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}b(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl[ h(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \\ &{} + \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2}) \tilde{\Psi } \bigl( u(\hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \end{aligned}$$

(2.12)

for $(\hat{\varsigma },\hat{\varrho }) \in \Omega$, then

$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \biggl\{ G^{-1} \biggl[ G \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) +A( \hat{\varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr] \biggr\} $$

(2.13)

for $0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho } \leq \hat{\varrho }_{1}$, where Λ̃ is defined by (2.7),

$$ \breve{A}(\hat{\varsigma },\hat{\varrho }) = \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}b(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl[ h(\hat{\xi }_{1},\hat{\xi }_{2})+ \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} ,$$

(2.14)

and $( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega$ is chosen so that

$$ \biggl( \tilde{\Lambda } \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) + \breve{A}(\hat{\varsigma },\hat{\varrho }) + \int _{0}^{ \hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f( \hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr) \in \operatorname{Dom}\bigl( \tilde{\Lambda }^{-1} \bigr) . $$

Proof

Assume that $a(\hat{\varsigma },\hat{\varrho }) >0$. Fixing an arbitrary $(\breve{\xi },\breve{\zeta }) \in \Omega$, we define a positive and nondecreasing function $z(\hat{\varsigma },\hat{\varrho })$ by

$$\begin{aligned} z(\hat{\varsigma },\hat{\varrho }) =&q(\breve{\xi },\breve{\zeta }) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2}\Delta \hat{ \xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}b(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl[ h(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \\ &{}+ \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2}) \tilde{\Psi } \bigl( u(\hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \end{aligned}$$

for $0\leq \hat{\varsigma }\leq \breve{\xi }\leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho }\leq \breve{\zeta }\leq y_{1}$, then $z(0,\hat{\varrho }) =z(\hat{\varsigma },0) =q(\breve{\xi },\breve{\zeta })$ and

$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \bigl( z( \hat{\varsigma },\hat{\varrho }) \bigr). $$

Now, by applying Theorem 2.1, we have

$$\begin{aligned}& z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) \\& \quad = \hat{\alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \bigl( u \bigl( \hat{ \alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \Delta \hat{ \xi }_{2}+\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}b \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \\& \qquad {} \times \biggl( h \bigl( \hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \bigl( u \bigl( \hat{ \alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) + \int _{0}^{ \hat{\varsigma } }g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\& \quad \leq \hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }( \hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \Delta \hat{\xi }_{2}+ \int _{0}^{ \hat{\beta }(\hat{\varrho })}b \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \\& \qquad {}\times \biggl( h \bigl(\hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{ \alpha }(\hat{\varsigma }), \hat{\xi }_{2} \bigr) \bigr) + \int _{0}^{\hat{\varsigma } }g(\hat{\zeta }, \hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\& \quad \leq \hat{\alpha }^{\Delta }(\hat{\varsigma })\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z(\hat{\varsigma },\hat{\varrho }) \bigr) \biggl[ \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) \Delta \hat{\xi }_{2} \\& \qquad {} + \int _{0}^{\hat{\beta }(\hat{\varrho })}b \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( h \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) + \int _{0}^{\hat{\varsigma } }g( \hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \biggr] \Delta \hat{\xi }_{2}. \end{aligned}$$

Since $\tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\varsigma }, \hat{\varrho }) ) \leq \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\varsigma },\hat{\varrho }) )$, we then get

$$\begin{aligned}& \frac{z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) }{\tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\varsigma },\hat{\varrho }) ) } \\& \quad \leq \hat{\alpha }^{\Delta }( \hat{ \varsigma }) \biggl[ \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \Delta \hat{\xi }_{2} \\& \qquad {}+ \int _{0}^{\hat{\beta }(\hat{\varrho })}b \bigl(\hat{\alpha }(\hat{ \varsigma }), \hat{\xi }_{2} \bigr) \biggl( h \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2}) \Delta \hat{\zeta } \biggr) \biggr] \Delta \hat{\xi }_{2}. \end{aligned}$$

(2.15)

Integrating (2.15), we get

$$ \tilde{\Lambda } \bigl( z(\hat{\varsigma },\hat{\varrho }) \bigr) \leq \tilde{ \Lambda } \bigl( q(\breve{\xi },\breve{\zeta }) \bigr) + \breve{A}(\hat{\varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}. $$

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

$$ z(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Lambda }^{-1} \biggl[ \tilde{\Lambda } \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) + \breve{A}(\hat{\varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr] . $$

(2.16)

Thus, from (2.16) and $u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} ( z( \hat{\varsigma },\hat{\varrho }) )$, we get the required inequality in (2.13). For $a(\hat{\varsigma },\hat{\varrho }) =0$, we carry out the above procedure with $\epsilon >0$ instead of $a(\hat{\varsigma },\hat{\varrho })$ and subsequently let $\epsilon \rightarrow 0$. This completes the proof. □

Remark 2.7

If we take $\hat{\alpha }(\hat{\varsigma })= \hat{\varsigma }$ and $\hat{\alpha }(\hat{\varrho })= \hat{\varrho }$, then Theorem 2.6 reduces to [1, Theorem 2.4].

Corollary 2.8

If we take $\mathbb{T}=\mathbb{R}$ in Theorem 2.6, then, with the help of relations (1.1), we have the following inequality due to Boudeliou [41]. If

$$\begin{aligned}& \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \\& \quad \leq a( \hat{ \varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })} \bigl[ f( \hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \,d \hat{\xi }_{2}\,d\hat{\xi }_{1} \\& \qquad {} + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}b(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl[ h(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) + \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2}) \tilde{\Psi } \bigl( u(\hat{\zeta },\hat{\xi }_{2}) \bigr) \,d \hat{\zeta } \biggr] \,d\hat{\xi }_{2}\,d \hat{\xi }_{1} \end{aligned}$$

holds for $(\hat{\varsigma },\hat{\varrho }) \in \Omega$, then

$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \biggl\{ G^{-1} \biggl[ G \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) +A( \hat{\varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\,d\hat{\xi }_{2}\,d\hat{\xi }_{1} \biggr] \biggr\} $$

for $0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho } \leq \hat{\varrho }_{1}$, where Λ̃ is defined by (2.7),

$$ \breve{A}(\hat{\varsigma },\hat{\varrho }) = \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}b(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl[ h(\hat{\xi }_{1},\hat{\xi }_{2})+ \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\,d\hat{\zeta } \biggr] \,d \hat{\xi }_{2}\,d\hat{\xi }_{1}, $$

and $( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega$ is chosen so that

$$ \biggl( \tilde{\Lambda } \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) + \breve{A}(\hat{\varsigma },\hat{\varrho }) + \int _{0}^{ \hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f( \hat{\xi }_{1}, \hat{\xi }_{2})\,d\hat{\xi }_{2}\,d\hat{\xi }_{1} \biggr) \in \operatorname{Dom}\bigl( \tilde{\Lambda }^{-1} \bigr) . $$

