Lemma 2.1
Suppose that (mathbb{T}_{1}), (mathbb{T}_{2}) are two times scales and (ain C (Omega = mathbb{T}_{1}times mathbb{T}_{2} ,mathbb{R}_{+})) is nondecreasing with respect to ((x,y) in Omega ). Assume that ϕ, u, (fin C (Omega ,mathbb{R}_{+})), (theta in C^{1} ( mathbb{T}_{1},mathbb{T}_{1} )), and (vartheta in C^{1} ( mathbb{T}_{2},mathbb{T}_{2} ) ) are nondecreasing functions with (theta (x)leq x) on (mathbb{T}_{1}), (vartheta (y)leq y) on (mathbb{T}_{2}). Furthermore, suppose that ψ, ({omega} in C(mathbb{R}_{+},mathbb{R}_{+})) are nondecreasing functions with ({ {psi} , {omega} } (u)>0) for (u>0), and (lim_{urightarrow +infty } {psi} (u)=+infty ). If (u(x,y) ) satisfies
$$ {psi} bigl( u(x,y) bigr) leq a(x,y)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi (s,t)f(s,t) {omega} bigl( u(s,t) bigr) nabla tDelta s $$
(2.1)
for ((x,y)in Omega ), then
$$ u(x,y)leq {psi} ^{-1} biggl{ G^{-1}G bigl( a ( x,y ) bigr) + int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)} phi (s,t)f(s,t)nabla tDelta s biggr} $$
(2.2)
for (0leq xleq x_{1}), (0leq yleq y_{1}), where
$$ G(v)= int _{v_{0}}^{v} frac{Delta s}{ {omega} ( {psi} ^{-1}(s) ) },quad vgeq v_{0}>0, qquad G(+infty )= int _{v_{0}}^{+infty } frac{Delta s}{ {omega} ( {psi} ^{-1}(s) ) }=+infty $$
(2.3)
and (( x_{1},y_{1} ) in Omega ) is chosen so that
$$ biggl( G bigl( a ( x,y ) bigr) + int _{x_{0}}^{ theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t)f(s,t) nabla t Delta s biggr) in operatorname {Dom}bigl( G^{-1} bigr) .$$
Proof
First we assume that (a ( x,y ) >0). Fixing an arbitrary ((x_{0},y_{0})in Omega ), we define a positive and nondecreasing function (z(x,y)) by
$$ z(x,y)=a(x_{0},y_{0})+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi (s,t)f(s,t) {omega} bigl( u(s,t) bigr) nabla tDelta s $$
(2.4)
for (0leq xleq x_{0}leq x_{1}), (0leq yleq y_{0}leq y_{1}), then (z(x_{0},y)=z(x,y_{0})=a(x_{0},y_{0})) and
$$ u(x,y)leq {psi} ^{-1} bigl( z(x,y) bigr). $$
(2.5)
Taking Δ-derivative for (2.4) with employing Theorem 1.5((i)), we have
$$begin{aligned} z^{Delta _{x}}(x,y) =&theta ^{Delta }(x) int _{y_{0}}^{vartheta (y)} phi bigl(theta (x),tbigr)f bigl(theta (x),tbigr) {omega} bigl( ubigl(theta (x),tbigr) bigr) nabla t \ leq &theta ^{Delta }(x) int _{y_{0}}^{vartheta (y)}phi bigl(theta (x),tbigr)f bigl( theta (x),tbigr) {omega} bigl( {psi} ^{-1} bigl( zbigl( theta (x),tbigr) bigr) bigr) nabla t \ leq & {omega} bigl( {psi} ^{-1} bigl( zbigl(theta (x), vartheta (y)bigr) bigr) bigr) theta ^{Delta }(x) int _{y_{0}}^{vartheta (y)} phi bigl(theta (x),tbigr)f bigl(theta (x),tbigr)nabla t. end{aligned}$$
(2.6)
Inequality (2.6) can be written in the form
$$ frac{z^{Delta _{x}}(x,y)}{ {omega} ( {psi} ^{-1} ( z(x,y) ) ) }leq theta ^{Delta }(x) int _{y_{0}}^{ vartheta (y)}phi bigl(theta (x),tbigr)f bigl(theta (x),tbigr)nabla t. $$
(2.7)
Taking Δ-integral for inequality (2.7) leads to
$$begin{aligned} G bigl( z(x,y) bigr) leq &G bigl( z(x_{0},y) bigr) + int _{x_{0}}^{ theta (x)} int _{y_{0}}^{vartheta (y)}phi (s,t)f(s,t)nabla t Delta s \ leq &G bigl( a(x_{0},y_{0}) bigr) + int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi (s,t)f(s,t)nabla tDelta s. end{aligned}$$
Since ((x_{0},y_{0})in Omega ) is chosen arbitrarily,
$$ z(x,y)leq G^{-1} biggl[ G bigl( a(x,y) bigr) + int _{x_{0}}^{ theta (x)} int _{y_{0}}^{vartheta (y)}phi (s,t)f(s,t)nabla t Delta s biggr] . $$
(2.8)
From (2.8) and (2.5) we obtain the desired result (2.2). We carry out the above procedure with (epsilon >0) instead of (a(x,y)) when (a(x,y)=0) and subsequently let (epsilon rightarrow 0). □
Now, as special cases of our results, we will give the continuous, discrete, and quantum inequalities. Namely, in the cases of time scales (mathbb{T}=mathbb{R}), (mathbb{T}=hmathbb{Z}), (mathbb{T}=mathbb{Z}), and (mathbb{T}=overline{q^{mathbb{Z}}}).
Remark 2.2
If we take (mathbb{T}=mathbb{R}), (x_{0}=0), and (y_{0}=0) in Lemma 2.1, then, by relation (1.1), inequality (2.1) becomes the inequality obtained in [15, Lemma 2.1].
Corollary 2.3
If we take (mathbb{T}=h mathbb{Z}) in Lemma 2.1by relation (1.3), then the following inequality
$$ {psi} bigl( u(x,y) bigr) leq a(x,y)+h^{2}sum _{s= frac{x_{0}}{h}}^{frac{theta (x)}{h}-1}sum_{t=frac{y_{0}}{h}}^{ frac{vartheta (y)}{h}+1} phi (sh,th)f(sh,th) {omega} bigl( u(sh,th) bigr) $$
for ((x,y)in Omega ) implies
$$ u(x,y)leq {psi} ^{-1} Biggl{ G^{-1}G bigl( a ( x,y ) bigr) +h^{2}sum_{s=frac{x_{0}}{h}}^{frac{theta (x)}{h}-1} sum _{t=frac{y_{0}}{h}}^{frac{vartheta (y)}{h}+1}phi (sh,th)f(sh,th) Biggr} $$
for (0leq xleq x_{1}), (0leq yleq y_{1}), where
$$ G(v)=sum_{s=frac{v_{0}}{h}}^{frac{v}{h}-1} frac{h}{ {omega} ( {psi} ^{-1}(sh) ) },quad vgeq v_{0}>0, qquad G(+infty )= sum_{s=frac{v_{0}}{h}}^{+infty } frac{h}{ {omega} ( {psi} ^{-1}(sh) ) }=+ infty $$
and (( x_{1},y_{1} ) in Omega ) is chosen so that
$$ Biggl( G bigl( a ( x,y ) bigr) +h^{2}sum _{s= frac{x_{0}}{h}}^{frac{theta (x)}{h}-1}sum_{t=frac{y_{0}}{h}}^{ frac{vartheta (y)}{h}+1} phi _{1}(sh,th)f(sh,th) Biggr) in operatorname {Dom}bigl( G^{-1} bigr) .$$
Remark 2.4
In Corollary 2.3, if we take (h=1), then the following inequality
$$ {psi} bigl( u(x,y) bigr) leq a(x,y)+sum_{s=x_{0}}^{theta (x)-1} sum_{t=y_{0}}^{vartheta (y)+1}phi (s,t)f(s,t) {omega} bigl( u(s,t) bigr) $$
