Some New Delay Integral Inequalities in Two Independent Variables on Time Scales

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Volume 2011, Article ID 659563,18pages doi:10.1155/2011/659563

Research Article

Some New Delay Integral Inequalities in Two Independent Variables on Time Scales

Bin Zheng, Yaoming Zhang, and Qinghua Feng

School of Science, Shandong University of Technology, Zibo, Shandong, 255049, China

Correspondence should be addressed to Bin Zheng,[email protected] Received 9 August 2011; Accepted 10 October 2011

Academic Editor: C. Conca

Copyrightq2011 Bin Zheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Some new Gronwall-Bellman type delay integral inequalities in two independent variables on time scales are established, which can be used as a handy tool in the research of boundedness of solutions of delay dynamic equations on time scales. Some of the established results are 2D extensions of several known results in the literature, while some results unify existing continuous and discrete analysis.

1. Introduction

In the research of solutions of certain differential and difference equations, if the solutions are unknown, then it is necessary to study their qualitative and quantitative properties such as boundedness, uniqueness, and continuous dependence on initial data. The Gronwall- Bellman inequality1,2and its various generalizations, which provide explicit bounds, play a fundamental role in the research of this domain. During the past decades, much effort has been done for developing such inequalitiese.g., see3–15and the references therein. On the other hand, Hilger16initiated the theory of time scales as a theory capable to contain both difference and differential calculus in a consistent way. Since then many authors have expounded on various aspects of the theory of dynamic equations on time scalese.g., see 17–19 and the references therein. In these investigations, integral inequalities on time scales have been paid much attention by many authors, which play a fundamental role in the research of quantitative as well as qualitative properties of solutions of certain dynamic equations on time scales. A lot of integral inequalities on time scales have been established e.g., see 20–26, which have been designed to unify continuous and discrete analysis.

But to our best knowledge, the Gronwall-Bellman-type delay integral inequalities on time scales have been paid little attention in the literature so far. Recent results in this direction include the work of Li27and that of Ma and Peˇcari´c28. Furthermore, nobody has studied

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the Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales.

The aim of this paper is to establish some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales, which provide new bounds for the unknown functions concerned. Some of our results are 2D extensions of many known inequalities in the literature, while some results unify existing continuous and discrete analysis. For illustrating the validity of the established results, we will present some applications of them.

First we will give some preliminaries on time scales and some universal symbols for further use.

Throughout this paper,Rdenotes the set of real numbers andR 0,∞, whileZ denotes the set of integers. For two given setsG, H, we denote the set of maps fromGtoH byG, H.

A time scale is an arbitrary nonempty closed subset of the real numbers. In this paper, Tdenotes an arbitrary time scale. OnTwe define the forward and backward jump operators σ∈T,Tandρ∈T,Tsuch thatσt inf{s∈T, s > t},ρt sup{s∈T, s < t}.

Definition 1.1. The graininessμ∈T,Ris defined byμt σt−t.

Remark 1.2. Obviously,μt 0 ifTRwhileμt 1 ifTZ.

Definition 1.3. A point t ∈ T is said to be left-dense ifρt t andt /infT, right-dense if σt tandt /supT, left-scattered ifρt< t, and right-scattered ifσt> t.

Definition 1.4. The setT^κ is defined to be T if Tdoes not have a left-scattered maximum;

otherwise it isTwithout the left-scattered maximum.

Definition 1.5. A functionf ∈T,Ris called rd-continuous if it is continuous at right-dense points and if the left-sided limits exist at left-dense points, while f is called regressive if 1μtft/0.Crddenotes the set of rd-continuous functions, whileRdenotes the set of all regressive and rd-continuous functions, andR {f|f ∈R,1μtft>0, ∀t∈T}.

Definition 1.6. For some t ∈ T^κ and a function f ∈ T,R, the delta derivative of f at t is denoted byf^Δt provided it existswith the property such that for everyε >0 there exists a neighborhoodUoftsatisfying

fσt−fs−f^Δtσt−s≤ε|σt−s| ∀s∈U. 1.1

Similarly, for somey ∈ T^κand a function f ∈ T×T, R, the partial delta off with respect toyis denoted byfx, y^Δ_y orf_y^Δx, yand satisfies

f x, σ

y

−fx, s−f_y^Δ x, y

σ y

−s≤εσ y

−s ∀ε >0, 1.2

wheres∈UandUis a neighborhood ofy.