Corollary 2.9

The discrete form can be obtained by letting $\mathbb{T}=\mathbb{Z}$, with the help of relations (1.2) and $\hat{\alpha }(\hat{\varsigma })=\hat{\varsigma }$, $\hat{\beta }(\hat{\varrho })=\hat{\varrho }$ in Theorem 2.6. If

$$\begin{aligned} \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &a( \hat{ \varsigma },\hat{\varrho }) +\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1} \bigl[ f( \hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \\ &{}+\sum_{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1}\sum _{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1}b(\hat{\xi }_{1},\hat{\xi }_{2}) \Biggl[ h(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) +\sum_{\hat{\zeta }=0}^{\hat{\xi }_{1}-1}g( \hat{\zeta }, \hat{\xi }_{2}) \tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Biggr] \end{aligned}$$

holds for $(\hat{\varsigma },\hat{\varrho }) \in \Omega$, then

$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \Biggl\{ G^{-1} \Biggl[ G \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) +A( \hat{\varsigma },\hat{\varrho }) +\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1}f( \hat{\xi }_{1},\hat{\xi }_{2}) \Biggr] \Biggr\} $$

for $0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho } \leq \hat{\varrho }_{1}$, where Λ̃ is defined by (2.7),

$$ \breve{A}(\hat{\varsigma },\hat{\varrho }) =\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1}b( \hat{\xi }_{1},\hat{\xi }_{2}) \Biggl[ h(\hat{\xi }_{1},\hat{\xi }_{2})+ \sum _{\hat{\zeta }=0}^{\hat{\xi }_{1}-1}g(\hat{\zeta },\hat{\xi }_{2}) \Biggr], $$

and $( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega$ is chosen so that

$$ \Biggl( \tilde{\Lambda } \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) + \breve{A}(\hat{\varsigma },\hat{\varrho }) +\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1}f( \hat{\xi }_{1},\hat{\xi }_{2}) \Biggr) \in \operatorname{Dom}\bigl( \tilde{ \Lambda }^{-1} \bigr) . $$

Theorem 2.10

Assume that g, a, u, f, p, Φ̃, and Ψ̃ are as in Theorem 2.3. If $u(\hat{\varsigma },\hat{\varrho })$ satisfies

$$\begin{aligned}& \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \\& \quad \leq a( \hat{ \varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \bigl[ f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \\& \qquad {} + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \biggl( \int _{0}^{\hat{\xi }_{1}}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}, \end{aligned}$$

(2.17)

for $(\hat{\varsigma },\hat{\varrho }) \in \Omega$, then

$$\begin{aligned}& u(\hat{\varsigma },\hat{\varrho }) \\& \quad \leq \tilde{\Phi } ^{-1} \biggl\{ \tilde{\Lambda }^{-1} \biggl( \tilde{\Theta }^{-1} \biggl[ \tilde{\Theta } \bigl( q_{1} ( \hat{\varsigma },\hat{\varrho } ) \bigr) \\& \qquad {}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \biggr] \biggr) \biggr\} , \end{aligned}$$

(2.18)

for $0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho } \leq \hat{\varrho }_{1}$, where

$$\begin{aligned}& q_{1} ( \hat{\varsigma },\hat{\varrho } ) =\tilde{\Lambda } \bigl( a(\hat{\varsigma },\hat{\varrho }) \bigr) + \int _{0}^{ \hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}p( \hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}, \end{aligned}$$

(2.19)

$$\begin{aligned}& \begin{aligned} &\tilde{\Theta }(r)= \int _{r_{0}}^{r} \frac{\Delta \hat{\xi }_{1}}{ ( ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) \circ \tilde{\Lambda }^{-1} ) (\hat{\xi }_{1} ) },\quad r \geq r_{0}>0, \\ &\tilde{\Theta }(+\infty )= \int _{r_{0}}^{+\infty } \frac{\Delta \hat{\xi }_{1}}{ ( \omega \circ \tilde{\Phi } ^{-1} ) \circ \tilde{\Lambda }^{-1}(\hat{\xi }_{1})}=+\infty , \end{aligned} \end{aligned}$$

(2.20)

and $( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega$ is chosen so that

$$ \biggl( \tilde{\Theta } \bigl( q_{1} ( \hat{\varsigma }, \hat{ \varrho } ) \bigr) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta }, \hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \biggr) \in \operatorname{Dom}\bigl( \tilde{\Theta }^{-1} \bigr) . $$

Proof

Suppose that $a(\breve{\xi },\breve{\zeta }) >0$. Fixing an arbitrary $(\breve{\xi },\breve{\zeta }) \in \Omega$, we define a positive and nondecreasing function $z(\hat{\varsigma },\hat{\varrho })$ by

$$\begin{aligned} z(\hat{\varsigma },\hat{\varrho }) =&a(\breve{\xi },\breve{\zeta })+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \bigl[ f(\hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1},\hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \biggl( \int _{0}^{\hat{\xi }_{1}}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}, \end{aligned}$$

for $0\leq \hat{\varsigma }\leq \breve{\xi }\leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho }\leq \breve{\zeta }\leq \hat{\varrho }_{1}$, then $z(0,\hat{\varrho }) =z(\hat{\varsigma },0) =a(\breve{\xi },\breve{\zeta })$ and

$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \bigl( z( \hat{\varsigma },\hat{\varrho }) \bigr) . $$

Now, by applying Theorem 2.1, we have

$$\begin{aligned}& z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) \\& \quad = \hat{\alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })} \tilde{\Psi } \bigl( u \bigl( \hat{ \alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \bigl[ f \bigl( \hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \bigl( u \bigl(\hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) +p(\hat{\varsigma },\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2} \\& \qquad {} +\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \bigl( u \bigl(\hat{ \alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \biggl( \int _{0}^{\hat{\varsigma } }g(\hat{\zeta } ,\hat{\xi }_{2}) \tilde{\Psi } \bigl( u(\hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\& \quad \leq \hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl( \hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \bigl[ f \bigl( \hat{\alpha }(\hat{\varsigma }), \hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) +p(\hat{\varsigma },\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2} \\& \qquad {} +\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }( \hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \\& \qquad {}\times\biggl( \int _{0}^{ \hat{\varsigma } }g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\& \quad \leq \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{ \alpha }( \hat{\varsigma }),\hat{\beta }(\hat{\varrho }) \bigr) \bigr) \hat{\alpha }^{ \Delta }(\hat{\varsigma }) \\& \qquad {}\times \int _{0}^{\hat{\beta }(\hat{\varrho })} \bigl[ f \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{ \Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) +p(\hat{\varsigma },\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2} \\& \qquad {} +\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{ \alpha }( \hat{\varsigma }),\hat{\beta }(\hat{\varrho }) \bigr) \bigr)\hat{\alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \\& \qquad {}\times\biggl( \int _{0}^{ \hat{\varsigma } }g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}, \end{aligned}$$

or

$$\begin{aligned}& \frac{z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) }{\tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\varsigma },\hat{\varrho }) ) } \\& \quad \leq \hat{\alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })} \bigl[ f \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{ \Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) +p(\hat{\varsigma },\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2} \\& \qquad {} +\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\varsigma } }g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}. \end{aligned}$$