for ((x,y)in Omega ) implies
$$ u(x,y)leq {psi} ^{-1} Biggl{ G^{-1}G bigl( a ( x,y ) bigr) +sum_{s=x_{0}}^{theta (x)-1}sum _{t=y_{0}}^{vartheta (y)+1} phi (s,t)f(s,t) Biggr} $$
for (0leq xleq x_{1}), (0leq yleq y_{1}), where
$$ G(v)=sum_{s=v_{0}}^{v-1} frac{1}{ {omega} ( {psi} ^{-1}(s) ) },quad vgeq v_{0}>0,qquad G(+infty )= sum_{s=v_{0}}^{+infty } frac{1}{ {omega} ( {psi} ^{-1}(s) ) }=+ infty $$
and (( x_{1},y_{1} ) in Omega ) is chosen so that
$$ Biggl( G bigl( a ( x,y ) bigr) +sum_{s=x_{0}}^{ theta (x)-1} sum_{t=y_{0}}^{vartheta (y)+1}phi _{1}(s,t)f(s,t) Biggr) in operatorname {Dom}bigl( G^{-1} bigr) .$$
Corollary 2.5
If we take (mathbb{T}=overline{q^{mathbb{Z}}}) in Lemma 2.1by relation (1.4), then the following inequality
$$ {psi} bigl( u(x,y) bigr) leq a(x,y)+(q-1)^{2}sum _{s=(log _{q}{x_{0}})}^{( log _{q}{theta (x)})-1}sum_{t=(log _{q}{y_{0}})}^{(log _{q}{ vartheta (y)})+1} q^{(s+t)}phi bigl(q^{s},q^{t}bigr)f bigl(q^{s},q^{t}bigr) {omega} bigl( u bigl(q^{s},q^{t}bigr) bigr) $$
for ((x,y)in Omega ) implies
$$ u(x,y)leq {psi} ^{-1} Biggl{ G^{-1}G bigl( a ( x,y ) bigr) +(q-1)^{2}sum_{s=(log _{q}{x_{0}})}^{(log _{q}{theta (x)})-1} sum_{t=(log _{q}{y_{0}})}^{(log _{q}{vartheta (y)})+1}q^{(s+t)} phi bigl(q^{s},q^{t}bigr)fbigl(q^{s},q^{t} bigr) Biggr} $$
for (0leq xleq x_{1}), (0leq yleq y_{1}), where
$$ G(v)=sum_{s=(log _{q}{v_{0}})}^{(log _{q}{v})-1} frac{(q-1)q^{s}}{ {omega} ( {psi} ^{-1}(q^{s}) ) },quad v geq v_{0}>0,qquad G(+infty )= sum_{s=(log _{q}{v_{0}})}^{+ infty } frac{(q-1)q^{s}}{ {omega} ( {psi} ^{-1}(q^{s}) ) }=+ infty $$
and (( x_{1},y_{1} ) in Omega ) is chosen so that
$$ Biggl( G bigl( a ( x,y ) bigr) +(q-1)^{2}sum _{s=( log _{q}{x_{0}})}^{(log _{q}{theta (x)})-1}sum_{t=(log _{q}{y_{0}})}^{( log _{q}{vartheta (y)})+1}q^{(s+t)} phi _{1}bigl(q^{s},q^{t}bigr)f bigl(q^{s},q^{t}bigr) Biggr) in operatorname {Dom}bigl( G^{-1} bigr) .$$
Theorem 2.6
Let u, a, f, θ, and ϑ be as in Lemma 2.1. Let (phi _{1},phi _{2}in C (Omega ,mathbb{R}_{+})). If (u(x,y)) satisfies
$$begin{aligned} {psi} bigl( u(x,y) bigr) leq &a(x,y)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t) biggl[ f(s,t) {omega} bigl( u(s,t) bigr) \ &{} + int _{x_{0}}^{s}phi _{2}(tau ,t) { omega} bigl( u( tau ,t) bigr) Delta tau biggr] nabla tDelta s end{aligned}$$
(2.9)
for ((x,y)in Omega ), then
$$ u(x,y)leq {psi} ^{-1} biggl{ G^{-1} biggl( p(x,y)+ int _{x_{0}}^{ theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t)f(s,t) nabla t Delta s biggr) biggr} $$
(2.10)
for (0leq xleq x_{1}), (0leq yleq y_{1}), where G is defined by (2.3) and
$$ p(x,y)=G bigl( a(x,y) bigr) + int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi _{1}(s,t) biggl( int _{x_{0}}^{s}phi _{2}(tau ,t) Delta tau biggr) nabla tDelta s $$
(2.11)
and (( x_{1},y_{1} ) in Omega ) is chosen so that
$$ biggl( p(x,y)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)} phi _{1}(s,t)f(s,t) nabla tDelta s biggr) in operatorname {Dom}bigl( G^{-1} bigr) .$$
Proof
By the same steps of the proof of Lemma 2.1, we can obtain (2.10) with suitable changes. □
Remark 2.7
If we take (phi _{2}(x,y)=0), then Theorem 2.6 reduces to Lemma 2.1.
Corollary 2.8
Let the functions u, f, (phi _{1}), (phi _{2}), a, θ, and ϑ be as in Theorem 2.6. Further, suppose that (q>p>0) are constants. If (u(x,y)) satisfies
$$begin{aligned} u^{q}(x,y) leq &a(x,y)+frac{q}{q-p} int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi _{1}(s,t) biggl[ f(s,t)u^{p}(s,t) \ &{} + int _{x_{0}}^{s}phi _{2}(tau ,t)u^{p}( tau ,t)Delta tau biggr] nabla tDelta s end{aligned}$$
(2.12)
for ((x,y)in Omega ), then
$$ u(x,y)leq biggl{ p(x,y)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi _{1}(s,t)f(s,t) nabla tDelta s biggr} ^{ frac{1}{q-p}}, $$
(2.13)
where
$$ p(x,y)= bigl( a(x,y) bigr) ^{frac{q-p}{q}}+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t) biggl( int _{x_{0}}^{s} phi _{2}(tau ,t) Delta tau biggr) nabla tDelta s. $$
Proof
In Theorem 2.6, by letting ({psi} (u)=u^{q}), ({omega} (u)=u^{p}), we have
$$ G(v)= int _{v_{0}}^{v} frac{Delta s}{ {omega} ( {psi} ^{-1}(s) ) }= int _{v_{0}}^{v}frac{Delta s}{s^{frac{p}{q}}}geq frac{q}{q-p} bigl( v^{frac{q-p}{q}}-v_{0}^{frac{q-p}{q}} bigr) ,quad vgeq v_{0}>0 $$
and
$$ G^{-1}(v)geq biggl{ v_{0}^{frac{q-p}{q}}+ frac{q-p}{q}v biggr} ^{ frac{1}{q-p}}. $$
We obtain inequality (2.13). □
Theorem 2.9
Under the hypotheses of Theorem 2.6, further, let ψ, ω, (eta in C(mathbb{R}_{+},mathbb{R}_{+})) be nondecreasing functions with ({ psi ,omega ,eta } (u)>0) for (u>0), and (lim_{urightarrow +infty } psi (u)=+infty ). If (u(x,y)) satisfies
$$begin{aligned} {psi} bigl( u(x,y) bigr) leq &a(x,y)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t) biggl[ f(s,t) {omega} bigl( u(s,t) bigr) eta bigl( u(s,t) bigr) \ &{} + int _{x_{0}}^{s}phi _{2}(tau ,t) { omega} bigl( u( tau ,t) bigr) Delta tau biggr] nabla tDelta s end{aligned}$$
(2.14)
for ((x,y)in Omega ), then
$$ u(x,y)leq {psi} ^{-1} biggl{ G^{-1} biggl( F^{-1} biggl[ F bigl( p(x,y) bigr) + int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)} phi _{1}(s,t)f(s,t) nabla tDelta s biggr] biggr) biggr} $$
(2.15)
for (0leq xleq x_{1}), (0leq yleq y_{1}), where G and p are as in (2.3), (2.11) respectively and