Remark 1.7. IfTR, thenf^Δtbecomes the usual derivativeft, whilef^Δt ft1−ft ifTZ, which represents the forward diﬀerence.

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For more details about the calculus of time scales, see29. In the rest of this paper, for the convenience of notation, we always assume thatT0 x0,∞

T,T0 y0,∞ T, wherex₀, y₀∈T^κand furthermore assumeT0⊆T^κ,T0⊆T^κ.

2. Main Results

We will give some lemmas for further use.

Lemma 2.1. SupposeX ∈T0 is a fixed number, anduX, y, aX, y, bX, y∈Crd, mX, y∈ Rwith respect toy,mX, y≥0, then

u X, y

≤a X, y

b X, y

y y0

mX, tuX, tΔt, y∈T0, 2.1

implies

u X, y

≤a X, y

b X, y

y y0

e_m y, σt

aX, tmX, tΔt, y∈T0, 2.2

wheremX, y mX, ybX, y: ande_my, y0is the unique solution of the following equation z^Δ_y

X, y m

X, y z

X, y , z

X, y₀

1. 2.3

The proof ofLemma 2.1is similar to26, Theorem 5.6.

Lemma 2.2. Under the conditions ofLemma 2.1, and furthermore assumingax, yis nondecreasing inyfor every fixedx,bx, y≡1, then one has

u X, y

≤a X, y

e_m y, y₀

. 2.4

Proof. Sinceax, yis nondecreasing inyfor every fixedx, then fromLemma 2.1we have

u X, y

≤a X, y

_y

y0

em

y, σt

aX, tmX, tΔt≤a X, y

1 _y

y0

em

y, σt

mX, tΔt . 2.5 On the other hand, from29, Theorems 2.39 and 2.36iwe have 1_y

y0e_my, σtmX, tΔt emy, y0. Then collecting the above information, we can obtain the desired inequality.

Lemma 2.3see11. Assume thata≥0, p≥q≥0, andp /0; then, for anyK >0

a^q/p ≤ q

pK^q−p/pa p−q

p K^q/p. 2.6

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Lemma 2.4. Leth:T×R → Rbe continuous and nondecreasing in the second variable, and assume Xis a fixed number inT. SupposevX, yandwX, ysatisfy the dynamics inequalities:

v^Δ_y ≤h y, v

, w^Δ_y ≥h y, w

. 2.7

ThenvX, y0≤wX, y0for somey₀∈TimpliesvX, y≤wX, yfor ally∈T.

The proof ofLemma 2.4is similar to26, Theorem 5.7.

Theorem 2.5. Suppose u, a, b, f ∈ CrdT0 ×T0,R, and ax, y, bx, y are nondecreasing.p is a constant, and p ≥ 1. τ₁ ∈ T0,T, τ1x ≤ x,−∞ < α inf{τ1x, x ∈ T0} ≤ x₀.τ₂ ∈ T0,T, τ2y≤y,−∞< βinf{τ2y, y∈T0} ≤y0.φ∈Crdα, x0×β, y0

T²,R. If for x, y∈T0×T0,ux, ysatisfies the following inequality:

u^p x, y

≤a x, y

b x, y

y y0

_x

x0

fs, tuτ1s, τ2t

ΔsΔt, 2.8

with the initial condition u

x, y φ

x, y

, ifx∈α, x0

Tor y∈

β, y₀ T, φ

τ₁x, τ2

y

≤a^1/p x, y

, ∀ x, y

∈T0×T0, ifτ₁x≤x₀ orτ₂ y

≤y₀,

2.9

then

u x, y

≤

H1

x, y b

x, y

y y0

e_H

2

y, σt

H2x, tH1x, tΔt

1/p

,

x, y

∈T0×T0, 2.10 where

H1

x, y a

x, y b

x, y

y y0

_x

x0

fs, tp−1

p K^1/pΔsΔt, ∀K >0, H2

x, y

_x

x0

f s, y1

pK^1−p/pΔs, H₂

x, y b

x, y H₂

x, y .

2.11

Proof. FixX∈T0, andx∈x0, X

T, y∈T0. Let

v x, y

a x, y

b x, y

y y0

_x

x0

fs, tuτ1s, τ2t

ΔsