Throughout the paper, we suppose that \(\mathbb{T}_{1}\) and \(\mathbb{T}_{2}\) are two time scales.
First, we prove the following result.
Theorem 2.1
(Leibniz integral rule on time scales)
In the following by \(f^{\Delta }(t,s)\) we mean the delta derivative of \(f(t,s)\) with respect to t. Similarly, \(f^{\nabla }(t,s)\) is understood. If f, \(f^{\Delta }\) and \(f^{\nabla }\) are continuous, and \(u,h:\mathbb{T}\rightarrow \mathbb{T}\) are delta differentiable functions, then the following formulas hold \(\forall t\in \mathbb{T^{\kappa }}\):
-
(i)
\([\int ^{h(t)}_{u(t)}f(t,s)\Delta s ]^{\Delta }=\int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + h^{\Delta }(t)f(\sigma (t),h(t))- u^{\Delta }(t)f(\sigma (t),u(t))\);
-
(ii)
\([\int ^{h(t)}_{u(t)}f(t,s)\Delta s ]^{\nabla }= \int ^{h(t)}_{u(t)}f^{\nabla }(t,s)\Delta s + h^{\nabla }(t)f(\rho (t),h(t))- u^{\nabla }(t)f(\rho (t),u(t))\);
-
(iii)
\([\int ^{h(t)}_{u(t)}f(t,s)\nabla s ]^{\Delta }= \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\nabla s + h^{\Delta }(t)f(\sigma (t),h(t))- u^{\Delta }(t)f(\sigma (t),u(t)) \);
-
(iv)
\([\int ^{h(t)}_{u(t)}f(t,s)\nabla s ]^{\nabla }= \int ^{h(t)}_{u(t)}f^{\nabla }(t,s)\nabla s + h^{\nabla }(t)f(\rho (t),h(t))- u^{\nabla }(t)f(\rho (t),u(t)) \).
Proof
We will only prove part (i); the others may be proved similarly. Define a function g by
$$ g(t) = \int _{u(t)}^{h(t)}f(t,s)\Delta s,\quad \text{for } t\in \mathbb{T^{\kappa }}. $$
(2.1)
We notice that g is a continuous function. Indeed, we have two cases for t. In the first case, if t is right-scattered, from (2.1), we get
$$\begin{aligned} g^{\Delta }(t) =& \frac{g(\sigma (t))-g(t)}{\sigma (t)-t} \\ =& \frac{1}{\sigma (t)-t} \biggl[ \int ^{h(\sigma (t))}_{u(\sigma (t))}f \bigl( \sigma (t),s \bigr)\Delta s - \int ^{h(t)}_{u(t)}f(t,s)\Delta s \biggr] \\ =& \frac{1}{\sigma (t)-t} \biggl[- \int _{u(t)}^{u(\sigma (t))}f \bigl( \sigma (t),s \bigr)\Delta s + \int ^{h(t)}_{u(t)}f \bigl(\sigma (t),s \bigr)\Delta s \\ &{}+ \int ^{h(\sigma (t))}_{h(t)}f \bigl(\sigma (t),s \bigr)\Delta s- \int ^{h(t)}_{u(t)}f(t,s) \Delta s \biggr] \\ =& \int ^{h(t)}_{u(t)}\frac{f(\sigma (t),s) - f(t,s)}{\sigma (t)-t} \Delta s + \frac{1}{\sigma (t)-t} \int ^{h(\sigma (t))}_{h(t)}f \bigl( \sigma (t),s \bigr)\Delta s \\ &{}- \frac{1}{\sigma (t)-t} \int _{u(t)}^{u(\sigma (t))}f \bigl(\sigma (t),s \bigr) \Delta s \\ =& \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + \frac{h(\sigma (t))-h(t)}{\sigma (t)-t}f \bigl(\sigma (t),h(t) \bigr) \\ &{}- \frac{u(\sigma (t))-u(t)}{\sigma (t)-t}f \bigl(\sigma (t),u(t) \bigr) \\ =& \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + h^{\Delta }(t)f \bigl(\sigma (t),h(t) \bigr) -u^{\Delta }(t)f \bigl( \sigma (t),u(t) \bigr). \end{aligned}$$
(2.2)
From (2.2), we get the required result.
Now consider the second case when t is right-dense. Since f is continuous, it is rd-continuous, hence it has a delta partial anti-derivative with respect to the second variable s, say \(F(t,s)\), that is, \(f(t,s)=F^{\Delta _{s}}(t,s)\), and then we have
$$\begin{aligned} \biggl[ \int ^{h(t)}_{u(t)}f(t,s)\Delta s \biggr]^{\Delta } =& g^{\Delta }(t) \\ =& \lim_{r\to t}\frac{g(t)-g(r)}{t - r} \\ =& \lim_{r\to t}\frac{1}{t - r} \biggl[ \int ^{h(t)}_{u(t)}f(t,s) \Delta s- \int ^{h(r)}_{u(r)}f(r,s)\Delta s \biggr] \\ =& \lim_{r\to t}\frac{1}{t - r} \biggl[ \int ^{h(t)}_{u(t)}f(t,s) \Delta s - \int _{u(r)}^{u(t)}f(r,s)\Delta s \\ &{}- \int _{u(t)}^{h(t)}f(r,s)\Delta s - \int _{h(t)}^{h(r)}f(r,s) \Delta s \biggr] \\ =& \lim_{r\to t} \int ^{h(t)}_{u(t)}\frac{f(t,s)-f(r,s)}{t - r} \Delta s + \lim_{r\to t}\frac{1}{t - r} \int _{h(r)}^{h(t)}F^{\Delta _{s}}(r,s) \Delta s \\ &{}- \lim_{r\to t}\frac{1}{t - r} \int _{u(r)}^{u(t)}F^{\Delta _{s}}(r,s) \Delta s. \end{aligned}$$
(2.3)
Thus, from (2.3), we get
$$\begin{aligned} \biggl[ \int ^{h(t)}_{u(t)}f(t,s)\Delta s \biggr]^{\Delta } =& \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + \lim_{r\to t}\frac{1}{t - r} \bigl[F \bigl(r,h(t) \bigr)-F \bigl(r,h(r) \bigr) \bigr] \\ &{}- \lim_{r\to t}\frac{1}{t - r} \bigl[F \bigl(r,u(t) \bigr)-F \bigl(r,u(r) \bigr) \bigr] \\ =& \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + \lim_{r\to t} \frac{h(t)-h(r)}{t - r}\frac{F(r,h(t))-F(r,h(r))}{h(t)-h(r)} \\ &{}- \lim_{r\to t}\frac{u(t)-u(r)}{t - r} \frac{F(r,u(t))-F(r,u(r))}{u(t)-u(r)} \\ =& \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + \lim_{r\to t} \frac{h(t)-h(r)}{t - r}\lim_{r\to t} \frac{F(r,h(t))-F(r,h(r))}{h(t)-h(r)} \\ &{}- \lim_{r\to t}\frac{u(t)-u(r)}{t - r}\lim_{r\to t} \frac{F(r,u(t))-F(r,u(r))}{u(t)-u(r)} \\ =& \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + h^{\Delta }(t)F^{ \Delta _{s}} \bigl(t,h(t) \bigr) - u^{\Delta }(t)F^{\Delta _{s}} \bigl(t,u(t) \bigr) \\ =& \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + h^{\Delta }(t)f \bigl(t,h(t) \bigr)- u^{\Delta }(t)f \bigl(t,u(t) \bigr). \end{aligned}$$
This completes the proof. □
Remark 2.2
If we take \(h(t)=t\) and \(u(t)=a\) (where a is constant), then Theorem 2.1 reduces to [4, Theorem 5.37, p. 139].