$$ F(v)= int _{v_{0}}^{v} frac{Delta s}{eta ( {psi} ^{-1} ( G^{-1}(s) ) ) },quad vgeq v_{0}>0,qquad F(+infty )=+infty , $$
(2.16)
and (( x_{1},y_{1} ) in Omega ) is chosen so that
$$ biggl[ F bigl( p(x,y) bigr) + int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi _{1}(s,t)f(s,t) nabla tDelta s biggr] in operatorname {Dom}bigl( F^{-1} bigr) .$$
Proof
Assume that (a(x,y)>0). Fixing arbitrary ((x_{0},y_{0})in Omega ), we define a positive and nondecreasing function (z(x,y)) by
$$begin{aligned} z(x,y) =&a(x_{0},y_{0})+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi _{1}(s,t) biggl[ f(s,t) {omega} bigl( u(s,t) bigr) eta bigl( u(s,t) bigr) end{aligned}$$
(2.17)
$$begin{aligned} & {}+ int _{x_{0}}^{s}phi _{2}(tau ,t) { omega} bigl( u( tau ,t) bigr) Delta tau biggr] nabla tDelta s end{aligned}$$
(2.18)
for (0leq xleq x_{0}leq x_{1}), (0leq yleq y_{0}leq y_{1}), then (z(x_{0},y)=z(x,y_{0})=a(x_{0},y_{0})) and
$$ u(x,y)leq {psi} ^{-1} bigl( z(x,y) bigr). $$
(2.19)
Taking Δ-derivative for (2.17) with employing Theorem 1.5(i) gives
$$begin{aligned} z^{Delta _{x}}(x,y) =&theta ^{Delta }(x) int _{y_{0}}^{vartheta (y)} phi _{1}bigl(theta (x),tbigr) biggl[ fbigl(theta (x),tbigr) {omega} bigl( ubigl( theta (x),tbigr) bigr) eta bigl( ubigl(theta (x),tbigr) bigr) end{aligned}$$
(2.20)
$$begin{aligned} &{} + int _{x_{0}}^{theta (x)}phi _{2}( tau ,t) { omega} bigl( u(tau ,t) bigr) Delta tau biggr] nabla t end{aligned}$$
(2.21)
$$begin{aligned} leq &theta ^{Delta }(x) int _{y_{0}}^{vartheta (y)}phi _{1}bigl( theta (x),tbigr) biggl[ fbigl(theta (x),tbigr) {omega} bigl( {psi} ^{-1} bigl( zbigl(theta (x),tbigr) bigr) bigr) end{aligned}$$
(2.22)
$$begin{aligned} & {}times etabigl( { psi} ^{-1} bigl( zbigl(theta (x),tbigr) bigr) bigr)+ int _{x_{0}}^{theta (x)}phi _{2}(tau ,t) { omega} bigl( {psi} ^{-1} bigl( z(tau ,t) bigr) bigr) Delta tau biggr] nabla t end{aligned}$$
(2.23)
$$begin{aligned} leq &theta ^{Delta }(x). {omega} bigl( {psi} ^{-1} bigl( zbigl( theta (x),vartheta (y)bigr) bigr) bigr) end{aligned}$$
(2.24)
$$begin{aligned} &{}times int _{y_{0}}^{vartheta (y)}phi _{1}bigl(theta (x),tbigr) biggl[ fbigl( theta (x),tbigr)eta bigl( {psi} ^{-1} bigl( zbigl(theta (x),tbigr) bigr) bigr) end{aligned}$$
(2.25)
$$begin{aligned} &{} + int _{x_{0}}^{theta (x)}phi _{2}(tau ,t) Delta tau biggr] nabla t. end{aligned}$$
(2.26)
From (2.20) we have
$$begin{aligned} frac{z^{Delta _{x}}(x,y)}{ {omega} ( {psi} ^{-1} ( z(x,y) ) ) } leq &theta ^{Delta }(x) int _{y_{0}}^{ vartheta (y)}phi _{1}bigl(theta (x),tbigr) biggl[ fbigl(theta (x),tbigr)eta bigl( {psi} ^{-1} bigl( zbigl(theta (x),tbigr) bigr) bigr) end{aligned}$$
(2.27)
$$begin{aligned} &{} + int _{x_{0}}^{theta (x)}phi _{2}(tau ,t) Delta tau biggr] nabla t. end{aligned}$$
(2.28)
Taking Δ-integral for (2.27) gives
$$begin{aligned} G bigl( z(x,y) bigr) leq &G bigl( z(x_{0},y) bigr) + int _{x_{0}}^{ theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t) biggl[ f(s,t) eta bigl( {psi} ^{-1} bigl( z(s,t) bigr) bigr) \ & {}+ int _{x_{0}}^{s}phi _{2}(tau ,t) Delta tau biggr] nabla tDelta s \ leq &G bigl( a(x_{0},y_{0}) bigr) + int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t) biggl[ f(s,t)eta bigl( { psi} ^{-1} bigl( z(s,t) bigr) bigr) \ & {}+ int _{x_{0}}^{s}phi _{2}(tau ,t) Delta tau biggr] nabla tDelta s. end{aligned}$$
Since ((x_{0},y_{0})in Omega ) is chosen arbitrarily, the last inequality can be rewritten as
$$ G bigl( z(x,y) bigr) leq p(x,y)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi _{1}(s,t)f(s,t) eta bigl( {psi} ^{-1} bigl( z(s,t) bigr) bigr) nabla tDelta s. $$
(2.29)
Since (p(x,y)) is a nondecreasing function, an application of Lemma 2.1 to (2.29) gives us
$$ z(x,y)leq G^{-1} biggl( F^{-1} biggl[ F bigl( p(x,y) bigr) + int _{x_{0}}^{ theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t)f(s,t) nabla t Delta s biggr] biggr) . $$
(2.30)
From (2.19) and (2.30) we obtain the desired inequality (2.15).
Now, we take the case (a(x,y)=0) for some ((x,y)in Omega ). Let (a_{epsilon }(x,y)=a(x,y)+epsilon ) for all ((x,y)in Omega ), where (epsilon >0) is arbitrary, then (a_{epsilon }(x,y)>0) and (a_{epsilon }(x,y)in C(Omega ,mathbb{R}_{+})) are nondecreasing with respect to ((x,y)in Omega ). We carry out the above procedure with (a_{epsilon }(x,y)>0) instead of (a(x,y)), and we get
$$ u(x,y)leq {psi} ^{-1} biggl{ G^{-1} biggl( F^{-1} biggl[ F bigl( p_{ epsilon }(x,y) bigr) + int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi _{1}(s,t)f(s,t) nabla tDelta s biggr] biggr) biggr} , $$
where
$$ p_{epsilon }(x,y)=G bigl( a_{epsilon }(x,y) bigr) + int _{x_{0}}^{ theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t) biggl( int _{x_{0}}^{s} phi _{2}(tau ,t) Delta tau biggr) nabla tDelta s. $$
Letting (epsilon rightarrow 0^{+}), we obtain (2.15). The proof is complete. □
Now, as special cases of our results, we will give the continuous, discrete, and quantum inequalities. Namely, in the cases of time scales (mathbb{T}=mathbb{R}), (mathbb{T}=hmathbb{Z}), (mathbb{T}=mathbb{Z}), and (mathbb{T}=overline{q^{mathbb{Z}}}).
Remark 2.10
If we take (mathbb{T}=mathbb{R}), (x_{0}=0), and (y_{0}=0) in Theorem 2.9, then, by relation (1.1), inequality (2.14) becomes the inequality obtained in [15, Theorem 2.2(A_2)].
