Skip to main content

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.

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 [536].

We use the following notations:

  1. (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)
  2. (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 [3740]. 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.

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 \(\forall t\in \mathbb{T^{\kappa }}\):

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

  4. (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

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

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 , 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). □

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.

Availability of data and materials

Not applicable.

References

  1. 1.

    El-Deeb, A.A., Khan, Z.A.: Certain new dynamic nonlinear inequalities in two independent variables and applications. Bound. Value Probl. 2020(1), 31 (2020)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Hilger, S.: Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Ph.D. thesis, Universität Würzburg (1988)

  3. 3.

    Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)

    Google Scholar 

  4. 4.

    Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)

    Google Scholar 

  5. 5.

    Abdeldaim, A., El-Deeb, A.A., Agarwal, P., El-Sennary, H.A.: On some dynamic inequalities of Steffensen type on time scales. Math. Methods Appl. Sci. 41(12), 4737–4753 (2018)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Agarwal, R., O’Regan, D., Saker, S.: Dynamic Inequalities on Time Scales. Springer, Cham (2014)

    Google Scholar 

  7. 7.

    Akin-Bohner, E., Bohner, M., Akin, F.: Pachpatte inequalities on time scales. JIPAM. J. Inequal. Pure Appl. Math. 6(1), Article ID 23 (2005)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Bohner, M., Matthews, T.: The Grüss inequality on time scales. Commun. Math. Anal. 3(1), 1–8 (2007)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Bohner, M., Matthews, T.: Ostrowski inequalities on time scales. JIPAM. J. Inequal. Pure Appl. Math. 9(1), Article ID 8 (2008)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Dinu, C.: Hermite–Hadamard inequality on time scales. J. Inequal. Appl. 2008, Article ID 287947 (2008)

    MathSciNet  Article  Google Scholar 

  11. 11.

    El-Deeb, A.A.: On some generalizations of nonlinear dynamic inequalities on time scales and their applications. Appl. Anal. Discrete Math. 13(2), 440–462 (2019)

    MathSciNet  Article  Google Scholar 

  12. 12.

    El-Deeb, A.A., Cheung, W.-S.: A variety of dynamic inequalities on time scales with retardation. J. Nonlinear Sci. Appl. 11(10), 1185–1206 (2018)

    MathSciNet  Article  Google Scholar 

  13. 13.

    El-Deeb, A.A., El-Sennary, H.A., Khan, Z.A.: Some Steffensen-type dynamic inequalities on time scales. Adv. Differ. Equ. 2019, 246 (2019)

    MathSciNet  Article  Google Scholar 

  14. 14.

    El-Deeb, A.A., Elsennary, H.A., Cheung, W.-S.: Some reverse Hölder inequalities with Specht’s ratio on time scales. J. Nonlinear Sci. Appl. 11(4), 444–455 (2018)

    MathSciNet  Article  Google Scholar 

  15. 15.

    El-Deeb, A.A., Elsennary, H.A., Nwaeze, E.R.: Generalized weighted Ostrowski, trapezoid and Grüss type inequalities on time scales. Fasc. Math. 60, 123–144 (2018)

    MATH  Google Scholar 

  16. 16.

    El-Deeb, A.A., Xu, H., Abdeldaim, A., Wang, G.: Some dynamic inequalities on time scales and their applications. Adv. Differ. Equ. 2019, 130 (2019)

    MathSciNet  Article  Google Scholar 

  17. 17.

    El-Deeb, A.A.: Some Gronwall–Bellman type inequalities on time scales for Volterra–Fredholm dynamic integral equations. J. Egypt. Math. Soc. 26(1), 1–17 (2018)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Hilscher, R.: A time scales version of a Wirtinger-type inequality and applications. J. Comput. Appl. Math. 141(1–2), 219–226 (2002)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Li, W.N.: Some delay integral inequalities on time scales. Comput. Math. Appl. 59(6), 1929–1936 (2010)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Řehák, P.: Hardy inequality on time scales and its application to half-linear dynamic equations. J. Inequal. Appl. 2005, 942973 (2005)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Saker, S.H., El-Deeb, A.A., Rezk, H.M., Agarwal, R.P.: On Hilbert’s inequality on time scales. Appl. Anal. Discrete Math. 11(2), 399–423 (2017)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Tian, Y., El-Deeb, A.A., Meng, F.: Some nonlinear delay Volterra–Fredholm type dynamic integral inequalities on time scales. Discrete Dyn. Nat. Soc. 2018, Article ID 5841985 (2018)

    MathSciNet  Article  Google Scholar 

  23. 23.