Now, by using the result of Theorem 2.1, we state and prove the rest of our main results:
Theorem 2.3
Suppose \(a\in C_{\mathrm{rd}}(\Omega ,\mathbb{R}_{+})\) is nondecreasing with respect to \((\hat{\varsigma },\hat{\varrho }) \in \Omega \), and g, u, p, \(f\in C_{\mathrm{rd}}(\Omega ,\mathbb{R}_{+})\). Also let \(\hat{\alpha }\in C^{1}_{\mathrm{rd}} ( \mathbb{T}_{1},\mathbb{T}_{1} )\) and \(\hat{\beta }\in C^{1}_{\mathrm{rd}} ( \mathbb{T}_{2},\mathbb{T}_{2} ) \) be nondecreasing functions with \(\hat{\alpha }(\hat{\varsigma })\leq \hat{\varsigma }\) on \(\mathbb{T}_{1}\), \(\hat{\beta }(\hat{\varrho })\leq \hat{\varrho }\) on \(\mathbb{T}_{2}\). Furthermore, suppose Φ̃, \(\tilde{\Psi } \in C(\mathbb{R}_{+},\mathbb{R}_{+})\) are nondecreasing functions with \(\{ \tilde{\Phi } ,\tilde{\Psi } \} (u)>0\) for \(u>0\), and \(\underset{u\rightarrow +\infty }{\lim }\tilde{\Phi } (u)=+\infty \). If \(u(\hat{\varsigma },\hat{\varrho }) \) satisfies
$$\begin{aligned} \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &a( \hat{ \varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })} \bigl[ f( \hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \end{aligned}$$
(2.4)
for \((\hat{\varsigma },\hat{\varrho }) \in \Omega \), then
$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \biggl\{ \tilde{\Lambda }^{-1} \biggl[ \tilde{\Lambda } \bigl( q(\hat{ \varsigma }, \hat{\varrho }) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2}) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr] \biggr\} $$
(2.5)
for \(0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varrho } \leq \hat{\varrho }_{1}\), where
$$\begin{aligned}& q(\hat{\varsigma },\hat{\varrho }) =a(\hat{\varsigma },\hat{ \varrho }) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}p(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} , \end{aligned}$$
(2.6)
$$\begin{aligned}& \tilde{\Lambda }(r)= \int _{r_{0}}^{r} \frac{\Delta \hat{\xi }_{1}}{\omega \circ \tilde{\Phi } ^{-1}(\hat{\xi }_{1})},\quad r \geq r_{0}>0,\qquad \tilde{\Lambda }(+\infty )= \int _{r_{0}}^{+\infty } \frac{\Delta \hat{\xi }_{1}}{\omega \circ \tilde{\Phi } ^{-1}(\hat{\xi }_{1})}=+\infty , \end{aligned}$$
(2.7)
and \(( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega \) is chosen so that
$$ \biggl( \tilde{\Lambda } \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \biggr) \in \operatorname{Dom}\bigl( G^{-1} \bigr) . $$
Proof
Assume that \(a ( \hat{\varsigma },\hat{\varrho } ) >0\). Since \(q\geq 0\) and it is nondecreasing, fixing an arbitrary point \((\breve{\xi },\breve{\zeta }) \in \Omega \) and defining \(z(\hat{\varsigma },\hat{\varrho }) \) by
$$\begin{aligned} z(\hat{\varsigma },\hat{\varrho }) =&q(\breve{\xi },\breve{\zeta }) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2}\Delta \hat{ \xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}, \end{aligned}$$
which is a positive and nondecreasing function for \(0\leq \hat{\varsigma }\leq \breve{\xi }\leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varsigma }\leq \breve{\zeta }\leq \hat{\varrho }_{1}\), we then get \(z(0,\hat{\varrho }) =z(\hat{\varsigma },0) =q(\breve{\xi },\breve{\zeta }) \) and
$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \bigl( z( \hat{\varsigma },\hat{\varrho }) \bigr) . $$
(2.8)
By applying Theorem 2.1, differentiating \(z(\hat{\varsigma },\hat{\varrho }) \) with respect to ς̂, and using (2.8), we get
$$\begin{aligned}& z^{\Delta }_{ \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) \\& \quad = \hat{ \alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl[ \tilde{\Psi } \bigl( u \bigl(\hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2}) \tilde{\Psi } \bigl( u(\hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2} \\& \quad \leq \hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl[ \tilde{\Psi } \circ \tilde{ \Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }( \hat{\varsigma }), \hat{\xi }_{2} \bigr) \bigr) \\& \qquad {}+ \int _{0}^{\hat{\alpha }( \hat{\varsigma })}g(\hat{\zeta } ,\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2}. \end{aligned}$$
Since \(\tilde{\Psi } \circ \tilde{\Phi } ^{-1}\) is nondecreasing with respect to \((\hat{\varsigma },\hat{\varrho }) \in \mathbb{R} _{+}\times \mathbb{R} _{+}\), we then have
$$\begin{aligned}& z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) \\& \quad \leq \hat{\alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl[ \tilde{\Psi } \circ \tilde{ \Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }(\hat{\varsigma }), \hat{\xi }_{2} \bigr) \bigr) \\& \qquad {}+\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl( \hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \int _{0}^{ \hat{\alpha }(\hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2} \\& \quad \leq \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{ \alpha }( \hat{\varsigma }),\hat{\beta }(\hat{\varrho }) \bigr) \bigr)\hat{\alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl[ 1+ \int _{0}^{ \hat{\alpha }(\hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2}, \end{aligned}$$
(2.9)
from which \(\tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\alpha }( \hat{\varsigma }),\hat{\beta }(\hat{\varrho })) )\leq \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\varsigma },\hat{\varrho }) )\), so from (2.9), we get
$$ \frac{z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) }{\tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\varsigma },\hat{\varrho }) ) }\leq \hat{\alpha }^{\Delta }( \hat{ \varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( 1+ \int _{0}^{\hat{\alpha }( \hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}. $$
(2.10)
Now from (2.10), we get
$$ \tilde{\Lambda } \bigl( z(\hat{\varsigma },\hat{\varrho }) \bigr) \leq \tilde{ \Lambda } \bigl( q(\breve{\xi },\breve{\zeta }) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}. $$
Since \((\breve{\xi },\breve{\zeta }) \in \Omega \) is chosen arbitrarily,
$$ z(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Lambda }^{-1} \biggl[ \tilde{\Lambda } \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \biggr] . $$
(2.11)
So from (2.11) and (2.8), we get the desired inequality in (2.5). For \(a(\hat{\varsigma },\hat{\varrho }) =0\), we carry out the above procedure with \(\epsilon >0\) instead of \(a(\hat{\varsigma },\hat{\varrho }) \) and subsequently let \(\epsilon \rightarrow 0\). This completes the proof. □
Remark 2.4
If we take \(\hat{\alpha }(\hat{\varsigma })= \hat{\varsigma }\) and \(\hat{\alpha }(\hat{\varrho })= \hat{\varrho }\), then Theorem 2.3 reduces to [1, Theorem 2.1].