Corollary 2.11
If we take (mathbb{T}=h mathbb{Z}) in Theorem 2.9by relation (1.3), then the following inequality
$$begin{aligned} {psi} bigl( u(x,y) bigr) leq &a(x,y)+h^{2}sum _{s= frac{x_{0}}{h}}^{frac{theta (x)}{h}-1}sum_{t=frac{y_{0}}{h}}^{ frac{vartheta (y)}{h}+1} phi _{1}(sh,th) Biggl[ f(sh,th) {omega} bigl( u(sh,th) bigr) eta bigl( u(sh,th) bigr) \ & {}+hsum_{t=frac{x_{0}}{h}}^{frac{s}{h}-1}phi _{2}(tau ,th) {omega} bigl( u(tau ,th) bigr) Biggr] end{aligned}$$
for ((x,y)in Omega ) implies
$$ u(x,y)leq {psi} ^{-1} Biggl{ G^{-1} Biggl( F^{-1} Biggl[ F bigl( p(x,y) bigr) +h^{2}sum _{s=frac{x_{0}}{h}}^{frac{theta (x)}{h}-1} sum_{t=frac{y_{0}}{h}}^{frac{vartheta (y)}{h}+1} phi _{1}(sh,th)f(sh,th) Biggr] Biggr) Biggr} $$
for (0leq xleq x_{1}), (0leq yleq y_{1}), where G and p are as in (2.3) and (2.11), respectively, and
$$ F(v)=sum_{s=frac{v_{0}}{h}}^{frac{v}{h}} frac{h}{eta ( {psi} ^{-1} ( G^{-1}(sh) ) ) },quad vgeq v_{0}>0,qquad F(+infty )=+infty $$
and (( x_{1},y_{1} ) in Omega ) is chosen so that
$$ Biggl[ F bigl( p(x,y) bigr) +h^{2}sum _{s=frac{x_{0}}{h}}^{ frac{theta (x)}{h}-1}sum_{t=frac{y_{0}}{h}}^{ frac{vartheta (y)}{h}+1} phi _{1}(sh,th)f(sh,th) Biggr] in operatorname {Dom}bigl( F^{-1} bigr) .$$
Remark 2.12
In Corollary 2.11, if we take (h=1), then the following inequality
$$begin{aligned} {psi} bigl( u(x,y) bigr) leq &a(x,y)+sum_{s=x_{0}}^{theta (x)-1} sum_{t=y_{0}}^{vartheta (y)+1}phi _{1}(s,t) Biggl[ f(s,t) {omega} bigl( u(s,t) bigr) eta bigl( u(s,t) bigr) \ & {}+sum_{t=x_{0}}^{s-1}phi _{2}(tau ,t) {omega} bigl( u( tau ,t) bigr) Biggr] end{aligned}$$
for ((x,y)in Omega ) implies
$$ u(x,y)leq {psi} ^{-1} Biggl{ G^{-1} Biggl( F^{-1} Biggl[ F bigl( p(x,y) bigr) +sum _{s=x_{0}}^{theta (x)-1}sum_{t=y_{0}}^{vartheta (y)+1} phi _{1}(s,t)f(s,t) s Biggr] Biggr) Biggr} $$
for (0leq xleq x_{1}), (0leq yleq y_{1}), where G and p are as in (2.3), and
$$ F(v)=sum_{s=v_{0}}^{v-1} frac{1}{eta ( {psi} ^{-1} ( G^{-1}(s) ) ) },quad vgeq v_{0}>0,qquad F(+infty )=+infty , $$
and (( x_{1},y_{1} ) in Omega ) is chosen so that
$$ Biggl[ F bigl( p(x,y) bigr) +sum_{s=x_{0}}^{theta (x)-1} sum_{t=y_{0}}^{ vartheta (y)+1}phi _{1}(s,t)f(s,t) Biggr] in operatorname {Dom}bigl( F^{-1} bigr) .$$
Corollary 2.13
If we take (mathbb{T}=overline{q^{mathbb{Z}}}) in Theorem 2.9by relation (1.4), then the following inequality
$$begin{aligned} {psi} bigl( u(x,y) bigr) leq &a(x,y)+(q-1)^{2}sum _{s=(log _{q}{x_{0}})}^{( log _{q}{theta (x)})-1}sum_{t=(log _{q}{y_{0}})}^{(log _{q}{ vartheta (y)})+1}q^{(s+t)} phi _{1}bigl(q^{s},q^{t}bigr) \ & {}timesBiggl[ f bigl(q^{s},q^{t}bigr) {omega} bigl( u bigl(q^{s},q^{t}bigr) bigr) eta bigl( u bigl(q^{s},q^{t}bigr) bigr) \ & {}+(q-1)sum_{t=(log _{q}{x_{0}})}^{(log _{q}{s})-1}q^{t} phi _{2}bigl(tau ,q^{t}bigr) {omega} bigl( u(tau ,t) bigr) Biggr] end{aligned}$$
for ((x,y)in Omega ), then
$$begin{aligned} u(x,y) leq& {psi} ^{-1} Biggl{ G^{-1} Biggl( F^{-1} Biggl[ F bigl( p(x,y) bigr)\ &{} +(q-1)^{2}sum _{s=(log _{q}{x_{0}})}^{(log _{q}{theta (x)})-1} sum _{t=(log _{q}{y_{0}})}^{(log _{q}{vartheta (y)})+1}q^{(s+t)} phi _{1}bigl(q^{s},q^{t}bigr)f bigl(q^{s},q^{t}bigr) s Biggr] Biggr) Biggr} end{aligned}$$
for (0leq xleq x_{1}), (0leq yleq y_{1}), where G and p are as in (2.3), and
$$ F(v)=sum_{s=(log _{q}{v_{0}})}^{(log _{q}{v})-1} frac{(q-1)q^{s}}{eta ( {psi} ^{-1} ( G^{-1}(q^{s}) ) ) },quad vgeq v_{0}>0,qquad F(+infty )=+infty , $$
and (( x_{1},y_{1} ) in Omega ) is chosen so that
$$ Biggl[ F bigl( p(x,y) bigr) +(q-1)^{2}sum _{s=(log _{q}{x_{0}})}^{( log _{q}{theta (x)})-1}sum_{t=(log _{q}{y_{0}})}^{(log _{q}{ vartheta (y)})+1}q^{(s+t)} phi _{1}bigl(q^{s},q^{t}bigr)f bigl(q^{s},q^{t}bigr) Biggr] in operatorname {Dom}bigl( F^{-1} bigr) .$$
Corollary 2.14
Let the functions u, a, f, (phi _{1}), (phi _{2}), θ, and ϑ be as in Theorem 2.6. Further, suppose that q, p, and r are constants with (p>0), (r>0), and (q>p+r). If (u(x,y)) satisfies
$$begin{aligned} u^{q}(x,y) leq &a(x,y)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi _{1}(s,t) biggl[ f(s,t)u^{p}(s,t)u^{r}(s,t) \ & {}+ int _{x_{0}}^{s}phi _{2}(tau ,t)u^{p}(tau ,t)Delta tau biggr] nabla tDelta s end{aligned}$$
(2.31)
for ((x,y)in Omega ), then
$$ u(x,y)leq biggl{ bigl[ p(x,y) bigr] ^{frac{q-p-r}{q-p}}+ frac{q-p-r}{q}int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t)f(s,t) nabla tDelta s biggr} ^{frac{1}{q-p-r}}, $$
(2.32)
where
$$ p(x,y)= bigl( a(x,y) bigr) ^{frac{q-p}{q}}+frac{q-p}{q} int _{x_{0}}^{ theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t) biggl( int _{x_{0}}^{s} phi _{2}(tau ,t) Delta tau biggr) nabla tDelta s. $$
Proof
An application of Theorem 2.9 with ({psi} ( u ) =u^{q}), ({omega} ( u ) =u^{p}), and (eta ( u ) =u^{r}) yields the desired inequality (2.32). □
Theorem 2.15
Under the hypotheses of Theorem 2.9. If (u(x,y)) satisfies
$$begin{aligned} {psi} bigl( u(x,y) bigr) leq &a(x,y)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t) biggl[ f(s,t) {omega} bigl( u(s,t) bigr) eta bigl( u(s,t) bigr) \ &{} + int _{x_{0}}^{s}phi _{2}(tau ,t) { omega} bigl( u(tau ,t) bigr) eta bigl( u(tau ,t) bigr) Delta tau biggr] nabla tDelta s end{aligned}$$
(2.33)
for ((x,y)in Omega ), then
$$ u(x,y)leq {psi} ^{-1} biggl{ G^{-1} biggl( F^{-1} biggl[ p_{0}(x,y)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t)f(s,t) nabla t Delta s biggr] biggr) biggr} $$
(2.34)
for (0leq xleq x_{1}), (0leq yleq y_{1}), where
$$ p_{0}(x,y)=Fbigl(G bigl( a(x,y) bigr) bigr)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t) biggl( int _{x_{0}}^{s} phi _{2}(tau ,t) Delta tau biggr) nabla tDelta s, $$
and (( x_{1},y_{1} ) in Omega ) is chosen so that
$$ biggl[ p_{0}(x,y)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi _{1}(s,t)f(s,t) nabla tDelta s biggr] in operatorname {Dom}bigl( F^{-1} bigr) .$$
Proof
Assume that (a(x,y)>0). Fixing arbitrary ((x_{0},y_{0})in Omega ), we define a positive and nondecreasing function (z(x,y)) by
$$begin{aligned} z(x,y) =&a(x_{0},y_{0})+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi _{1}(s,t) biggl[ f(s,t) {omega} bigl( u(s,t) bigr) eta bigl( u(s,t) bigr) \ & {}+ int _{x_{0}}^{s}phi _{2}(tau ,t) { omega} bigl( u( tau ,t) bigr) eta bigl( u(tau ,t) bigr) Delta tau biggr] nabla tDelta s end{aligned}$$
