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

Aug 25, 2022

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].

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