Corollary 2.5
The discrete form can be obtained by letting \(\mathbb{T}=\mathbb{Z}\), with the help of relations (1.2), and \(\hat{\alpha }(\hat{\varsigma })=\hat{\varsigma }\), \(\hat{\beta }(\hat{\varrho })=\hat{\varrho }\) in Theorem 2.3. If
$$\begin{aligned} \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &a( \hat{ \varsigma },\hat{\varrho }) +\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1} \bigl[ f( \hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \\ &{}+\sum_{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1}\sum _{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1}f(\hat{\xi }_{1},\hat{\xi }_{2}) \Biggl( \sum_{ \hat{\zeta }=0}^{\hat{\xi }_{1}-1} g(\hat{\zeta },\hat{\xi }_{2}) \tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Biggr) \end{aligned}$$
holds for \((\hat{\varsigma },\hat{\varrho }) \in \Omega \), then
$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \Biggl\{ \tilde{\Lambda }^{-1} \Biggl[ \tilde{\Lambda } \bigl( q(\hat{ \varsigma }, \hat{\varrho }) \bigr) +\sum_{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1}f(\hat{\xi }_{1},\hat{\xi }_{2}) \Biggl( 1+ \sum _{\hat{\zeta }=0}^{\hat{\xi }_{1}-1}g(\hat{\zeta }, \hat{\xi }_{2}) \Biggr) \Biggr] \Biggr\} $$
for \(0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varrho } \leq \hat{\varrho }_{1}\), where
$$\begin{aligned}& q(\hat{\varsigma },\hat{\varrho }) =a(\hat{\varsigma },\hat{\varrho }) + \sum _{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1}\sum _{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1}p(\hat{\xi }_{1},\hat{\xi }_{2}), \\& \tilde{\Lambda }(r)=\sum_{\hat{\xi }_{1}=r_{0}}^{r-1} \frac{1}{\omega \circ \tilde{\Phi } ^{-1}(\hat{\xi }_{1})},\quad r\geq r_{0}>0,\qquad \tilde{\Lambda }(+\infty )=\sum_{\hat{\xi }_{1}=r_{0}}^{+ \infty } \frac{1}{\omega \circ \tilde{\Phi } ^{-1}(\hat{\xi }_{1})}=+\infty , \end{aligned}$$
and \(( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega \) is chosen so that
$$ \Biggl( \tilde{\Lambda } \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) +\sum _{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1}\sum _{ \breve{st}=0}^{\hat{\varrho }-1}f(\hat{\xi }_{1},\hat{\xi }_{2}) \Biggl( 1+ \sum_{\hat{\zeta }0}^{\hat{\xi }_{1}}g( \hat{\zeta },\hat{\xi }_{2}) \Biggr) \Biggr) \in \operatorname{Dom}\bigl( G^{-1} \bigr) . $$
Theorem 2.6
Assume that h, \(b\in C_{\mathrm{rd}}(\Omega ,\mathbb{R} _{+})\). Let g, f, p, a, u, Φ̃, and Ψ̃ be as in Theorem 2.3. If \(u(\hat{\varsigma },\hat{\varrho }) \) satisfies
$$\begin{aligned} \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &a( \hat{ \varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })} \bigl[ f( \hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}b(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl[ h(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \\ &{} + \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2}) \tilde{\Psi } \bigl( u(\hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \end{aligned}$$
(2.12)
for \((\hat{\varsigma },\hat{\varrho }) \in \Omega \), then
$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \biggl\{ G^{-1} \biggl[ G \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) +A( \hat{\varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr] \biggr\} $$
(2.13)
for \(0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varrho } \leq \hat{\varrho }_{1}\), where Λ̃ is defined by (2.7),
$$ \breve{A}(\hat{\varsigma },\hat{\varrho }) = \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}b(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl[ h(\hat{\xi }_{1},\hat{\xi }_{2})+ \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} ,$$
(2.14)
and \(( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega \) is chosen so that
$$ \biggl( \tilde{\Lambda } \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) + \breve{A}(\hat{\varsigma },\hat{\varrho }) + \int _{0}^{ \hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f( \hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr) \in \operatorname{Dom}\bigl( \tilde{\Lambda }^{-1} \bigr) . $$
Proof
Assume that \(a(\hat{\varsigma },\hat{\varrho }) >0\). Fixing an arbitrary \((\breve{\xi },\breve{\zeta }) \in \Omega \), we define a positive and nondecreasing function \(z(\hat{\varsigma },\hat{\varrho }) \) by
$$\begin{aligned} z(\hat{\varsigma },\hat{\varrho }) =&q(\breve{\xi },\breve{\zeta }) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2}\Delta \hat{ \xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}b(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl[ h(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \\ &{}+ \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2}) \tilde{\Psi } \bigl( u(\hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \end{aligned}$$
for \(0\leq \hat{\varsigma }\leq \breve{\xi }\leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varrho }\leq \breve{\zeta }\leq y_{1}\), then \(z(0,\hat{\varrho }) =z(\hat{\varsigma },0) =q(\breve{\xi },\breve{\zeta }) \) and
$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \bigl( z( \hat{\varsigma },\hat{\varrho }) \bigr). $$
Now, by applying Theorem 2.1, we have
$$\begin{aligned}& z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) \\& \quad = \hat{\alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \bigl( u \bigl( \hat{ \alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \Delta \hat{ \xi }_{2}+\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}b \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \\& \qquad {} \times \biggl( h \bigl( \hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \bigl( u \bigl( \hat{ \alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) + \int _{0}^{ \hat{\varsigma } }g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\& \quad \leq \hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }( \hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \Delta \hat{\xi }_{2}+ \int _{0}^{ \hat{\beta }(\hat{\varrho })}b \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \\& \qquad {}\times \biggl( h \bigl(\hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{ \alpha }(\hat{\varsigma }), \hat{\xi }_{2} \bigr) \bigr) + \int _{0}^{\hat{\varsigma } }g(\hat{\zeta }, \hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\& \quad \leq \hat{\alpha }^{\Delta }(\hat{\varsigma })\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z(\hat{\varsigma },\hat{\varrho }) \bigr) \biggl[ \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) \Delta \hat{\xi }_{2} \\& \qquad {} + \int _{0}^{\hat{\beta }(\hat{\varrho })}b \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( h \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) + \int _{0}^{\hat{\varsigma } }g( \hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \biggr] \Delta \hat{\xi }_{2}. \end{aligned}$$
Since \(\tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\varsigma }, \hat{\varrho }) ) \leq \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\varsigma },\hat{\varrho }) ) \), we then get
$$\begin{aligned}& \frac{z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) }{\tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\varsigma },\hat{\varrho }) ) } \\& \quad \leq \hat{\alpha }^{\Delta }( \hat{ \varsigma }) \biggl[ \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \Delta \hat{\xi }_{2} \\& \qquad {}+ \int _{0}^{\hat{\beta }(\hat{\varrho })}b \bigl(\hat{\alpha }(\hat{ \varsigma }), \hat{\xi }_{2} \bigr) \biggl( h \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2}) \Delta \hat{\zeta } \biggr) \biggr] \Delta \hat{\xi }_{2}. \end{aligned}$$
(2.15)
Integrating (2.15), we get
$$ \tilde{\Lambda } \bigl( z(\hat{\varsigma },\hat{\varrho }) \bigr) \leq \tilde{ \Lambda } \bigl( q(\breve{\xi },\breve{\zeta }) \bigr) + \breve{A}(\hat{\varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}. $$
Since \((\breve{\xi },\breve{\zeta }) \in \Omega \) is chosen arbitrarily,
$$ z(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Lambda }^{-1} \biggl[ \tilde{\Lambda } \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) + \breve{A}(\hat{\varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr] . $$
(2.16)
Thus, from (2.16) and \(u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} ( z( \hat{\varsigma },\hat{\varrho }) ) \), we get the required inequality in (2.13). For \(a(\hat{\varsigma },\hat{\varrho }) =0\), we carry out the above procedure with \(\epsilon >0\) instead of \(a(\hat{\varsigma },\hat{\varrho }) \) and subsequently let \(\epsilon \rightarrow 0\). This completes the proof. □
Remark 2.7
If we take \(\hat{\alpha }(\hat{\varsigma })= \hat{\varsigma }\) and \(\hat{\alpha }(\hat{\varrho })= \hat{\varrho }\), then Theorem 2.6 reduces to [1, Theorem 2.4].