for (0leq xleq x_{0}leq x_{1}), (0leq yleq y_{0}leq y_{1}), then (z(x_{0},y)=z(x,y_{0})=a(x_{0},y_{0})), and
$$ u(x,y)leq {psi} ^{-1} bigl( z(x,y) bigr) . $$
(2.35)
By the same steps as the proof of Theorem 2.9, we obtain
$$begin{aligned} z(x,y) leq &G^{-1} biggl{ G bigl( a(x_{0},y_{0}) bigr) + int _{x_{0}}^{ theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t) biggl[ f(s,t) eta bigl( {psi} ^{-1} bigl( z(s,t) bigr) bigr) \ & {}+ int _{x_{0}}^{s}phi _{2}(tau ,t)eta bigl( { psi} ^{-1} bigl( z(tau ,t) bigr) bigr) Delta tau biggr] nabla tDelta s biggr} . end{aligned}$$
We define a nonnegative and nondecreasing function (v(x,y)) by
$$begin{aligned} v(x,y) =&G bigl( a(x_{0},y_{0}) bigr) + int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t) biggl[ bigl[ f(s,t) eta bigl( {psi} ^{-1} bigl( z(s,t) bigr) bigr) bigr] \ & {}+ int _{x_{0}}^{s}phi _{2}(tau ,t)eta bigl( {psi} ^{-1} bigl( z(tau ,t) bigr) bigr) Delta tau biggr] nabla t Delta s, end{aligned}$$
then (v(x_{0},y)=v(x,y_{0})=G ( a(x_{0},y_{0}) )),
$$ z(x,y)leq G^{-1} bigl[ v(x,y) bigr], $$
(2.36)
and then
$$begin{aligned} v^{Delta x}(x,y) leq &theta ^{Delta }(x) int _{y_{0}}^{ vartheta (y)}phi _{1}bigl(theta (x),tbigr) biggl[ fbigl(theta (x),tbigr)eta bigl( {psi} ^{-1} bigl( G^{-1} bigl( vbigl(theta (x),ybigr) bigr) bigr) bigr) \ &{} + int _{x_{0}}^{theta (x)}phi _{2}(tau ,t) eta bigl( {psi} ^{-1} bigl( G^{-1} bigl( v(tau ,y) bigr) bigr) bigr) Delta tau biggr] nabla t \ leq &theta ^{Delta }(x)eta bigl( {psi} ^{-1} bigl( G^{-1} bigl( vbigl(theta (x),vartheta (y)bigr) bigr) bigr) bigr) int _{y_{0}}^{ vartheta (y)}phi _{1}bigl(theta (x),tbigr) biggl[ fbigl(theta (x),tbigr) \ & {}+ int _{x_{0}}^{theta (x)}phi _{2}(tau ,t) Delta tau biggr] nabla t end{aligned}$$
or
$$begin{aligned} frac{v^{Delta x}(x,y)}{eta ( {psi} ^{-1} ( G^{-1} ( v(x,y) ) ) ) } leq &theta ^{ Delta }(x) int _{y_{0}}^{vartheta (y)}phi _{1}bigl(theta (x),tbigr) biggl[ fbigl(theta (x),tbigr) + int _{x_{0}}^{theta (x)}phi _{2}(tau ,t) Delta tau biggr] nabla t. end{aligned}$$
Taking Δ-integral for the above inequality gives
$$ Fbigl(v(x,y)bigr)leq Fbigl(v(x_{0},y)bigr)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi _{1}(s,t) biggl[ f(s,t)+ int _{x_{0}}^{s}phi _{2}( tau ,t) Delta tau biggr] nabla tDelta s $$
or
$$begin{aligned} v(x,y) leq &F^{-1} biggl{ Fbigl(G bigl( a(x_{0},y_{0}) bigr) bigr)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t) biggl[ f(s,t) \ & {}+ int _{x_{0}}^{s}phi _{2}(tau ,t) Delta tau biggr] nabla tDelta s biggr} . end{aligned}$$
(2.37)
From (2.35)–(2.37), and since ((x_{0},y_{0})in Omega ) is chosen arbitrarily, we obtain the desired inequality (2.34). If (a(x,y)=0), we carry out the above procedure with (epsilon >0) instead of (a(x,y)) and subsequently let (epsilon rightarrow 0). The proof is complete. □
Now, as special cases of our results, we will give the continuous, discrete, and quantum inequalities. Namely, in the cases of time scales (mathbb{T}=mathbb{R}), (mathbb{T}=hmathbb{Z}), (mathbb{T}=mathbb{Z}), and (mathbb{T}=overline{q^{mathbb{Z}}}).
Remark 2.16
If we take (mathbb{T}=mathbb{R}) and (x_{0}=0) and (y_{0}=0) in Theorem 2.15, then, by relation (1.1), inequality (2.33) becomes the inequality obtained in [15, Theorem 2.2(A3)].
Corollary 2.17
If we take (mathbb{T}=h mathbb{Z}) in Theorem 2.15by relation (1.3), then the following inequality
$$begin{aligned} {psi} bigl( u(x,y) bigr) leq &a(x,y)+h^{2}sum _{s= frac{x_{0}}{h}}^{frac{theta (x)}{h}-1}sum_{t=frac{y_{0}}{h}}^{ frac{vartheta (y)}{h}+1} phi _{1}(sh,th) Biggl[ f(sh,th) {omega} bigl( u(sh,th) bigr) eta bigl( u(sh,th) bigr) \ &{} +hsum_{t=x_{0}}^{frac{s}{h}-1}phi _{2}(tau ,th) { omega} bigl( u(tau ,th) bigr) eta bigl( u( tau ,th) bigr) Biggr] end{aligned}$$
for ((x,y)in Omega ) implies
$$ u(x,y)leq {psi} ^{-1} Biggl{ G^{-1} Biggl( F^{-1} Biggl[ p_{0}(x,y)+h^{2} sum _{s=frac{x_{0}}{h}}^{frac{theta (x)}{h}-1}sum _{t= frac{y_{0}}{h}}^{frac{vartheta (y)}{h}+1}phi _{1}(sh,th)f(sh,th) Biggr] Biggr) Biggr} $$
for (0leq xleq x_{1}), (0leq yleq y_{1}), where
$$ p_{0}(x,y)=Fbigl(G bigl( a(x,y) bigr) bigr)+h^{2} sum_{s=frac{x_{0}}{h}}^{ frac{theta (x)}{h}-1}sum _{t=frac{y_{0}}{h}}^{ frac{vartheta (y)}{h}+1}phi _{1}(sh,th) Biggl( sum_{t= frac{x_{0}}{h}}^{frac{s}{h}}phi _{2}( tau ,th) Biggr), $$
and (( x_{1},y_{1} ) in Omega ) is chosen so that
$$ Biggl[ p_{0}(x,y)+h^{2}sum _{s=frac{x_{0}}{h}}^{ frac{theta (x)}{h}-1}sum_{t=frac{y_{0}}{h}}^{ frac{vartheta (y)}{h}+1} phi _{1}(sh,th)f(sh,th) Biggr] in operatorname {Dom}bigl( F^{-1} bigr) .$$
Remark 2.18
In Corollary 2.17, if we take (h=1), then the following inequality
$$begin{aligned} {psi} bigl( u(x,y) bigr) leq &a(x,y)+sum_{s=x_{0}}^{theta (x)-1} sum_{t=y_{0}}^{vartheta (y)+1}phi _{1}(s,t) Biggl[ f(s,t) {omega} bigl( u(s,t) bigr) eta bigl( u(s,t) bigr) \ &{} +sum_{s=x_{0}}^{s-1}phi _{2}(tau ,t) {omega} bigl( u(tau ,t) bigr) eta bigl( u(tau ,t) bigr) Biggr] end{aligned}$$
for ((x,y)in Omega ) implies
$$ u(x,y)leq {psi} ^{-1} Biggl{ G^{-1} Biggl( F^{-1} Biggl[ p_{0}(x,y)+ sum _{s=x_{0}}^{theta (x)-1}sum_{t=y_{0}}^{vartheta (y)+1} phi _{1}(s,t)f(s,t) Biggr] Biggr) Biggr} $$
for (0leq xleq x_{1}), (0leq yleq y_{1}), where
$$ p_{0}(x,y)=Fbigl(G bigl( a(x,y) bigr) bigr)+sum _{s=x_{0}}^{theta (x)-1} sum_{t=y_{0}}^{vartheta (y)+1} phi _{1}(s,t) Biggl( sum_{t=x_{0}}^{s-1} phi _{2}(tau ,t) Biggr), $$
and (( x_{1},y_{1} ) in Omega ) is chosen so that
$$ Biggl[ p_{0}(x,y)+sum_{s=x_{0}}^{theta (x)-1} sum_{t=y_{0}}^{ vartheta (y)+1}phi _{1}(s,t)f(s,t) Biggr] in operatorname {Dom}bigl( F^{-1} bigr) .$$
Corollary 2.19
If we take (mathbb{T}=overline{q^{mathbb{Z}}}) in Theorem 2.15by relation (1.4), then the following inequality
$$begin{aligned} {psi} bigl( u(x,y) bigr) leq &a(x,y)+(q-1)^{2}sum _{s=(log _{q}{x_{0}})}^{( log _{q}{theta (x)})-1}sum_{t=(log _{q}{y_{0}})}^{(log _{q}{ vartheta (y)})+1} q^{(s+t)} \ &{}times phi _{1}bigl(q^{s},q^{t} bigr) Biggl[ fbigl(q^{s},q^{t}bigr){omega} bigl( u bigl(q^{s},q^{t}bigr) bigr) eta bigl( u bigl(q^{s},q^{t}bigr) bigr) \ &{} +(q-1)sum_{s=(log _{q}{x_{0}})}^{(log _{q}{s})-1}q^{t} phi _{2}bigl(tau ,q^{t}bigr) {omega} bigl( u bigl(tau ,q^{t}bigr) bigr) eta bigl( ubigl(tau ,q^{t}bigr) bigr) Biggr] end{aligned}$$
for ((x,y)in Omega ) implies