(2.21)

Integrating (2.21), we get

$$\begin{aligned} \tilde{\Lambda } \bigl( z(\hat{\varsigma },\hat{\varrho }) \bigr) \leq & \tilde{\Lambda } \bigl( a(\breve{\xi },\breve{\zeta }) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })} \bigl[ f(\hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta } ,\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}. \end{aligned}$$

If $(\breve{\xi },\breve{\zeta }) \in \Omega$ is chosen arbitrarily, then

$$\begin{aligned} \tilde{\Lambda } \bigl( z(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &q_{1} (\hat{\varsigma },\hat{\varrho } ) + \int _{0}^{ \hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f( \hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta } ,\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}. \end{aligned}$$

Since $q_{1} (\hat{\varsigma },\hat{\varrho } )>0$ is a nondecreasing function, fixing an arbitrary point $( \breve{\xi },\breve{\zeta } ) \in \Omega$ and defining $v(\hat{\varsigma },\hat{\varrho }) >0$ to be a nondecreasing function given by

$$\begin{aligned} v(\hat{\varsigma },\hat{\varrho }) =&q_{1} ( \breve{\xi }, \breve{ \zeta } ) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta } ,\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}, \end{aligned}$$

for $0\leq \hat{\varsigma }\leq \breve{\xi }\leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho }\leq \breve{\zeta }\leq y_{1}$, we obtain $v(0,\hat{\varrho }) =v(\hat{\varsigma },0) =q_{1}(\breve{\xi }, \breve{\zeta })$ and

$$ z(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Lambda }^{-1} \bigl( v( \hat{\varsigma },\hat{\varrho }) \bigr) . $$

(2.22)

Now, by applying Theorem 2.1, we have

$$\begin{aligned} v^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) =&\hat{\alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \Delta \hat{\xi }_{2} \\ &{}+\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\varsigma } }g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\ \leq &\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( G^{-1} \bigl( v \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \bigr) \bigr) \Delta \hat{\xi }_{2} \\ &{}+\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\varsigma } }g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( G^{-1} \bigl( v(\hat{\zeta },\hat{\xi }_{2}) \bigr) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\ \leq & \bigl( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigr) \circ \tilde{\Lambda }^{-1} \bigl(v \bigl( \hat{\alpha }(\hat{\varsigma }), \hat{\beta }(\hat{\varrho }) \bigr) \bigr)\hat{\alpha }^{\Delta }( \hat{ \varsigma }) \\ &{} \times \biggl[ \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \Delta \hat{\xi }_{2}+ \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }), \hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\varsigma } }g(\hat{\zeta }, \hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \biggr] , \end{aligned}$$

or

$$\begin{aligned}& \frac{v^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) }{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) \circ \tilde{\Lambda }^{-1}(v ( \hat{\varsigma },\hat{\varrho } ) )} \\& \quad \leq \hat{\alpha }^{\Delta }(\hat{ \varsigma }) \biggl[ \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \Delta \hat{\xi }_{2}+ \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\varsigma } }g( \hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \biggr] . \end{aligned}$$

(2.23)

Integrating (2.23), we get

$$ \tilde{\Theta } \bigl( v ( \hat{\varsigma },\hat{\varrho } ) \bigr) \leq \tilde{ \Theta } \bigl( q_{1}(\breve{\xi },\breve{\zeta }) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl[ 1+ \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta } ,\hat{\xi }_{2})\Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}. $$

Since we chose $(\breve{\xi },\breve{\zeta }) \in \Omega$ arbitrarily,

$$\begin{aligned}& v ( \hat{\varsigma },\hat{\varrho } ) \\& \quad \leq \tilde{\Theta }^{-1} \biggl[ \tilde{\Theta } \bigl( q_{1}(\hat{\varsigma },\hat{\varrho }) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl[ 1+ \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta } ,\hat{\xi }_{2})\Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \biggr] . \end{aligned}$$

(2.24)

From (2.24), (2.22), and $u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} ( z( \hat{\varsigma },\hat{\varrho }) )$, we get the desired inequality in (2.18). For $a(\hat{\varsigma },\hat{\varrho }) =0$, we carry out the above procedure with $\epsilon >0$ instead of $a(\hat{\varsigma },\hat{\varrho })$ and subsequently let $\epsilon \rightarrow 0$. This completes the proof. □

Remark 2.11

If we take $\hat{\alpha }(\hat{\varsigma })= \hat{\varsigma }$ and $\hat{\alpha }(\hat{\varrho })= \hat{\varrho }$, then Theorem 2.10 reduces to [1, Theorem 2.7].

Corollary 2.12

If we take $\mathbb{T}=\mathbb{R}$ in Theorem 2.10, then, with the help of relations (1.1), we get the following inequality due to Boudeliou [41]. If

$$\begin{aligned}& \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \\& \quad \leq a( \hat{ \varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \bigl[ f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \,d\hat{\xi }_{2}\,d \hat{\xi }_{1} \\& \qquad {} + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \biggl( \int _{0}^{\hat{\xi }_{1}}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \,d\hat{\xi }_{2}\,d \hat{\xi }_{1} \end{aligned}$$

holds for $(\hat{\varsigma },\hat{\varrho }) \in \Omega$, then

$$\begin{aligned} u(\hat{\varsigma },\hat{\varrho }) \leq& \tilde{\Phi } ^{-1} \biggl\{ \tilde{\Lambda }^{-1} \biggl( \tilde{\Theta }^{-1} \biggl[ \tilde{\Theta } \bigl( q_{2} ( \hat{\varsigma },\hat{\varrho } ) \bigr) \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\,d\hat{\zeta } \biggr) \,d \hat{\xi }_{2}\,d\hat{\xi }_{1} \biggr] \biggr) \biggr\} , \end{aligned}$$

for $0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho } \leq \hat{\varrho }_{1}$, where

$$\begin{aligned}& q_{2} ( \hat{\varsigma },\hat{\varrho } ) =\tilde{\Lambda } \bigl( a(\hat{\varsigma },\hat{\varrho }) \bigr) + \int _{0}^{ \hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}p( \hat{\xi }_{1}, \hat{\xi }_{2})\,d\hat{\xi }_{2}\,d\hat{\xi }_{1}, \\& \tilde{\Theta }(r)= \int _{r_{0}}^{r} \frac{d\hat{\xi }_{1}}{ ( ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) \circ \tilde{\Lambda }^{-1} ) (\hat{\xi }_{1} ) },\quad r \geq r_{0}>0, \\& \tilde{\Theta }(+\infty )= \int _{r_{0}}^{+\infty } \frac{d\hat{\xi }_{1}}{ ( \omega \circ \tilde{\Phi } ^{-1} ) \circ \tilde{\Lambda }^{-1}(\hat{\xi }_{1})}=+\infty , \end{aligned}$$

and $( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega$ is chosen so that

$$ \biggl( \tilde{\Theta } \bigl( q_{2} ( \hat{\varsigma }, \hat{ \varrho } ) \bigr) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })} f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta }, \hat{\xi }_{2})\,d\hat{\zeta } \biggr) \,d\hat{\xi }_{2}\,d\hat{\xi }_{1} \biggr) \in \operatorname{Dom}\bigl( \tilde{\Theta }^{-1} \bigr) . $$