Corollary 2.8
If we take \(\mathbb{T}=\mathbb{R}\) in Theorem 2.6, then, with the help of relations (1.1), we have the following inequality due to Boudeliou [41]. If
$$\begin{aligned}& \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \\& \quad \leq a( \hat{ \varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })} \bigl[ f( \hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \,d \hat{\xi }_{2}\,d\hat{\xi }_{1} \\& \qquad {} + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}b(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl[ h(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) + \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2}) \tilde{\Psi } \bigl( u(\hat{\zeta },\hat{\xi }_{2}) \bigr) \,d \hat{\zeta } \biggr] \,d\hat{\xi }_{2}\,d \hat{\xi }_{1} \end{aligned}$$
holds for \((\hat{\varsigma },\hat{\varrho }) \in \Omega \), then
$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \biggl\{ G^{-1} \biggl[ G \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) +A( \hat{\varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\,d\hat{\xi }_{2}\,d\hat{\xi }_{1} \biggr] \biggr\} $$
for \(0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varrho } \leq \hat{\varrho }_{1}\), where Λ̃ is defined by (2.7),
$$ \breve{A}(\hat{\varsigma },\hat{\varrho }) = \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}b(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl[ h(\hat{\xi }_{1},\hat{\xi }_{2})+ \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\,d\hat{\zeta } \biggr] \,d \hat{\xi }_{2}\,d\hat{\xi }_{1}, $$
and \(( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega \) is chosen so that
$$ \biggl( \tilde{\Lambda } \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) + \breve{A}(\hat{\varsigma },\hat{\varrho }) + \int _{0}^{ \hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f( \hat{\xi }_{1}, \hat{\xi }_{2})\,d\hat{\xi }_{2}\,d\hat{\xi }_{1} \biggr) \in \operatorname{Dom}\bigl( \tilde{\Lambda }^{-1} \bigr) . $$
Corollary 2.9
The discrete form can be obtained by letting \(\mathbb{T}=\mathbb{Z}\), with the help of relations (1.2) and \(\hat{\alpha }(\hat{\varsigma })=\hat{\varsigma }\), \(\hat{\beta }(\hat{\varrho })=\hat{\varrho }\) in Theorem 2.6. If
$$\begin{aligned} \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &a( \hat{ \varsigma },\hat{\varrho }) +\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1} \bigl[ f( \hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \\ &{}+\sum_{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1}\sum _{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1}b(\hat{\xi }_{1},\hat{\xi }_{2}) \Biggl[ h(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) +\sum_{\hat{\zeta }=0}^{\hat{\xi }_{1}-1}g( \hat{\zeta }, \hat{\xi }_{2}) \tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Biggr] \end{aligned}$$
holds for \((\hat{\varsigma },\hat{\varrho }) \in \Omega \), then
$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \Biggl\{ G^{-1} \Biggl[ G \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) +A( \hat{\varsigma },\hat{\varrho }) +\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1}f( \hat{\xi }_{1},\hat{\xi }_{2}) \Biggr] \Biggr\} $$
for \(0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varrho } \leq \hat{\varrho }_{1}\), where Λ̃ is defined by (2.7),
$$ \breve{A}(\hat{\varsigma },\hat{\varrho }) =\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1}b( \hat{\xi }_{1},\hat{\xi }_{2}) \Biggl[ h(\hat{\xi }_{1},\hat{\xi }_{2})+ \sum _{\hat{\zeta }=0}^{\hat{\xi }_{1}-1}g(\hat{\zeta },\hat{\xi }_{2}) \Biggr], $$
and \(( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega \) is chosen so that
$$ \Biggl( \tilde{\Lambda } \bigl( q(\hat{\varsigma },\hat{\varrho }) \bigr) + \breve{A}(\hat{\varsigma },\hat{\varrho }) +\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1}f( \hat{\xi }_{1},\hat{\xi }_{2}) \Biggr) \in \operatorname{Dom}\bigl( \tilde{ \Lambda }^{-1} \bigr) . $$
Theorem 2.10
Assume that g, a, u, f, p, Φ̃, and Ψ̃ are as in Theorem 2.3. If \(u(\hat{\varsigma },\hat{\varrho }) \) satisfies
$$\begin{aligned}& \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \\& \quad \leq a( \hat{ \varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \bigl[ f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \\& \qquad {} + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \biggl( \int _{0}^{\hat{\xi }_{1}}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}, \end{aligned}$$
(2.17)
for \((\hat{\varsigma },\hat{\varrho }) \in \Omega \), then
$$\begin{aligned}& u(\hat{\varsigma },\hat{\varrho }) \\& \quad \leq \tilde{\Phi } ^{-1} \biggl\{ \tilde{\Lambda }^{-1} \biggl( \tilde{\Theta }^{-1} \biggl[ \tilde{\Theta } \bigl( q_{1} ( \hat{\varsigma },\hat{\varrho } ) \bigr) \\& \qquad {}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \biggr] \biggr) \biggr\} , \end{aligned}$$
(2.18)
for \(0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varrho } \leq \hat{\varrho }_{1}\), where
$$\begin{aligned}& q_{1} ( \hat{\varsigma },\hat{\varrho } ) =\tilde{\Lambda } \bigl( a(\hat{\varsigma },\hat{\varrho }) \bigr) + \int _{0}^{ \hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}p( \hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}, \end{aligned}$$
(2.19)
$$\begin{aligned}& \begin{aligned} &\tilde{\Theta }(r)= \int _{r_{0}}^{r} \frac{\Delta \hat{\xi }_{1}}{ ( ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) \circ \tilde{\Lambda }^{-1} ) (\hat{\xi }_{1} ) },\quad r \geq r_{0}>0, \\ &\tilde{\Theta }(+\infty )= \int _{r_{0}}^{+\infty } \frac{\Delta \hat{\xi }_{1}}{ ( \omega \circ \tilde{\Phi } ^{-1} ) \circ \tilde{\Lambda }^{-1}(\hat{\xi }_{1})}=+\infty , \end{aligned} \end{aligned}$$
(2.20)
and \(( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega \) is chosen so that
$$ \biggl( \tilde{\Theta } \bigl( q_{1} ( \hat{\varsigma }, \hat{ \varrho } ) \bigr) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta }, \hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \biggr) \in \operatorname{Dom}\bigl( \tilde{\Theta }^{-1} \bigr) . $$
Proof
Suppose that \(a(\breve{\xi },\breve{\zeta }) >0\). Fixing an arbitrary \((\breve{\xi },\breve{\zeta }) \in \Omega \), we define a positive and nondecreasing function \(z(\hat{\varsigma },\hat{\varrho }) \) by
$$\begin{aligned} z(\hat{\varsigma },\hat{\varrho }) =&a(\breve{\xi },\breve{\zeta })+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \bigl[ f(\hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1},\hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \biggl( \int _{0}^{\hat{\xi }_{1}}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}, \end{aligned}$$
for \(0\leq \hat{\varsigma }\leq \breve{\xi }\leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varrho }\leq \breve{\zeta }\leq \hat{\varrho }_{1}\), then \(z(0,\hat{\varrho }) =z(\hat{\varsigma },0) =a(\breve{\xi },\breve{\zeta })\) and
$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \bigl( z( \hat{\varsigma },\hat{\varrho }) \bigr) . $$
Now, by applying Theorem 2.1, we have