$$begin{aligned} u(x,y) leq& {psi} ^{-1} Biggl{ G^{-1} Biggl( F^{-1} Biggl[ p_{0}(x,y)\ &{}+(q-1)^{2} sum _{s=(log _{q}{x_{0}})}^{(log _{q}{theta (x)})-1}sum _{t=( log _{q}{y_{0}})}^{(log _{q}{vartheta (y)})+1} q^{(s+t)}phi _{1}bigl(q^{s},q^{t}bigr)f bigl(q^{s},q^{t}bigr) Biggr] Biggr) Biggr} end{aligned}$$
for (0leq xleq x_{1}), (0leq yleq y_{1}), where
$$ p_{0}(x,y)=Fbigl(G bigl( a(x,y) bigr) bigr)+(q-1)^{2} sum_{s=(log _{q}{x_{0}})}^{( log _{q}{theta (x)})-1}sum _{t=(log _{q}{y_{0}})}^{(log _{q}{ vartheta (y)})+1} q^{(s+t)}phi _{1}bigl(q^{s},q^{t}bigr) Biggl( sum _{t=( log _{q}{x_{0}})}^{(log _{q}{s})-1}phi _{2}bigl( tau ,q^{t}bigr) Biggr) $$
and (( x_{1},y_{1} ) in Omega ) is chosen so that
$$ Biggl[ p_{0}(x,y)+(q-1)^{2}sum _{s=(log _{q}{x_{0}})}^{(log _{q}{ theta (x)})-1}sum_{t=(log _{q}{y_{0}})}^{(log _{q}{vartheta (y)})+1} q^{(s+t)}phi _{1}bigl(q^{s},q^{t} bigr)fbigl(q^{s},q^{t}bigr) Biggr] in operatorname {Dom}bigl( F^{-1} bigr) .$$
Corollary 2.20
Under the hypotheses of Corollary 2.14. If (u(x,y)) satisfies
$$begin{aligned} u^{q}(x,y) leq &a(x,y)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi _{1}(s,t) biggl[ f(s,t)u^{p}(s,t)u^{r}(s,t) \ & {}+ int _{x_{0}}^{s}phi _{2}(tau ,t)u^{p}(tau ,t)u^{r}( tau ,t)Delta tau biggr] nabla tDelta s end{aligned}$$
(2.38)
for ((x,y)in Omega ), then
$$ u(x,y)leq biggl{ p_{0}(x,y)+frac{q-p-r}{q} int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t)f(s,t) nabla tDelta s biggr} ^{frac{1}{q-p-r}}, $$
(2.39)
where
$$ p_{0}(x,y)= bigl( a ( x,y ) bigr) ^{frac{q-p-r}{q}}+ frac{q-p-r}{q} int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t) biggl( int _{x_{0}}^{s}phi _{2}(tau ,t) Delta tau biggr) nabla tDelta s. $$
Proof
An application of Theorem 2.15 with ({psi} ( u ) =u^{q}), ({omega} ( u ) =u^{p}), and (eta ( u ) =u^{r}) yields the desired inequality (2.39). □
Theorem 2.21
Under the hypotheses of Theorem 2.9. If (u(x,y)) satisfies
$$begin{aligned} {psi} bigl( u(x,y) bigr) leq &a(x,y)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t)eta bigl( u ( s,t ) bigr) \ & {}timesbiggl[ f(s,t) {omega} bigl( u(s,t) bigr) + int _{x_{0}}^{s} phi _{2}(tau ,t) Delta tau biggr] nabla tDelta s end{aligned}$$
(2.40)
for ((x,y)in Omega ), then
$$ u ( x,y ) leq {psi} ^{-1} biggl{ G_{1}^{-1} biggl( F_{1}^{-1} biggl[ F_{1} bigl( p_{1} ( x,y ) bigr) + int _{x_{0}}^{ theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t)f(s,t) nabla t Delta s biggr] biggr) biggr} $$
(2.41)
for (0leq xleq x_{2}), (0leq yleq y_{2}), where
$$begin{aligned}& G_{1}(v)= int _{v_{0}}^{v} frac{Delta s}{eta ( {psi} ^{-1}(s) ) },quad vgeq v_{0}>0,qquad G_{1}(+ infty )= int _{v_{0}}^{+infty } frac{Delta s}{eta ( {psi} ^{-1}(s) ) }=+ infty \& F_{1}(v)= int _{v_{0}}^{v} frac{Delta s}{ {omega} [ {psi} ^{-1} ( G_{1}^{-1} ( s ) ) ] },quad vgeq v_{0}>0,qquad F_{1}(+ infty )=+infty \& p_{1} ( x,y ) =G_{1}bigl(a(x,y)bigr)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t) biggl( int _{x_{0}}^{s} phi _{2}(tau ,t) Delta tau biggr) nabla tDelta s, end{aligned}$$
and (( x_{2},y_{2} ) in Omega ) is chosen so that
$$ biggl[ F_{1} bigl( p_{1} ( x,y ) bigr) + int _{x_{0}}^{ theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t)f(s,t) nabla t Delta s biggr] in operatorname {Dom}bigl( F_{1}^{-1} bigr). $$
Proof
Suppose that (a(x,y)>0). Fixing an arbitrary ((x_{0},y_{0})in Omega ), we define a positive and nondecreasing function (z(x,y)) by
$$begin{aligned} z ( x,y ) =&a(x_{0},y_{0})+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t)eta bigl( u ( s,t ) bigr) biggl[ f(s,t) {omega} bigl( u(s,t) bigr) \ & {}+ int _{x_{0}}^{s}phi _{2}(tau ,t) Delta tau biggr] nabla tDelta s end{aligned}$$
for (0leq xleq x_{0}leq x_{2}), (0leq yleq y_{0}leq y_{2}), then (z ( x_{0},y ) =z(x,y_{0})=a(x_{0},y_{0})),
$$ u(x,y)leq {psi} ^{-1} bigl( z(x,y) bigr) $$
(2.42)
and
$$begin{aligned} z^{Delta _{x}}(x,y) leq &theta ^{Delta }(x) int _{y_{0}}^{ vartheta (y)}phi _{1}bigl(theta (x),tbigr)eta bigl[ {psi} ^{-1} bigl( zbigl( theta (x),tbigr) bigr) bigr] biggl[ fbigl(theta (x),tbigr) {omega} bigl( {psi} ^{-1} bigl( zbigl(theta (x),tbigr) bigr) bigr) \ & {}+ int _{x_{0}}^{theta (x)}phi _{2}(tau ,t) Delta tau biggr] nabla t \ leq &theta ^{Delta }(x)eta bigl[ {psi} ^{-1} bigl( z bigl( theta (x),vartheta (y) bigr) bigr) bigr] int _{y_{0}}^{ vartheta (y)}phi _{1}bigl(theta (x),tbigr) biggl[ fbigl(theta (x),tbigr) {omega} bigl( {psi} ^{-1} bigl( zbigl(theta (x),tbigr) bigr) bigr) \ & {}+ int _{x_{0}}^{theta (x)}phi _{2}(tau ,t) Delta tau biggr] nabla t, end{aligned}$$
then
$$begin{aligned} frac{z^{Delta _{x}}(x,y)}{eta [ {psi} ^{-1} ( z ( x,y ) ) ] } leq &theta ^{Delta }(x) int _{y_{0}}^{vartheta (y)}phi _{1}bigl(theta (x),tbigr) biggl[ fbigl(theta (x),tbigr) {omega} bigl( {psi} ^{-1} bigl( zbigl(theta (x),tbigr) bigr) bigr) \ & {}+ int _{x_{0}}^{theta (x)}phi _{2}(tau ,t) Delta tau biggr] nabla t. end{aligned}$$
Taking Δ-integral for the above inequality gives
$$begin{aligned} G_{1} bigl( z ( x,y ) bigr) leq &G_{1} bigl( z ( 0,y ) bigr) + int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi _{1}(s,t) biggl[ f(s,t) {omega} bigl( {psi} ^{-1} bigl( z(s,t) bigr) bigr) \ & {}+ int _{x_{0}}^{s}phi _{2}(tau ,t) Delta tau biggr] nabla tDelta s, end{aligned}$$
then
$$begin{aligned} G_{1} bigl( z ( x,y ) bigr) leq &G_{1} bigl( a(x_{0},y_{0}) bigr) + int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)} phi _{1}(s,t) biggl[ f(s,t) {omega} bigl( {psi} ^{-1} bigl( z(s,t) bigr) bigr) \ & {}+ int _{x_{0}}^{s}phi _{2}(tau ,t) Delta tau biggr] nabla tDelta s. end{aligned}$$
Since ((x_{0},y_{0})in Omega ) is chosen arbitrarily, the last inequality can be restated as
$$ G_{1} bigl( z ( x,y ) bigr) leq p_{1} ( x,y ) + int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)} phi _{1}(s,t)f(s,t) { omega} bigl( {psi} ^{-1} bigl( z(s,t) bigr) bigr) nabla tDelta s. $$
(2.43)
It is easy to observe that (p_{1} ( x,y ) ) is a positive and nondecreasing function for all ((x,y)in Omega ), then an application of Lemma 2.1 to (2.43) yields the inequality
$$ z ( x,y ) leq G_{1}^{-1} biggl( F_{1}^{-1} biggl[ F_{1} bigl( p_{1} ( x,y ) bigr) + int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t)f(s,t) nabla tDelta s biggr] biggr) . $$
(2.44)
From (2.44) and (2.42) we get the desired inequality (2.41).