Corollary 2.13

The discrete form, due to El-Deeb et al. [1], can be obtained by letting $\mathbb{T}=\mathbb{Z}$ in Theorem 2.10, with the help of relations (1.2) and $\hat{\alpha }(\hat{\varsigma })=\hat{\varsigma }$, $\hat{\beta }(\hat{\varrho })=\hat{\varrho }$ as follows. If

$$\begin{aligned} \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &a( \hat{ \varsigma },\hat{\varrho }) +\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1} \tilde{\Psi } \bigl( u( \hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \bigl[ f( \hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \\ &{}+\sum_{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1}\sum _{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1}f(\hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \Biggl( \sum_{\hat{\zeta }=0}^{ \hat{\xi }_{1}-1}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\zeta }, \hat{\xi }_{2}) \bigr) \Biggr) , \end{aligned}$$

holds for $(\hat{\varsigma },\hat{\varrho }) \in \Omega$, then

$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \Biggl\{ \bar{G}^{-1} \Biggl( \bar{F}^{-1} \Biggl[ \bar{F} \bigl( \bar{q}_{2} ( \hat{\varsigma },\hat{\varrho } ) \bigr) +\sum _{ \hat{\xi }_{1}=0}^{\hat{\varsigma }-1}\sum_{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1}f( \hat{\xi }_{1},\hat{\xi }_{2}) \Biggl( 1+\sum _{ \hat{\zeta }=0}^{\hat{\xi }_{1}-1}g(\hat{\zeta },\hat{\xi }_{2}) \Biggr) \Biggr] \Biggr) \Biggr\} , $$

for $0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho } \leq \hat{\varrho }_{1}$, where

$$\begin{aligned}& \bar{q}_{2} ( \hat{\varsigma },\hat{\varrho } ) = \tilde{\Lambda } \bigl( a(\hat{\varsigma },\hat{\varrho }) \bigr) + \sum _{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1}p( \hat{\xi }_{1},\hat{\xi }_{2}), \\& \bar{F}(r)=\sum_{\hat{\xi }_{1}=r_{0}}^{r-1} \frac{1}{ ( ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) \circ \bar{G}^{-1} ) (\hat{\xi }_{1} ) },\quad r\geq r_{0}>0, \\& \bar{F}(+\infty )=\sum_{\hat{\xi }_{1}=r_{0}}^{+\infty } \frac{1}{ ( \omega \circ \tilde{\Phi } ^{-1} ) \circ \bar{G}^{-1}(\hat{\xi }_{1})}=+ \infty , \end{aligned}$$

and $( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega$ is chosen so that

$$ \Biggl( \bar{F} \bigl( \bar{q}_{2} ( \hat{\varsigma }, \hat{\varrho } ) \bigr) +\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1}\sum _{\hat{\xi }_{2}=0}^{\hat{\varrho }-1} f( \hat{\xi }_{1},\hat{ \xi }_{2}) \Biggl( 1+\sum_{\hat{\zeta }=0}^{ \hat{\xi }_{1}-1}g( \hat{\zeta },\hat{\xi }_{2}) \Biggr) \Biggr) \in \operatorname{Dom}\bigl( \bar{F}^{-1} \bigr) . $$

Theorem 2.14

Assume that g, a, f, u, Φ̃, and Ψ̃ are as in Theorem 2.3. If $u(\hat{\varsigma },\hat{\varrho })$ satisfies

$$\begin{aligned}& \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \\& \quad \leq a( \hat{ \varsigma },\hat{\varrho }) + \biggl( \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2}\Delta \hat{ \xi }_{1} \biggr) ^{2} \\& \qquad {} + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \biggl( \int _{0}^{\hat{\xi }_{1}}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}, \end{aligned}$$

(2.25)

for $(\hat{\varsigma },\hat{\varrho }) \in \Omega$, then

$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \biggl\{ \breve{H}^{-1} \biggl[ \breve{H} \bigl( a ( \hat{\varsigma }, \hat{ \varrho } ) \bigr) +\breve{B}(\hat{\varsigma }, \hat{\varrho }) + \biggl( \int _{0}^{\hat{\beta }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr) ^{2} \biggr] \biggr\} , $$

(2.26)

for $0\leq \hat{\varsigma } \leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho } \leq \hat{\varrho }_{1}$, where

$$\begin{aligned}& \breve{B}(\hat{\varsigma },\hat{\varrho }) = \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta }, \hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}, \end{aligned}$$

(2.27)

$$\begin{aligned}& \begin{aligned} &\breve{H}(r)= \int _{r_{0}}^{r} \frac{\Delta \hat{\xi }_{1}}{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) ^{2} ( \hat{\xi }_{1} ) },\quad r\geq r_{0}>0, \\ &\tilde{\Theta }(+\infty )= \int _{r_{0}}^{+\infty } \frac{\Delta \hat{\xi }_{1}}{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) ^{2} ( \hat{\xi }_{1} ) }=+\infty , \end{aligned} \end{aligned}$$

(2.28)

and $( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega$ is chosen so that

$$ \biggl( \breve{H} \bigl( a ( \hat{\varsigma },\hat{\varrho } ) \bigr) +B(\hat{ \varsigma },\hat{\varrho }) +2 \biggl( \int _{0}^{ \sigma (\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f( \hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr) ^{2} \biggr) \in \operatorname{Dom}\bigl( \breve{H}^{-1} \bigr) . $$

Proof

Assume that $a(\hat{\varsigma },\hat{\varrho }) >0$. Taking $(\breve{\xi },\breve{\zeta })\in \Omega$ as a fixed arbitrary point, we define $z(\hat{\varsigma },\hat{\varrho }) >0$ to be a nondecreasing function by

$$\begin{aligned}& z(\hat{\varsigma },\hat{\varrho }) \\& \quad =a(\breve{\xi },\breve{\zeta })+ \biggl( \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \biggr) ^{2} \end{aligned}$$

(2.29)

$$\begin{aligned}& \qquad {}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \biggl( \int _{0}^{\hat{\xi }_{1}}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}, \end{aligned}$$

(2.30)

for $0\leq \hat{\varsigma }\leq \breve{\xi }\leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho }\leq \breve{\zeta }\leq \hat{\varrho }_{1}$, hence $z(0,\hat{\varrho }) =z(\hat{\varsigma },0) =a(\breve{\xi },\breve{\zeta })$ and

$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \bigl( z( \hat{\varsigma },\hat{\varrho }) \bigr) . $$