$$\begin{aligned}& z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) \\& \quad = \hat{\alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })} \tilde{\Psi } \bigl( u \bigl( \hat{ \alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \bigl[ f \bigl( \hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \bigl( u \bigl(\hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) +p(\hat{\varsigma },\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2} \\& \qquad {} +\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \bigl( u \bigl(\hat{ \alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \biggl( \int _{0}^{\hat{\varsigma } }g(\hat{\zeta } ,\hat{\xi }_{2}) \tilde{\Psi } \bigl( u(\hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\& \quad \leq \hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl( \hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \bigl[ f \bigl( \hat{\alpha }(\hat{\varsigma }), \hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) +p(\hat{\varsigma },\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2} \\& \qquad {} +\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }( \hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \\& \qquad {}\times\biggl( \int _{0}^{ \hat{\varsigma } }g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\& \quad \leq \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{ \alpha }( \hat{\varsigma }),\hat{\beta }(\hat{\varrho }) \bigr) \bigr) \hat{\alpha }^{ \Delta }(\hat{\varsigma }) \\& \qquad {}\times \int _{0}^{\hat{\beta }(\hat{\varrho })} \bigl[ f \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{ \Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) +p(\hat{\varsigma },\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2} \\& \qquad {} +\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{ \alpha }( \hat{\varsigma }),\hat{\beta }(\hat{\varrho }) \bigr) \bigr)\hat{\alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \\& \qquad {}\times\biggl( \int _{0}^{ \hat{\varsigma } }g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}, \end{aligned}$$
or
$$\begin{aligned}& \frac{z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) }{\tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\varsigma },\hat{\varrho }) ) } \\& \quad \leq \hat{\alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })} \bigl[ f \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{ \Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) +p(\hat{\varsigma },\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2} \\& \qquad {} +\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\varsigma } }g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}. \end{aligned}$$
(2.21)
Integrating (2.21), we get
$$\begin{aligned} \tilde{\Lambda } \bigl( z(\hat{\varsigma },\hat{\varrho }) \bigr) \leq & \tilde{\Lambda } \bigl( a(\breve{\xi },\breve{\zeta }) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })} \bigl[ f(\hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta } ,\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}. \end{aligned}$$
If \((\breve{\xi },\breve{\zeta }) \in \Omega \) is chosen arbitrarily, then
$$\begin{aligned} \tilde{\Lambda } \bigl( z(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &q_{1} (\hat{\varsigma },\hat{\varrho } ) + \int _{0}^{ \hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f( \hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta } ,\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}. \end{aligned}$$
Since \(q_{1} (\hat{\varsigma },\hat{\varrho } )>0 \) is a nondecreasing function, fixing an arbitrary point \(( \breve{\xi },\breve{\zeta } ) \in \Omega \) and defining \(v(\hat{\varsigma },\hat{\varrho }) >0\) to be a nondecreasing function given by
$$\begin{aligned} v(\hat{\varsigma },\hat{\varrho }) =&q_{1} ( \breve{\xi }, \breve{ \zeta } ) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta } ,\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1}, \end{aligned}$$
for \(0\leq \hat{\varsigma }\leq \breve{\xi }\leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varrho }\leq \breve{\zeta }\leq y_{1}\), we obtain \(v(0,\hat{\varrho }) =v(\hat{\varsigma },0) =q_{1}(\breve{\xi }, \breve{\zeta })\) and
$$ z(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Lambda }^{-1} \bigl( v( \hat{\varsigma },\hat{\varrho }) \bigr) . $$
(2.22)
Now, by applying Theorem 2.1, we have
$$\begin{aligned} v^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) =&\hat{\alpha }^{ \Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \Delta \hat{\xi }_{2} \\ &{}+\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\varsigma } }g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\ \leq &\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( G^{-1} \bigl( v \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \bigr) \bigr) \Delta \hat{\xi }_{2} \\ &{}+\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\varsigma } }g(\hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( G^{-1} \bigl( v(\hat{\zeta },\hat{\xi }_{2}) \bigr) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\ \leq & \bigl( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigr) \circ \tilde{\Lambda }^{-1} \bigl(v \bigl( \hat{\alpha }(\hat{\varsigma }), \hat{\beta }(\hat{\varrho }) \bigr) \bigr)\hat{\alpha }^{\Delta }( \hat{ \varsigma }) \\ &{} \times \biggl[ \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl( \hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \Delta \hat{\xi }_{2}+ \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }), \hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\varsigma } }g(\hat{\zeta }, \hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \biggr] , \end{aligned}$$
or
$$\begin{aligned}& \frac{v^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) }{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) \circ \tilde{\Lambda }^{-1}(v ( \hat{\varsigma },\hat{\varrho } ) )} \\& \quad \leq \hat{\alpha }^{\Delta }(\hat{ \varsigma }) \biggl[ \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \Delta \hat{\xi }_{2}+ \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl(\hat{\alpha }( \hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\varsigma } }g( \hat{\zeta },\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \biggr] . \end{aligned}$$
(2.23)
Integrating (2.23), we get
$$ \tilde{\Theta } \bigl( v ( \hat{\varsigma },\hat{\varrho } ) \bigr) \leq \tilde{ \Theta } \bigl( q_{1}(\breve{\xi },\breve{\zeta }) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl[ 1+ \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta } ,\hat{\xi }_{2})\Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}. $$
Since we chose \((\breve{\xi },\breve{\zeta }) \in \Omega \) arbitrarily,
$$\begin{aligned}& v ( \hat{\varsigma },\hat{\varrho } ) \\& \quad \leq \tilde{\Theta }^{-1} \biggl[ \tilde{\Theta } \bigl( q_{1}(\hat{\varsigma },\hat{\varrho }) \bigr) + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl[ 1+ \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta } ,\hat{\xi }_{2})\Delta \hat{\zeta } \biggr] \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \biggr] . \end{aligned}$$
(2.24)
From (2.24), (2.22), and \(u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} ( z( \hat{\varsigma },\hat{\varrho }) ) \), we get the desired inequality in (2.18). For \(a(\hat{\varsigma },\hat{\varrho }) =0\), we carry out the above procedure with \(\epsilon >0\) instead of \(a(\hat{\varsigma },\hat{\varrho }) \) and subsequently let \(\epsilon \rightarrow 0\). This completes the proof. □
Remark 2.11
If we take \(\hat{\alpha }(\hat{\varsigma })= \hat{\varsigma }\) and \(\hat{\alpha }(\hat{\varrho })= \hat{\varrho }\), then Theorem 2.10 reduces to [1, Theorem 2.7].