If (a(x,y)=0), we carry out the above procedure with (epsilon >0) instead of (a(x,y)) and subsequently let (epsilon rightarrow 0). The proof is complete. □
Now, as special cases of our results, we will give the continuous, discrete, and quantum inequalities. Namely, in the cases of time scales (mathbb{T}=mathbb{R}), (mathbb{T}=hmathbb{Z}), (mathbb{T}=mathbb{Z}), and (mathbb{T}=overline{q^{mathbb{Z}}}).
Remark 2.22
If we take (mathbb{T}=mathbb{R}) and (x_{0}=0) and (y_{0}=0) in Theorem 2.21, then, by relation (1.1), inequality (2.41) becomes the inequality obtained in [15, Theorem 2.7].
Corollary 2.23
If we take (mathbb{T}=h mathbb{Z}) in Theorem 2.15by relation (1.3), then the following inequality
$$begin{aligned} {psi} bigl( u(x,y) bigr) leq &a(x,y)+h^{2}sum _{s= frac{x_{0}}{h}}^{frac{theta (x)}{h}-1}sum_{t=frac{y_{0}}{h}}^{ frac{vartheta (y)}{h}+1} phi _{1}(sh,th)eta bigl( u ( sh,th ) bigr) \ & {}timesBiggl[ f(sh,th) {omega} bigl( u(sh,th) bigr) +sum _{t= frac{x_{0}}{h}}^{frac{s}{h}-1}phi _{2}(tau ,th) Biggr] end{aligned}$$
for ((x,y)in Omega ), then
$$ u ( x,y ) leq {psi} ^{-1} Biggl{ G_{1}^{-1} Biggl( F_{1}^{-1} Biggl[ F_{1} bigl( p_{1} ( x,y ) bigr) +h^{2}sum _{s= frac{x_{0}}{h}}^{frac{theta (x)}{h}-1}sum_{t=frac{y_{0}}{h}}^{ frac{vartheta (y)}{h}+1} phi _{1}(sh,th)f(sh,th) Biggr] Biggr) Biggr} $$
for (0leq xleq x_{2}), (0leq yleq y_{2}), where
$$begin{aligned}& G_{1}(v)=sum_{s=frac{v_{0}}{h}}^{frac{v}{h}-1} frac{h}{eta ( {psi} ^{-1}(sh) ) },quad vgeq v_{0}>0,qquad G_{1}(+ infty )=sum _{s=frac{v_{0}}{h}}^{+infty } frac{h}{eta ( {psi} ^{-1}(sh) ) }=+infty \& F_{1}(v)=sum_{s=frac{v_{0}}{h}}^{frac{v}{h}-1} frac{h}{ {omega} [ {psi} ^{-1} ( G_{1}^{-1} ( sh ) ) ] },quad vgeq v_{0}>0,qquad F_{1}(+ infty )=+ infty \& p_{1} ( x,y ) =G_{1}bigl(a(x,y)bigr)+h^{2} sum_{s=frac{x_{0}}{h}}^{ frac{theta (x)}{h}-1}sum _{t=frac{y_{0}}{h}}^{ frac{vartheta (y)}{h}+1}phi _{1}(sh,th) Biggl( h sum_{t= frac{x_{0}}{h}}^{frac{s}{h}-1}phi _{2}( tau ,th) Biggr), end{aligned}$$
and (( x_{2},y_{2} ) in Omega ) is chosen so that
$$ Biggl[ F_{1} bigl( p_{1} ( x,y ) bigr) +h^{2}sum_{s= frac{x_{0}}{h}}^{frac{theta (x)}{h}-1}sum _{t=frac{y_{0}}{h}}^{ frac{vartheta (y)}{h}+1}phi _{1}(sh,th)f(sh,th) Biggr] in operatorname {Dom}bigl( F_{1}^{-1} bigr). $$
Corollary 2.24
In Corollary 2.23, if we take (h=1), then the following inequality
$$begin{aligned} {psi} bigl( u(x,y) bigr) leq &a(x,y)+sum_{s=x_{0}}^{theta (x)-1} sum_{t=y_{0}}^{vartheta (y)+1}phi _{1}(s,t) eta bigl( u ( s,t ) bigr) \ & {}timesBiggl[ f(s,t) {omega} bigl( u(s,t) bigr) +sum _{t=x_{0}}^{s-1} phi _{2}(tau ,t) Biggr] end{aligned}$$
for ((x,y)in Omega ), then
$$ u ( x,y ) leq {psi} ^{-1} Biggl{ G_{1}^{-1} Biggl( F_{1}^{-1} Biggl[ F_{1} bigl( p_{1} ( x,y ) bigr) +sum_{s=x_{0}}^{ theta (x)-1} sum_{t=y_{0}}^{vartheta (y)+1}phi _{1}(s,t)f(s,t) Biggr] Biggr) Biggr} $$
for (0leq xleq x_{2}), (0leq yleq y_{2}), where
$$begin{aligned}& G_{1}(v)=sum_{s=v_{0}}^{v-1} frac{1}{eta ( {psi} ^{-1}(s) ) },quad vgeq v_{0}>0,qquad G_{1}(+ infty )=sum _{s=v_{0}}^{+infty } frac{1}{eta ( {psi} ^{-1}(s) ) }=+ infty, \& F_{1}(v)=sum_{s=v_{0}}^{v-1} frac{1}{ {omega} [ {psi} ^{-1} ( G_{1}^{-1} ( s ) ) ] },quad vgeq v_{0}>0,qquad F_{1}(+ infty )=+ infty,\& p_{1} ( x,y ) =G_{1}bigl(a(x,y)bigr)+sum _{s=x_{0}}^{theta (x)-1} sum_{t=y_{0}}^{vartheta (y)+1} phi _{1}(s,t) Biggl( sum_{t=x_{0}}^{s-1} phi _{2}(tau ,t) Biggr) end{aligned}$$
and (( x_{2},y_{2} ) in Omega ) is chosen so that
$$ Biggl[ F_{1} bigl( p_{1} ( x,y ) bigr) +sum _{s=x_{0}}^{ theta (x)-1}sum_{t=y_{0}}^{vartheta (y)+1} phi _{1}(s,t)f(s,t) Biggr] in operatorname {Dom}bigl( F_{1}^{-1} bigr). $$
Corollary 2.25
If we take (mathbb{T}=overline{q^{mathbb{Z}}}) in Theorem 2.21by relation (1.4), then the following inequality
$$begin{aligned} {psi} bigl( u(x,y) bigr) leq &a(x,y)+(q-1)^{2}sum _{s=(log _{q}{x_{0}})}^{( log _{q}{theta (x)})-1}sum_{t=(log _{q}{y_{0}})}^{(log _{q}{ vartheta (y)})+1} q^{(s+t)}phi _{1}bigl(q^{s},q^{t} bigr)eta bigl( u bigl( q^{s},q^{t} bigr) bigr) \ &{}times Biggl[ fbigl(q^{s},q^{t}bigr) {omega} bigl( ubigl(q^{s},q^{t}bigr) bigr) +(q-1) sum _{t=(log _{q}{x_{0}})}^{(log _{q}{s})-1}q^{t}phi _{2}bigl(tau ,q^{t}bigr) Biggr] end{aligned}$$
for ((x,y)in Omega ), then
$$begin{aligned} u ( x,y ) leq &{psi} ^{-1} Biggl{ G_{1}^{-1} Biggl( F_{1}^{-1} Biggl[ F_{1} bigl( p_{1} ( x,y ) bigr) \ &{}+(q-1)^{2} sum _{s=(log _{q}{x_{0}})}^{(log _{q}{theta (x)})-1}sum_{t=( log _{q}{y_{0}})}^{(log _{q}{vartheta (y)})+1} q^{(s+t)}phi _{1}bigl(q^{s},q^{t} bigr)fbigl(q^{s},q^{t}bigr) Biggr] Biggr) Biggr} end{aligned}$$