From (2.29), and applying the chain rule on time scales (1.2), we get

$$\begin{aligned}& z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) \\& \quad = 2 \biggl( \int _{0}^{\hat{\alpha }(c)} \int _{0}^{\hat{\beta }(\hat{\varrho })}f( \hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2}\Delta \hat{ \xi }_{1} \biggr) \\& \qquad {}\times\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \bigl( u \bigl(\hat{ \alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \Delta \hat{ \xi }_{2} \\& \qquad {} +\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })} f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \bigl( u \bigl(\hat{ \alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \biggl( \int _{0}^{\hat{\alpha }(\hat{\varsigma })}g( \hat{\zeta } ,\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\& \quad \leq 2 \biggl( \int _{0}^{\hat{\alpha }(c)} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr) \\& \qquad {} \times \hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }( \hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \Delta \hat{\xi }_{2} \\& \qquad {} +\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }( \hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \\& \qquad {}\times\biggl( \int _{0}^{ \hat{\alpha }(\hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2}) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\& \quad \leq 2 \bigl( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl( \hat{\alpha }(\hat{\varsigma }),\hat{\beta }(\hat{\varrho }) \bigr) \bigr) \bigr) ^{2}\hat{\alpha }^{\Delta }(\hat{\varsigma }) \biggl( \int _{0}^{ \hat{\alpha }(c)} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr) \\& \qquad {}\times\int _{0}^{ \hat{\beta }(\hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \Delta \hat{\xi }_{2} \\& \qquad {} + \bigl( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl( \hat{\alpha }(\hat{\varsigma }),\hat{\beta }(\hat{\varrho }) \bigr) \bigr) \bigr) ^{2}\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{ \hat{\beta }(\hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\varsigma } }g(\hat{\zeta },\hat{\xi }_{2}) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}, \end{aligned}$$

thus we have

$$\begin{aligned} \frac{z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) }{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\varsigma },\hat{\varrho }) ) ) ^{2}} \leq &2 \biggl( \int _{0}^{\hat{\alpha }(c)} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \biggr) \hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }), \hat{\xi }_{2} \bigr) \Delta \hat{\xi }_{2} \\ &{}+\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\alpha }(\hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2}) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}, \\ =& \biggl[ \biggl( \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr)^{2} \biggr]^{\Delta _{ \hat{\varsigma }}} \\ &{}+\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\alpha }(\hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2}) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}. \end{aligned}$$

(2.31)

Integrating (2.31), we get

$$\begin{aligned} \breve{H} \bigl( z(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &\breve{H} \bigl( a(\breve{\xi },\breve{\zeta }) \bigr) + \biggl( \int _{0}^{ \hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f( \hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr) ^{2} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta } ,\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}. \end{aligned}$$

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

$$ z(\hat{\varsigma },\hat{\varrho }) \leq \breve{H}^{-1} \biggl[ \breve{H} \bigl( a(\hat{\varsigma },\hat{\varrho }) \bigr) +\breve{B}( \hat{ \varsigma },\hat{\varrho }) + \biggl( \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr) ^{2} \biggr] . $$

(2.32)

From (2.32) and $u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} ( z( \hat{\varsigma },\hat{\varrho }) )$, we get the desired inequality (2.26). For $a(\hat{\varsigma },\hat{\varrho }) =0$, we carry out the above procedure with $\epsilon >0$ instead of $a(\hat{\varsigma },\hat{\varrho })$ and subsequently let $\epsilon \rightarrow 0$. This completes the proof. □

Remark 2.15

If we take $\hat{\alpha }(\hat{\varsigma })= \hat{\varsigma }$ and $\hat{\alpha }(\hat{\varrho })= \hat{\varrho }$, then Theorem 2.14 reduces to [1, Theorem 10].

Theorem 2.16

If we take $\mathbb{T}=\mathbb{R}$ in Theorem 2.14, with the help of relations (1.1), we have the following inequality due to Boudeliou. If

$$\begin{aligned} \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &a( \hat{ \varsigma },\hat{\varrho }) + \biggl( \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \,d\hat{\xi }_{2}\,d\hat{\xi }_{1} \biggr) ^{2} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \biggl( \int _{0}^{\hat{\xi }_{1}}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \,d\hat{\zeta } \biggr) \,d\hat{\xi }_{2}\,d \hat{\xi }_{1}, \end{aligned}$$

for $(\hat{\varsigma },\hat{\varrho }) \in \Omega$, then

$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \biggl\{ \breve{H}^{-1} \biggl[ \breve{H} \bigl( a ( \hat{\varsigma }, \hat{ \varrho } ) \bigr) +\breve{B}(\hat{\varsigma }, \hat{\varrho }) + \biggl( \int _{0}^{\hat{\beta }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\,d\hat{\xi }_{2}\,d \hat{\xi }_{1} \biggr) ^{2} \biggr] \biggr\} , $$

for $0\leq \hat{\varsigma } \leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho } \leq \hat{\varrho }_{1}$, where

$$\begin{aligned}& \breve{B}(\hat{\varsigma },\hat{\varrho }) = \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta }, \hat{\xi }_{2})\,d\hat{\zeta } \biggr) \,d\hat{\xi }_{2}\,d\hat{\xi }_{1}, \\& \breve{H}(r)= \int _{r_{0}}^{r} \frac{d\hat{\xi }_{1}}{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) ^{2} ( \hat{\xi }_{1} ) },\quad r\geq r_{0}>0,\qquad \tilde{\Theta }(+\infty )= \int _{r_{0}}^{+\infty } \frac{d\hat{\xi }_{1}}{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) ^{2} ( \hat{\xi }_{1} ) }=+\infty , \end{aligned}$$

and $( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega$ is chosen so that

$$ \biggl( \breve{H} \bigl( a ( \hat{\varsigma },\hat{\varrho } ) \bigr) +B(\hat{ \varsigma },\hat{\varrho }) +2 \biggl( \int _{0}^{ \sigma (\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f( \hat{\xi }_{1}, \hat{\xi }_{2})\,d\hat{\xi }_{2}\,d\hat{\xi }_{1} \biggr) ^{2} \biggr) \in \operatorname{Dom}\bigl( \breve{H}^{-1} \bigr) . $$

Corollary 2.17

The discrete form, due to El-Deeb et al. [1], can be obtained by letting $\mathbb{T}=\mathbb{Z}$ and $\hat{\alpha }(\hat{\varsigma })=\hat{\varsigma }$, $\hat{\beta }(\hat{\varrho })=\hat{\varrho }$ in Theorem 2.14as follows. If

$$\begin{aligned} \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &a( \hat{ \varsigma },\hat{\varrho }) + \Biggl( \sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1} f( \hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \Biggr) ^{2} \\ &{}+\sum_{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1}\sum _{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1}f(\hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \Biggl( \sum_{\hat{\zeta }=0}^{ \hat{\xi }_{1}-1}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\zeta }, \hat{\xi }_{2}) \bigr) \Biggr) \end{aligned}$$

holds for $(\hat{\varsigma },\hat{\varrho }) \in \Omega$, then

$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \Biggl\{ \breve{H}^{-1} \Biggl[ \breve{H} \bigl( a ( \hat{\varsigma }, \hat{ \varrho } ) \bigr) +\breve{B}(\hat{\varsigma }, \hat{\varrho }) + \Biggl( \sum _{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1} \sum _{\hat{\xi }_{2}=0}^{\hat{\varrho }-1}f(\hat{\xi }_{1},\hat{\xi }_{2}) \Biggr) ^{2} \Biggr] \Biggr\} , $$