Corollary 2.12
If we take \(\mathbb{T}=\mathbb{R}\) in Theorem 2.10, then, with the help of relations (1.1), we get the following inequality due to Boudeliou [41]. If
$$\begin{aligned}& \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \\& \quad \leq a( \hat{ \varsigma },\hat{\varrho }) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \bigl[ f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \,d\hat{\xi }_{2}\,d \hat{\xi }_{1} \\& \qquad {} + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \biggl( \int _{0}^{\hat{\xi }_{1}}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \,d\hat{\xi }_{2}\,d \hat{\xi }_{1} \end{aligned}$$
holds for \((\hat{\varsigma },\hat{\varrho }) \in \Omega \), then
$$\begin{aligned} u(\hat{\varsigma },\hat{\varrho }) \leq& \tilde{\Phi } ^{-1} \biggl\{ \tilde{\Lambda }^{-1} \biggl( \tilde{\Theta }^{-1} \biggl[ \tilde{\Theta } \bigl( q_{2} ( \hat{\varsigma },\hat{\varrho } ) \bigr) \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta },\hat{\xi }_{2})\,d\hat{\zeta } \biggr) \,d \hat{\xi }_{2}\,d\hat{\xi }_{1} \biggr] \biggr) \biggr\} , \end{aligned}$$
for \(0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varrho } \leq \hat{\varrho }_{1}\), where
$$\begin{aligned}& q_{2} ( \hat{\varsigma },\hat{\varrho } ) =\tilde{\Lambda } \bigl( a(\hat{\varsigma },\hat{\varrho }) \bigr) + \int _{0}^{ \hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}p( \hat{\xi }_{1}, \hat{\xi }_{2})\,d\hat{\xi }_{2}\,d\hat{\xi }_{1}, \\& \tilde{\Theta }(r)= \int _{r_{0}}^{r} \frac{d\hat{\xi }_{1}}{ ( ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) \circ \tilde{\Lambda }^{-1} ) (\hat{\xi }_{1} ) },\quad r \geq r_{0}>0, \\& \tilde{\Theta }(+\infty )= \int _{r_{0}}^{+\infty } \frac{d\hat{\xi }_{1}}{ ( \omega \circ \tilde{\Phi } ^{-1} ) \circ \tilde{\Lambda }^{-1}(\hat{\xi }_{1})}=+\infty , \end{aligned}$$
and \(( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega \) is chosen so that
$$ \biggl( \tilde{\Theta } \bigl( q_{2} ( \hat{\varsigma }, \hat{ \varrho } ) \bigr) + \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })} f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( 1+ \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta }, \hat{\xi }_{2})\,d\hat{\zeta } \biggr) \,d\hat{\xi }_{2}\,d\hat{\xi }_{1} \biggr) \in \operatorname{Dom}\bigl( \tilde{\Theta }^{-1} \bigr) . $$
Corollary 2.13
The discrete form, due to El-Deeb et al. [1], can be obtained by letting \(\mathbb{T}=\mathbb{Z}\) in Theorem 2.10, with the help of relations (1.2) and \(\hat{\alpha }(\hat{\varsigma })=\hat{\varsigma }\), \(\hat{\beta }(\hat{\varrho })=\hat{\varrho }\) as follows. If
$$\begin{aligned} \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &a( \hat{ \varsigma },\hat{\varrho }) +\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1} \tilde{\Psi } \bigl( u( \hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \bigl[ f( \hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) +p(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr] \\ &{}+\sum_{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1}\sum _{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1}f(\hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \Biggl( \sum_{\hat{\zeta }=0}^{ \hat{\xi }_{1}-1}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\zeta }, \hat{\xi }_{2}) \bigr) \Biggr) , \end{aligned}$$
holds for \((\hat{\varsigma },\hat{\varrho }) \in \Omega \), then
$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \Biggl\{ \bar{G}^{-1} \Biggl( \bar{F}^{-1} \Biggl[ \bar{F} \bigl( \bar{q}_{2} ( \hat{\varsigma },\hat{\varrho } ) \bigr) +\sum _{ \hat{\xi }_{1}=0}^{\hat{\varsigma }-1}\sum_{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1}f( \hat{\xi }_{1},\hat{\xi }_{2}) \Biggl( 1+\sum _{ \hat{\zeta }=0}^{\hat{\xi }_{1}-1}g(\hat{\zeta },\hat{\xi }_{2}) \Biggr) \Biggr] \Biggr) \Biggr\} , $$
for \(0\leq \hat{\varsigma }\leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varrho } \leq \hat{\varrho }_{1}\), where
$$\begin{aligned}& \bar{q}_{2} ( \hat{\varsigma },\hat{\varrho } ) = \tilde{\Lambda } \bigl( a(\hat{\varsigma },\hat{\varrho }) \bigr) + \sum _{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1}p( \hat{\xi }_{1},\hat{\xi }_{2}), \\& \bar{F}(r)=\sum_{\hat{\xi }_{1}=r_{0}}^{r-1} \frac{1}{ ( ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) \circ \bar{G}^{-1} ) (\hat{\xi }_{1} ) },\quad r\geq r_{0}>0, \\& \bar{F}(+\infty )=\sum_{\hat{\xi }_{1}=r_{0}}^{+\infty } \frac{1}{ ( \omega \circ \tilde{\Phi } ^{-1} ) \circ \bar{G}^{-1}(\hat{\xi }_{1})}=+ \infty , \end{aligned}$$
and \(( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega \) is chosen so that
$$ \Biggl( \bar{F} \bigl( \bar{q}_{2} ( \hat{\varsigma }, \hat{\varrho } ) \bigr) +\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1}\sum _{\hat{\xi }_{2}=0}^{\hat{\varrho }-1} f( \hat{\xi }_{1},\hat{ \xi }_{2}) \Biggl( 1+\sum_{\hat{\zeta }=0}^{ \hat{\xi }_{1}-1}g( \hat{\zeta },\hat{\xi }_{2}) \Biggr) \Biggr) \in \operatorname{Dom}\bigl( \bar{F}^{-1} \bigr) . $$
Theorem 2.14
Assume that g, a, f, u, Φ̃, and Ψ̃ are as in Theorem 2.3. If \(u(\hat{\varsigma },\hat{\varrho }) \) satisfies
$$\begin{aligned}& \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \\& \quad \leq a( \hat{ \varsigma },\hat{\varrho }) + \biggl( \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2}\Delta \hat{ \xi }_{1} \biggr) ^{2} \\& \qquad {} + \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \biggl( \int _{0}^{\hat{\xi }_{1}}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}, \end{aligned}$$
(2.25)
for \((\hat{\varsigma },\hat{\varrho }) \in \Omega \), then
$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \biggl\{ \breve{H}^{-1} \biggl[ \breve{H} \bigl( a ( \hat{\varsigma }, \hat{ \varrho } ) \bigr) +\breve{B}(\hat{\varsigma }, \hat{\varrho }) + \biggl( \int _{0}^{\hat{\beta }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr) ^{2} \biggr] \biggr\} , $$
(2.26)
for \(0\leq \hat{\varsigma } \leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varrho } \leq \hat{\varrho }_{1}\), where
$$\begin{aligned}& \breve{B}(\hat{\varsigma },\hat{\varrho }) = \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta }, \hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}, \end{aligned}$$
(2.27)
$$\begin{aligned}& \begin{aligned} &\breve{H}(r)= \int _{r_{0}}^{r} \frac{\Delta \hat{\xi }_{1}}{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) ^{2} ( \hat{\xi }_{1} ) },\quad r\geq r_{0}>0, \\ &\tilde{\Theta }(+\infty )= \int _{r_{0}}^{+\infty } \frac{\Delta \hat{\xi }_{1}}{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) ^{2} ( \hat{\xi }_{1} ) }=+\infty , \end{aligned} \end{aligned}$$