for (0leq xleq x_{2}), (0leq yleq y_{2}), where
$$begin{aligned}& G_{1}(v)=sum_{s=(log _{q}{v_{0}})}^{(log _{q}{v})-1} frac{(q-1)q^{s}}{eta ( {psi} ^{-1}(q^{s}) ) },quad vgeq v_{0}>0,\& G_{(q-1)q^{s}}(+ infty )= sum_{s=(log _{q}{v_{0}})}^{+infty } frac{(q-1)q^{s}}{eta ( {psi} ^{-1}(q^{s}) ) }=+ infty ,\& F_{1}(v)=sum_{s=(log _{q}{v_{0}})}^{(log _{q}{v})-1} frac{(q-1)q^{s}}{ {omega} [ {psi} ^{-1} ( G_{1}^{-1} ( q^{s} ) ) ] },quad vgeq v_{0}>0,qquad F_{1}(+ infty )=+ infty,\& p_{1} ( x,y ) =G_{1}bigl(a(x,y)bigr)+(q-1)^{2} sum_{s=(log _{q}{x_{0}})}^{( log _{q}{theta (x)})-1}sum _{t=(log _{q}{y_{0}})}^{(log _{q}{ vartheta (y)})+1} q^{(s+t)}phi _{1}bigl(q^{s},q^{t}bigr) Biggl( sum _{t=x_{0}}^{s-1} phi _{2}bigl( tau ,q^{t}bigr) Biggr) end{aligned}$$
and (( x_{2},y_{2} ) in Omega ) is chosen so that
$$ Biggl[ F_{1} bigl( p_{1} ( x,y ) bigr) +(q-1)^{2} sum_{s=(log _{q}{x_{0}})}^{(log _{q}{theta (x)})-1} sum_{t=( log _{q}{y_{0}})}^{(log _{q}{vartheta (y)})+1} q^{(s+t)}phi _{1}bigl(q^{s},q^{t}bigr)f bigl(q^{s},q^{t}bigr) Biggr] in operatorname {Dom}bigl( F_{1}^{-1} bigr). $$
Theorem 2.26
Under the hypotheses of Theorem 2.9, and let p be a nonnegative constant. If (u(x,y)) satisfies
$$begin{aligned} {psi} bigl( u(x,y) bigr) leq &a(x,y)+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t)u^{p} ( s,t ) \ &{}times biggl[ f(s,t) {omega} bigl( u(s,t) bigr) + int _{x_{0}}^{s} phi _{2}(tau ,t) Delta tau biggr] nabla tDelta s end{aligned}$$
(2.45)
for ((x,y)in Omega ), then
$$ u ( x,y ) leq {psi} ^{-1} biggl{ G_{1}^{-1} biggl( F_{1}^{-1} biggl[ F_{1} bigl( p_{1} ( x,y ) bigr) + int _{x_{0}}^{ theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t)f(s,t) nabla t Delta s biggr] biggr) biggr} $$
(2.46)
for (0leq xleq x_{2}), (0leq yleq y_{2}), where
$$ G_{1}(v)= int _{v_{0}}^{v} frac{Delta s}{ [ {psi} ^{-1}(s) ] ^{p}},quad vgeq v_{0}>0,qquad G_{1}(+ infty )= int _{v_{0}}^{+infty } frac{Delta s}{ [ {psi} ^{-1}(s) ] ^{p}}=+infty , $$
(2.47)
and (F_{1}), (p_{1}) are as in Theorem 2.21and (( x_{2},y_{2} ) in Omega ) is chosen so that
$$ biggl[ F_{1} bigl( p_{1} ( x,y ) bigr) + int _{x_{0}}^{ theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t)f(s,t) nabla t Delta s biggr] in operatorname {Dom}bigl( F_{1}^{-1} bigr) . $$
Proof
An application of Theorem 2.21 with (eta ( u ) =u^{p}) yields the desired inequality (2.46). □
Remark 2.27
Taking (mathbb{T}=mathbb{R}). The inequality established in Theorem 2.26 generalizes [38, Theorem 1] (with (p=1), (a(x,y)=b(x)+c(y)), (x_{0}=0), (y_{0}=0), (phi _{1}(s,t)f(s,t)=h(s,t)), and (phi _{1}(s,t) ( int _{x_{0}}^{s}phi _{2}(tau ,t)Delta tau ) =g(s,t))).
Corollary 2.28
Under the hypotheses of Theorem 2.26, and let (q>p>0) be constants. If (u(x,y)) satisfies
$$begin{aligned} u^{q}(x,y) leq &a(x,y)+frac{p}{p-q} int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi _{1}(s,t)u^{p} ( s,t ) \ &{} timesbiggl[ f(s,t) {omega} bigl( u(s,t) bigr) + int _{x_{0}}^{s} phi _{2}(tau ,t) Delta tau biggr] nabla tDelta s end{aligned}$$
(2.48)
for ((x,y)in Omega ), then
$$ u(x,y)leq biggl{ F_{1}^{-1} biggl[ F_{1} bigl( p_{1} ( x,y ) bigr) + int _{x_{0}}^{theta (x)} int _{y_{0}}^{ vartheta (y)}phi _{1}(s,t)f(s,t) nabla tDelta s biggr] biggr} ^{frac{1}{q-p}} $$
(2.49)
for (0leq xleq x_{2}), (0leq yleq y_{2}), where
$$ p_{1} ( x,y ) = bigl[ a(x,y) bigr] ^{frac{q-p}{q}}+ int _{x_{0}}^{theta (x)} int _{y_{0}}^{vartheta (y)}phi _{1}(s,t) biggl( int _{x_{0}}^{s}phi _{2}(tau ,t) Delta tau biggr) nabla tDelta s $$
and (F_{1}) is defined in Theorem 2.21.
Proof
An application of Theorem 2.26 with ({psi} ( u(x,y) ) =u^{p}) to (2.48) yields inequality (2.49); to save space, we omit the details. □
Remark 2.29
Taking (mathbb{T}=mathbb{R}), (x_{0}=0), (y_{0}=0), (a(x,y)=b(x)+c(y)), (phi _{1}(s,t)f(s,t)=h(s,t)), and (phi _{1}(s,t) ( int _{x_{0}}^{s}phi _{2}(tau ,t)Delta tau ) =g(s,t) ) in Corollary 2.28, we obtain [39, Theorem 1].
Remark 2.30
Taking (mathbb{T}=mathbb{R}), (x_{0}=0), (y_{0}=0), (a(x,y)=c^{frac{p}{p-q}}), (phi _{1}(s,t)f(s,t)=h(t)), and (phi _{1}(s, t) ( int _{x_{0}}^{s}phi _{2}(tau ,t)Delta tau ) =g(t)) and keeping y fixed in Corollary 2.28, we obtain [25, Theorem 2.1].
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/.
Disclaimer:
This article is autogenerated using RSS feeds and has not been created or edited by OA JF.
Click here for Source link (https://www.springeropen.com/)