for $0\leq \hat{\varsigma } \leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho } \leq \hat{\varrho }_{1}$, where

$$\begin{aligned}& \breve{B}(\hat{\varsigma },\hat{\varrho }) =\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \Biggl( \sum _{\hat{\zeta }=0}^{\hat{\xi }_{1}-1}g( \hat{\zeta },\hat{\xi }_{2}) \Biggr), \\& \breve{H}(r)=\sum_{\hat{\xi }_{1}=r_{0}}^{r-1} \frac{1}{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) ^{2} ( \hat{\xi }_{1} ) },\quad r\geq r_{0}>0,\qquad \tilde{\Theta }(+\infty )=\sum_{\hat{\xi }_{1}=r_{0}}^{+\infty } \frac{1}{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) ^{2} ( \hat{\xi }_{1} ) }=+\infty , \end{aligned}$$

and $( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega$ is chosen so that

$$ \Biggl( \breve{H} \bigl( a ( \hat{\varsigma },\hat{\varrho } ) \bigr) +B(\hat{ \varsigma },\hat{\varrho }) + \Biggl( \sum_{ \hat{\xi }_{1}=0}^{\hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1} f(\hat{\xi }_{1},\hat{\xi }_{2}) \Biggr) ^{2} \Biggr) \in \operatorname{Dom}\bigl( \breve{H}^{-1} \bigr). $$

3 Applications

In this section we would like to show the beauty behind our results by applying Theorems 2.10 and 2.3 to study the boundedness of the solutions of some delay initial boundary value problems.

Consider the problem

$$\begin{aligned}& u^{\Delta \hat{\varsigma }\Delta \hat{\varrho }}(\hat{\varsigma }, \hat{\varrho }) =\tilde{\Theta } \biggl( \hat{\varsigma },\hat{\varrho },u \bigl( \hat{\alpha }(\hat{\varsigma }),\hat{ \beta }(\hat{\varrho }) \bigr) , \int _{0}^{\hat{\alpha }(\hat{\varsigma })}\breve{k} \bigl( \hat{\xi }_{1},\hat{\varrho },u ( s,\hat{\varrho } ) \bigr) \Delta \hat{\xi }_{1} \biggr) , \end{aligned}$$

(3.1)

$$\begin{aligned}& u ( \hat{\varsigma },0 ) =a_{1}(\hat{\varsigma }),\qquad u(0, \hat{\varrho }) =a_{2}(\hat{\varrho }),\qquad a_{1}(0)=a_{2}(0)=0, \end{aligned}$$

(3.2)

for any $(\hat{\varsigma },\hat{\varrho }) \in \Omega$, where $\breve{k}\in C_{\mathrm{rd}} ( \Omega \times \mathbb{R} ,\mathbb{R} )$, $\tilde{\Theta }\in C_{\mathrm{rd}} ( \Omega \times \mathbb{R} \times \mathbb{R} ,\mathbb{R} )$, $a_{1}\in C_{\mathrm{rd}} ( \mathbb{T}_{1},\mathbb{R} )$, and $a_{2}\in C_{\mathrm{rd}} ( \mathbb{T}_{2},\mathbb{R} )$.

Theorem 3.1

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

$$\begin{aligned}& \bigl\vert \tilde{\Theta } ( \hat{\varsigma },\hat{\varrho },u \bigl( \hat{ \alpha }(\hat{\varsigma }),\hat{\beta }(\hat{\varrho }),v \bigr) \bigr\vert \\& \quad \leq \tilde{\Psi } \bigl( \bigl\vert u \bigl( \hat{\alpha }(\hat{\varsigma }),\hat{ \beta }(\hat{\varrho } ) \bigr\vert \bigr) \bigl[ f ( \hat{\varsigma },\hat{ \varrho } ) \tilde{\Psi } ( \bigl\vert u \bigl(\hat{\alpha }( \hat{\varsigma }), \hat{\beta }(\hat{\varrho } ) \bigr\vert \bigr) +p ( \hat{\varsigma },\hat{ \varrho } ) \bigr] \\& \qquad {}+f ( \hat{\varsigma },\hat{\varrho } ) \tilde{\Psi } ( \bigl\vert u \bigl( \hat{\alpha }(\hat{\varsigma }),\hat{\beta }( \hat{\varrho } ) \bigr\vert \bigr) v, \end{aligned}$$

(3.3)

$$\begin{aligned}& \bigl\vert \breve{k} ( \hat{\varsigma },\hat{\varrho },u \bigl( \hat{\alpha }( \hat{\varsigma }),\hat{\beta }(\hat{\varrho } ) \bigr) \bigr\vert \leq g ( \hat{ \varsigma },\hat{\varrho } ) \tilde{\Psi } \bigl( \bigl\vert u \bigl( \hat{ \alpha }( \hat{\varsigma }),\hat{\beta }(\hat{\varrho } ) \bigr\vert \bigr) , \end{aligned}$$

(3.4)

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

(3.5)

where the functions p, g, a, f, α̂, β̂, and Ψ̃ are defined as in Theorem 2.10with $a(\hat{\varsigma },\hat{\varrho }) >0$, for all $(\hat{\varsigma },\hat{\varrho }) \in \Omega$. Then

$$\begin{aligned} \bigl\vert u ( \hat{\varsigma },\hat{\varrho } ) \bigr\vert \leq& \tilde{ \Lambda }^{-1} \biggl( \tilde{\Theta }^{-1} \biggl[ \tilde{\Theta } \bigl( q_{2}(\hat{\varsigma },\hat{\varrho }) \bigr) + \int _{0}^{\hat{\varsigma }} \int _{0}^{\hat{\varrho }} \frac{f(\hat{\alpha }^{-1}(\hat{\xi }_{1}), \hat{\beta }^{-1}(\hat{\xi }_{2}))}{\hat{\alpha }'(\hat{\alpha }^{-1} (\hat{\xi }_{1}))\hat{\beta }'(\hat{\beta }^{-1}(\hat{\xi }_{2}))} \\ &{}\times \biggl[ 1+ \int _{0}^{\hat{\xi }_{1}}g ( \hat{\zeta },\hat{\xi }_{2} ) \Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr] \biggr) , \end{aligned}$$

(3.6)

for $0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho } \leq \hat{\varrho }_{1}$, where F and G are defined as in Theorem 2.10,

$$ q_{2} \bigl(\hat{\varsigma },\hat{\varrho }=G \bigl(a(\hat{ \varsigma },\hat{\varrho }) \bigr) \bigr)+ \int _{0}^{\hat{\varsigma }} \int _{0}^{\hat{\varrho }} \frac{p(\hat{\alpha }^{-1}(\hat{\xi }_{1}), \hat{\beta }^{-1}(\hat{\xi }_{2}))}{\hat{\alpha }'(\hat{\alpha }^{-1}(\hat{\xi }_{1})) \hat{\beta }'(\hat{\beta }^{-1}(\hat{\xi }_{2}))} \Delta t \Delta s, $$