(2.28)
and \(( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega \) is chosen so that
$$ \biggl( \breve{H} \bigl( a ( \hat{\varsigma },\hat{\varrho } ) \bigr) +B(\hat{ \varsigma },\hat{\varrho }) +2 \biggl( \int _{0}^{ \sigma (\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f( \hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr) ^{2} \biggr) \in \operatorname{Dom}\bigl( \breve{H}^{-1} \bigr) . $$
Proof
Assume that \(a(\hat{\varsigma },\hat{\varrho }) >0\). Taking \((\breve{\xi },\breve{\zeta })\in \Omega \) as a fixed arbitrary point, we define \(z(\hat{\varsigma },\hat{\varrho }) >0\) to be a nondecreasing function by
$$\begin{aligned}& z(\hat{\varsigma },\hat{\varrho }) \\& \quad =a(\breve{\xi },\breve{\zeta })+ \biggl( \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \biggr) ^{2} \end{aligned}$$
(2.29)
$$\begin{aligned}& \qquad {}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \biggl( \int _{0}^{\hat{\xi }_{1}}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}, \end{aligned}$$
(2.30)
for \(0\leq \hat{\varsigma }\leq \breve{\xi }\leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varrho }\leq \breve{\zeta }\leq \hat{\varrho }_{1}\), hence \(z(0,\hat{\varrho }) =z(\hat{\varsigma },0) =a(\breve{\xi },\breve{\zeta })\) and
$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \bigl( z( \hat{\varsigma },\hat{\varrho }) \bigr) . $$
From (2.29), and applying the chain rule on time scales (1.2), we get
$$\begin{aligned}& z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) \\& \quad = 2 \biggl( \int _{0}^{\hat{\alpha }(c)} \int _{0}^{\hat{\beta }(\hat{\varrho })}f( \hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2}\Delta \hat{ \xi }_{1} \biggr) \\& \qquad {}\times\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \bigl( u \bigl(\hat{ \alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \Delta \hat{ \xi }_{2} \\& \qquad {} +\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })} f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \bigl( u \bigl(\hat{ \alpha }(\hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \biggl( \int _{0}^{\hat{\alpha }(\hat{\varsigma })}g( \hat{\zeta } ,\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\& \quad \leq 2 \biggl( \int _{0}^{\hat{\alpha }(c)} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z(\hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr) \\& \qquad {} \times \hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }( \hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \Delta \hat{\xi }_{2} \\& \qquad {} +\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl(\hat{\alpha }( \hat{\varsigma }),\hat{\xi }_{2} \bigr) \bigr) \\& \qquad {}\times\biggl( \int _{0}^{ \hat{\alpha }(\hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2}) \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z( \hat{\zeta },\hat{\xi }_{2}) \bigr) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \\& \quad \leq 2 \bigl( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl( \hat{\alpha }(\hat{\varsigma }),\hat{\beta }(\hat{\varrho }) \bigr) \bigr) \bigr) ^{2}\hat{\alpha }^{\Delta }(\hat{\varsigma }) \biggl( \int _{0}^{ \hat{\alpha }(c)} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr) \\& \qquad {}\times\int _{0}^{ \hat{\beta }(\hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \Delta \hat{\xi }_{2} \\& \qquad {} + \bigl( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} \bigl( z \bigl( \hat{\alpha }(\hat{\varsigma }),\hat{\beta }(\hat{\varrho }) \bigr) \bigr) \bigr) ^{2}\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{ \hat{\beta }(\hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\varsigma } }g(\hat{\zeta },\hat{\xi }_{2}) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}, \end{aligned}$$
thus we have
$$\begin{aligned} \frac{z^{\Delta \hat{\varsigma }}(\hat{\varsigma },\hat{\varrho }) }{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ( z(\hat{\varsigma },\hat{\varrho }) ) ) ^{2}} \leq &2 \biggl( \int _{0}^{\hat{\alpha }(c)} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1} \biggr) \hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }(\hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }), \hat{\xi }_{2} \bigr) \Delta \hat{\xi }_{2} \\ &{}+\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\alpha }(\hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2}) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}, \\ =& \biggl[ \biggl( \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr)^{2} \biggr]^{\Delta _{ \hat{\varsigma }}} \\ &{}+\hat{\alpha }^{\Delta }(\hat{\varsigma }) \int _{0}^{\hat{\beta }( \hat{\varrho })}f \bigl(\hat{\alpha }(\hat{ \varsigma }),\hat{\xi }_{2} \bigr) \biggl( \int _{0}^{\hat{\alpha }(\hat{\varsigma })}g(\hat{\zeta },\hat{\xi }_{2}) \Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2}. \end{aligned}$$
(2.31)
Integrating (2.31), we get
$$\begin{aligned} \breve{H} \bigl( z(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &\breve{H} \bigl( a(\breve{\xi },\breve{\zeta }) \bigr) + \biggl( \int _{0}^{ \hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f( \hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr) ^{2} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{ \hat{\xi }_{1}}g(\hat{\zeta } ,\hat{\xi }_{2})\Delta \hat{\zeta } \biggr) \Delta \hat{\xi }_{2} \Delta \hat{\xi }_{1}. \end{aligned}$$
Since \((\breve{\xi },\breve{\zeta })\in \Omega \) is chosen arbitrarily,
$$ z(\hat{\varsigma },\hat{\varrho }) \leq \breve{H}^{-1} \biggl[ \breve{H} \bigl( a(\hat{\varsigma },\hat{\varrho }) \bigr) +\breve{B}( \hat{ \varsigma },\hat{\varrho }) + \biggl( \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\Delta \hat{\xi }_{2}\Delta \hat{\xi }_{1} \biggr) ^{2} \biggr] . $$
(2.32)
From (2.32) and \(u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} ( z( \hat{\varsigma },\hat{\varrho }) ) \), we get the desired inequality (2.26). For \(a(\hat{\varsigma },\hat{\varrho }) =0\), we carry out the above procedure with \(\epsilon >0\) instead of \(a(\hat{\varsigma },\hat{\varrho }) \) and subsequently let \(\epsilon \rightarrow 0\). This completes the proof. □
Remark 2.15
If we take \(\hat{\alpha }(\hat{\varsigma })= \hat{\varsigma }\) and \(\hat{\alpha }(\hat{\varrho })= \hat{\varrho }\), then Theorem 2.14 reduces to [1, Theorem 10].