    El-Deeb, A.A., Kh, F.M., Ismail, G.A.F., Khan, Z.A.: Weighted dynamic inequalities of Opial-type on time scales. Adv. Differ. Equ. 2019(1), 393 (2019)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Kh, F.M., El-Deeb, A.A., Abdeldaim, A., Khan, Z.A.: On some generalizations of dynamic Opial-type inequalities on time scales. Adv. Differ. Equ. 2019(1), 323 (2019)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Abdeldaim, A., El-Deeb, A.A.: Some new retarded nonlinear integral inequalities with iterated integrals and their applications in retarded differential equations and integral equations. J. Fract. Calc. Appl. 5(suppl. 3S), Paper no. 9 (2014)

    MathSciNet  Google Scholar 

  26. 26.

    Abdeldaim, A., El-Deeb, A.A.: On generalized of certain retarded nonlinear integral inequalities and its applications in retarded integro-differential equations. Appl. Math. Comput. 256, 375–380 (2015)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Abdeldaim, A., El-Deeb, A.A.: On some generalizations of certain retarded nonlinear integral inequalities with iterated integrals and an application in retarded differential equation. J. Egypt. Math. Soc. 23(3), 470–475 (2015)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Abdeldaim, A., El-Deeb, A.A.: On some new nonlinear retarded integral inequalities with iterated integrals and their applications in integro-differential equations. Br. J. Math. Comput. Sci. 5(4), 479–491 (2015)

    Article  Google Scholar 

  29. 29.

    Agarwal, R.P., Lakshmikantham, V.: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. Series in Real Analysis, vol. 6. World Scientific, Singapore (1993)

    Google Scholar 

  30. 30.

    El-Deeb, A.A.: On Integral Inequalities and Their Applications. LAP Lambert Academic Publishing, Saarbrücken (2017)

    Google Scholar 

  31. 31.

    El-Deeb, A.A.: A variety of nonlinear retarded integral inequalities of Gronwall type and their applications. In: Advances in Mathematical Inequalities and Applications, pp. 143–164. Springer, Berlin (2018)

    Google Scholar 

  32. 32.

    El-Deeb, A.A., Ahmed, R.G.: On some explicit bounds on certain retarded nonlinear integral inequalities with applications. Adv. Inequal. Appl. 2016, Article ID 15 (2016)

    Article  Google Scholar 

  33. 33.

    El-Deeb, A.A., Ahmed, R.G.: On some generalizations of certain nonlinear retarded integral inequalities for Volterra–Fredholm integral equations and their applications in delay differential equations. J. Egypt. Math. Soc. 25(3), 279–285 (2017)

    MathSciNet  Article  Google Scholar 

  34. 34.

    El-Owaidy, H., Abdeldaim, A., El-Deeb, A.A.: On some new retarded nonlinear integral inequalities and their applications. Math. Sci. Lett. 3(3), 157–164 (2014)

    Article  Google Scholar 

  35. 35.

    El-Owaidy, H.M., Ragab, A.A., Eldeeb, A.A., Abuelela, W.M.K.: On some new nonlinear integral inequalities of Gronwall–Bellman type. Kyungpook Math. J. 54(4), 555–575 (2014)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Li, J.D.: Opial-type integral inequalities involving several higher order derivatives. J. Math. Anal. Appl. 167(1), 98–110 (1992)

    MathSciNet  Article  Google Scholar 

  37. 37.

    El-Deeb, A.A., Makharesh, S.D., Baleanu, D.: Dynamic Hilbert-type inequalities with Fenchel–Legendre transform. Symmetry 12(4), 582 (2020)

    Article  Google Scholar 

  38. 38.

    El-Deeb, A.A., Baleanu, D.: New weighted Opial-type inequalities on time scales for convex functions. Symmetry 12(5), 842 (2020)

    Article  Google Scholar 

  39. 39.

    El-Deeb, A.A., El-Sennary, H.A., Khan, Z.A.: Some reverse inequalities of Hardy type on time scales. Adv. Differ. Equ. 2020(1), 402 (2020)

    MathSciNet  Article  Google Scholar 

  40. 40.

    El-Deeb, A.A., Elsennary, H.A., Baleanu, D.: Some new Hardy-type inequalities on time scales. Adv. Differ. Equ. 2020(1), 441 (2020)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Ammar, B.: On certain new nonlinear retarded integral inequalities in two independent variables and applications. Appl. Math. Comput. 2019(335), 103–111 (2018)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editor and reviewers for their valuable comments which improved the paper.

Funding

Not applicable.

Author information

Affiliations

Authors

Contributions

The authors have read and finalized the manuscript with equal contribution. All authors read and approved the final manuscript.

Corresponding author

Correspondence to A. A. El-Deeb.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

El-Deeb, A.A., Rashid, S. On some new double dynamic inequalities associated with Leibniz integral rule on time scales. Adv Differ Equ 2021, 125 (2021). https://doi.org/10.1186/s13662-021-03282-3

Download citation

Keywords

  • Gronwall-type inequality
  • Boundedness
  • Time scales
\