(3.7)

and $(\hat{\varsigma },\hat{\varrho }) \in \Omega$ is chosen so that

$$\begin{aligned}& \tilde{\Theta } \bigl( q_{2}(\hat{\varsigma },\hat{\varrho }) \bigr) + \int _{0}^{\hat{\varsigma }} \int _{0}^{\hat{\varrho }} \frac{f(\hat{\alpha }^{-1}(\hat{\xi }_{1}),\hat{\beta }^{-1}(\hat{\xi }_{2}))}{\hat{\alpha }'(\hat{\alpha }^{-1}(\hat{\xi }_{1}))\hat{\beta }'(\hat{\beta }^{-1}(\hat{\xi }_{2}))} \biggl[ 1+ \int _{0}^{\hat{\xi }_{1}}g ( \hat{\zeta },\hat{\xi }_{2} ) \Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \\& \quad \in \operatorname{Dom}\bigl(F^{-1} \bigr). \end{aligned}$$

Proof

If the problem (3.1) and (3.2) has a solution $u(\hat{\varsigma },\hat{\varrho })$, it can be written as

$$\begin{aligned} u(\hat{\varsigma },\hat{\varrho }) =&a_{1}(\hat{\varsigma })+a_{2}( \hat{\varrho }) \\ &{}+ \int _{0}^{\hat{\varsigma }} \int _{0}^{\hat{\varrho }} \tilde{\Theta } \biggl( \hat{\xi }_{1},\hat{\xi }_{2},u \bigl( \hat{\alpha }(\hat{\xi }_{1}),\hat{\beta }(\hat{\xi }_{2}) \bigr) , \int _{0}^{ \hat{\xi }_{1}}\breve{k} \bigl( \hat{\zeta }, \hat{\xi }_{2},u ( \hat{\zeta } ,t ) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}, \end{aligned}$$

(3.8)

for any $(\hat{\varsigma },\hat{\varrho }) \in \Omega$. Using the conditions (3.3), (3.4), and (3.5) in (3.8), we get

$$\begin{aligned} \bigl\vert u ( \hat{\varsigma },\hat{\varrho } ) \bigr\vert \leq &a(\hat{\varsigma },\hat{\varrho }) + \int _{0}^{ \hat{\varsigma }} \int _{0}^{\hat{\varrho }}\tilde{\Psi } \bigl( \bigl\vert u \bigl( \hat{\alpha }(\hat{\xi }_{1}),\hat{\beta }(\hat{\xi }_{2}) \bigr) \bigr\vert \bigr) \\ &{} \times \bigl[ f ( \hat{\xi }_{1},\hat{\xi }_{2} ) \tilde{\Psi } \bigl( \bigl\vert u \bigl(\hat{\alpha }( \hat{\xi }_{1}), \hat{\beta }(\hat{\xi }_{2}) \bigr) \bigr\vert \bigr)+p ( \hat{\xi }_{1},\hat{\xi }_{2} ) \bigr] \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \\ &{}+ \int _{0}^{\hat{\varsigma }} \int _{0}^{\hat{\varrho }}f ( s,t ) \tilde{\Psi } \bigl( \bigl\vert u \bigl(\hat{\alpha }( \hat{\xi }_{1}),\hat{\beta }( \hat{\xi }_{2}) \bigr) \bigr\vert \bigr) \\ &{}\times \biggl( \int _{0}^{\hat{\xi }_{1}}g ( \hat{\zeta }, \hat{\xi }_{2} ) \tilde{\Psi } \bigl( \bigl\vert u ( \hat{\zeta },\hat{\xi }_{2} ) \bigr\vert \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{ \xi }_{2}\Delta \hat{\xi }_{1}. \end{aligned}$$

(3.9)

Now, from (3.9), we get

$$\begin{aligned} \bigl\vert u (\hat{\varsigma },\hat{\varrho } ) \bigr\vert \leq &a(\hat{\varsigma },\hat{\varrho })+ \int _{0}^{ \hat{\alpha } (\hat{\varsigma })} \int _{0}^{\hat{\beta } (\hat{\varrho })} \frac{\varphi ( \vert u ( \hat{\xi }_{1},\hat{\xi }_{2} ) \vert ) }{\hat{\alpha } ^{\prime } ( \hat{\alpha } ^{-1}(\hat{\xi }_{1}) ) \hat{\beta } ^{\prime } ( \hat{\beta } ^{-1} ( \hat{\xi }_{2} ) ) } \\ &{}\times\bigl[ f \bigl( \hat{\alpha } ^{-1}(\hat{\xi }_{1}),\hat{\beta } ^{-1} ( \hat{\xi }_{2} ) \bigr) \varphi \bigl( \bigl\vert u ( \hat{\xi }_{1},\hat{\xi }_{2} ) \bigr\vert \bigr) \\ &{} +p \bigl( \hat{\alpha } ^{-1}(\hat{\xi }_{1}),\hat{ \beta } ^{-1} ( \hat{\xi }_{2} ) \bigr) \bigr] \Delta t \Delta s \\ &{}+ \int _{0}^{\hat{\alpha } (\hat{\varsigma })} \int _{0}^{\hat{\beta } ( \hat{\varrho })} \frac{f ( \hat{\alpha } ^{-1}(\hat{\xi }_{1}),\hat{\beta } ^{-1} ( \hat{\xi }_{2} ) ) }{\hat{\alpha } ^{\prime } ( \hat{\alpha } ^{-1}(\hat{\xi }_{1}) ) \hat{\beta } ^{\prime } ( \hat{\beta } ^{-1} ( \hat{\xi }_{2} ) ) }\varphi \bigl( \bigl\vert u ( \hat{\xi }_{1},\hat{\xi }_{2} ) \bigr\vert \bigr) \\ &{} \times \biggl( \int _{0}^{\hat{\xi }_{1}}g ( \hat{\zeta } , \hat{\xi }_{2} ) \varphi \bigl( \bigl\vert u ( \hat{\zeta } ,\hat{\xi }_{2} ) \bigr\vert \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{ \xi }_{2}\Delta \hat{\xi }_{1}, \end{aligned}$$

(3.10)

for any $(\hat{\varsigma },\hat{\varrho }) \in \Omega$. Now, an application of Theorem 2.10 to (3.10) yields the required inequality in (3.6). □

Consider the initial boundary value problem of the form

$$\begin{aligned}& \bigl(z^{q} \bigr)^{\Delta \hat{\varsigma } \Delta \hat{\varrho }}( \hat{\varsigma },\hat{\varrho }) =\breve{A} \biggl( \hat{\varsigma }, \hat{\varrho },z \bigl( \hat{\alpha }( \hat{\varsigma }),\hat{\beta }( \hat{\varrho }) \bigr) , \int _{0}^{\hat{\alpha }(\hat{\varsigma })}h \bigl( \hat{\xi }_{1},\hat{\varrho },z ( \hat{\xi }_{1}, \hat{\varrho } ) \bigr) \Delta \hat{\xi }_{1} \biggr) , \end{aligned}$$

(3.11)

$$\begin{aligned}& z ( \hat{\varsigma },0 ) =a_{1}(\hat{\varsigma }),\qquad z(0, \hat{\varrho }) =a_{2}(\hat{\varrho }),\qquad a_{1}(0)=a_{2}(0)=0, \end{aligned}$$

(3.12)

for any $(\hat{\varsigma },\hat{\varrho }) \in \Omega$.