Theorem 2.16
If we take \(\mathbb{T}=\mathbb{R}\) in Theorem 2.14, with the help of relations (1.1), we have the following inequality due to Boudeliou. If
$$\begin{aligned} \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &a( \hat{ \varsigma },\hat{\varrho }) + \biggl( \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \,d\hat{\xi }_{2}\,d\hat{\xi }_{1} \biggr) ^{2} \\ &{}+ \int _{0}^{\hat{\alpha }(\hat{\varsigma })} \int _{0}^{\hat{\beta }( \hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \biggl( \int _{0}^{\hat{\xi }_{1}}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\zeta }, \hat{\xi }_{2}) \bigr) \,d\hat{\zeta } \biggr) \,d\hat{\xi }_{2}\,d \hat{\xi }_{1}, \end{aligned}$$
for \((\hat{\varsigma },\hat{\varrho }) \in \Omega \), then
$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \biggl\{ \breve{H}^{-1} \biggl[ \breve{H} \bigl( a ( \hat{\varsigma }, \hat{ \varrho } ) \bigr) +\breve{B}(\hat{\varsigma }, \hat{\varrho }) + \biggl( \int _{0}^{\hat{\beta }(\hat{\varsigma })} \int _{0}^{ \hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2})\,d\hat{\xi }_{2}\,d \hat{\xi }_{1} \biggr) ^{2} \biggr] \biggr\} , $$
for \(0\leq \hat{\varsigma } \leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varrho } \leq \hat{\varrho }_{1}\), where
$$\begin{aligned}& \breve{B}(\hat{\varsigma },\hat{\varrho }) = \int _{0}^{\hat{\alpha }( \hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \biggl( \int _{0}^{\hat{\xi }_{1}}g(\hat{\zeta }, \hat{\xi }_{2})\,d\hat{\zeta } \biggr) \,d\hat{\xi }_{2}\,d\hat{\xi }_{1}, \\& \breve{H}(r)= \int _{r_{0}}^{r} \frac{d\hat{\xi }_{1}}{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) ^{2} ( \hat{\xi }_{1} ) },\quad r\geq r_{0}>0,\qquad \tilde{\Theta }(+\infty )= \int _{r_{0}}^{+\infty } \frac{d\hat{\xi }_{1}}{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) ^{2} ( \hat{\xi }_{1} ) }=+\infty , \end{aligned}$$
and \(( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega \) is chosen so that
$$ \biggl( \breve{H} \bigl( a ( \hat{\varsigma },\hat{\varrho } ) \bigr) +B(\hat{ \varsigma },\hat{\varrho }) +2 \biggl( \int _{0}^{ \sigma (\hat{\varsigma })} \int _{0}^{\hat{\beta }(\hat{\varrho })}f( \hat{\xi }_{1}, \hat{\xi }_{2})\,d\hat{\xi }_{2}\,d\hat{\xi }_{1} \biggr) ^{2} \biggr) \in \operatorname{Dom}\bigl( \breve{H}^{-1} \bigr) . $$
Corollary 2.17
The discrete form, due to El-Deeb et al. [1], can be obtained by letting \(\mathbb{T}=\mathbb{Z}\) and \(\hat{\alpha }(\hat{\varsigma })=\hat{\varsigma }\), \(\hat{\beta }(\hat{\varrho })=\hat{\varrho }\) in Theorem 2.14as follows. If
$$\begin{aligned} \tilde{\Phi } \bigl( u(\hat{\varsigma },\hat{\varrho }) \bigr) \leq &a( \hat{ \varsigma },\hat{\varrho }) + \Biggl( \sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }-1} f( \hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u(\hat{\xi }_{1}, \hat{\xi }_{2}) \bigr) \Biggr) ^{2} \\ &{}+\sum_{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1}\sum _{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1}f(\hat{\xi }_{1},\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\xi }_{1},\hat{\xi }_{2}) \bigr) \Biggl( \sum_{\hat{\zeta }=0}^{ \hat{\xi }_{1}-1}g( \hat{\zeta },\hat{\xi }_{2})\tilde{\Psi } \bigl( u( \hat{\zeta }, \hat{\xi }_{2}) \bigr) \Biggr) \end{aligned}$$
holds for \((\hat{\varsigma },\hat{\varrho }) \in \Omega \), then
$$ u(\hat{\varsigma },\hat{\varrho }) \leq \tilde{\Phi } ^{-1} \Biggl\{ \breve{H}^{-1} \Biggl[ \breve{H} \bigl( a ( \hat{\varsigma }, \hat{ \varrho } ) \bigr) +\breve{B}(\hat{\varsigma }, \hat{\varrho }) + \Biggl( \sum _{\hat{\xi }_{1}=0}^{\hat{\varsigma }-1} \sum _{\hat{\xi }_{2}=0}^{\hat{\varrho }-1}f(\hat{\xi }_{1},\hat{\xi }_{2}) \Biggr) ^{2} \Biggr] \Biggr\} , $$
for \(0\leq \hat{\varsigma } \leq \hat{\varsigma }_{1}\), \(0\leq \hat{\varrho } \leq \hat{\varrho }_{1}\), where
$$\begin{aligned}& \breve{B}(\hat{\varsigma },\hat{\varrho }) =\sum_{\hat{\xi }_{1}=0}^{ \hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{\hat{\varrho }}f(\hat{\xi }_{1}, \hat{\xi }_{2}) \Biggl( \sum _{\hat{\zeta }=0}^{\hat{\xi }_{1}-1}g( \hat{\zeta },\hat{\xi }_{2}) \Biggr), \\& \breve{H}(r)=\sum_{\hat{\xi }_{1}=r_{0}}^{r-1} \frac{1}{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) ^{2} ( \hat{\xi }_{1} ) },\quad r\geq r_{0}>0,\qquad \tilde{\Theta }(+\infty )=\sum_{\hat{\xi }_{1}=r_{0}}^{+\infty } \frac{1}{ ( \tilde{\Psi } \circ \tilde{\Phi } ^{-1} ) ^{2} ( \hat{\xi }_{1} ) }=+\infty , \end{aligned}$$
and \(( \hat{\varsigma }_{1},\hat{\varrho }_{1} ) \in \Omega \) is chosen so that
$$ \Biggl( \breve{H} \bigl( a ( \hat{\varsigma },\hat{\varrho } ) \bigr) +B(\hat{ \varsigma },\hat{\varrho }) + \Biggl( \sum_{ \hat{\xi }_{1}=0}^{\hat{\varsigma }-1} \sum_{\hat{\xi }_{2}=0}^{ \hat{\varrho }-1} f(\hat{\xi }_{1},\hat{\xi }_{2}) \Biggr) ^{2} \Biggr) \in \operatorname{Dom}\bigl( \breve{H}^{-1} \bigr). $$