Theorem 3.2

Assume that the functions h, Ă, $a_{2}$, $a_{1}$ in (3.11) and (3.12) satisfy the conditions

$$\begin{aligned}& \bigl\vert \breve{A} ( \hat{\varsigma },\hat{\varrho },z \bigl( \hat{\alpha }( \hat{\varsigma }),\hat{\beta }(\hat{\varrho }),v \bigr) \bigr\vert \leq f ( \hat{\varsigma },\hat{\varrho } ) \bigl\vert z^{r} \bigl(\hat{ \alpha }( \hat{\varsigma }),\hat{\beta }( \hat{\varrho }) \bigr) \bigr\vert +f ( \hat{\varsigma }, \hat{\varrho } ) v, \end{aligned}$$

(3.13)

$$\begin{aligned}& \bigl\vert h \bigl( \hat{\varsigma },\hat{\varrho },z ( \hat{\varsigma },\hat{ \varrho } ) \bigr) \bigr\vert \leq g ( \hat{\varsigma },\hat{\varrho } ) \bigl\vert z^{r} ( \hat{\varsigma },\hat{\varrho } ) \bigr\vert , \end{aligned}$$

(3.14)

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

(3.15)

where $r\geq q>0$. Then

$$\begin{aligned} \bigl\vert z(\hat{\varsigma },\hat{\varrho }) \bigr\vert \leq &\biggl[ \bigl( a(\hat{\varsigma },\hat{\varrho }) \bigr) ^{\frac{q-r}{q}}+ \frac{q-r}{q} \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })} \frac{f ( \hat{\alpha } ^{-1}(\hat{\xi }_{1}),\hat{\beta } ^{-1} ( \hat{\xi }_{2} ) ) }{\hat{\alpha } ^{\prime } ( \hat{\alpha } ^{-1}(\hat{\xi }_{1}) ) \hat{\beta } ^{\prime } ( \hat{\beta } ^{-1} ( \hat{\xi }_{2} ) ) } \\ &{}\times\biggl( 1+ \int _{0}^{\hat{\xi }_{1}}g ( \hat{\zeta },\hat{\xi }_{2} ) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr]^{ \frac{1}{q-r}}, \end{aligned}$$

(3.16)

for $0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}$, $0\leq \hat{\varrho } \leq \hat{\varrho }_{1}$.

Proof

If the problem (3.11) and (3.12) has a solution $z(\hat{\varsigma },\hat{\varrho })$, it can be written as

$$\begin{aligned} z^{q}(\hat{\varsigma },\hat{\varrho }) =&a_{1}(x)+a_{2}(y)+ \int _{0}^{ \hat{\varsigma }} \int _{0}^{\hat{\varrho }}\tilde{\Theta } \biggl( \hat{\xi }_{1},\hat{\xi }_{1},u \bigl( \hat{\alpha }(\hat{\xi }_{1}), \hat{\beta }(\hat{\xi }_{2}) \bigr) , \\ & \int _{0}^{\hat{\alpha }(\hat{\xi }_{1})} \breve{k} \bigl( \hat{\zeta }, \hat{\xi }_{2},u ( \hat{\zeta }, \hat{\xi }_{2} ) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}, \end{aligned}$$

(3.17)

for any $(\hat{\varsigma },\hat{\varrho }) \in \Omega$. Using the conditions (3.13), (3.14), and (3.15) in (3.17), we get

$$\begin{aligned} \bigl\vert z^{q} ( \hat{\varsigma },\hat{\varrho } ) \bigr\vert \leq &a(\hat{\varsigma },\hat{\varrho }) + \int _{0}^{ \hat{\varsigma }} \int _{0}^{\hat{\varrho }}f ( \hat{\xi }_{1}, \hat{\xi }_{2} ) \bigl\vert z^{r} \bigl( \hat{\alpha }(s), \hat{\beta }(t) \bigr) \bigr\vert \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \\ &{}+ \int _{0}^{\hat{\varsigma }} \int _{0}^{\hat{\varrho }}f ( \hat{\xi }_{1}, \hat{\xi }_{2} ) \biggl( \int _{0}^{\hat{\xi }_{1}}g ( \hat{\zeta },\hat{\xi }_{2} ) \bigl\vert z^{r} ( \hat{\zeta },\hat{\xi }_{2} ) \bigr\vert \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}. \end{aligned}$$

(3.18)

From (3.18), we get

$$\begin{aligned} \bigl\vert z^{q} ( \hat{\varsigma },\hat{\varrho } ) \bigr\vert \leq &a(\hat{\varsigma },\hat{\varrho }) + \int _{0}^{ \hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\alpha }(\hat{\varrho })} \frac{f ( \hat{\alpha } ^{-1}(\hat{\xi }_{1}),\hat{\beta } ^{-1} ( \hat{\xi }_{2} ) ) }{\hat{\alpha } ^{\prime } ( \hat{\alpha } ^{-1}(\hat{\xi }_{1}) ) \hat{\beta } ^{\prime } ( \hat{\beta } ^{-1} ( \hat{\xi }_{2} ) ) } \bigl\vert z^{r} ( \hat{\xi }_{1},\hat{\xi }_{2} ) \bigr\vert \Delta \hat{\xi }_{2}\Delta \hat{ \xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })} \frac{f ( \hat{\alpha } ^{-1}(\hat{\xi }_{1}),\hat{\beta } ^{-1} ( \hat{\xi }_{2} ) ) }{\hat{\alpha } ^{\prime } ( \hat{\alpha } ^{-1}(\hat{\xi }_{1}) ) \hat{\beta } ^{\prime } ( \hat{\beta } ^{-1} ( \hat{\xi }_{2} ) ) } \\ &{}\times \biggl( \int _{0}^{\hat{\xi }_{1}}g ( \hat{\zeta },\hat{\xi }_{2} ) \bigl\vert z^{r} ( \hat{\zeta },\hat{\xi }_{2} ) \bigr\vert \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}, \end{aligned}$$

(3.19)

for any $(\hat{\varsigma },\hat{\varrho }) \in \Omega$. A suitable application of Theorem 2.3 to (3.19) with $\tilde{\Phi } (u)=u^{q}$, $\tilde{\Psi } ( u ) =u^{r}$ and $p(\hat{\varsigma },\hat{\varrho }) =0$ gives the required inequality in (3.16). □

4 Conclusion

In this work, by using a new technique, we proved several nonlinear retarded dynamic inequalities in two independent variables of Gronwall type on time scales. We also gave a new proof and formula of Leibniz integral rule on time scales. Further, we also applied our inequalities to discrete and continuous calculus to obtain some new inequalities as special cases. Furthermore, we studied the boundedness of some delay initial value problems by applying our results.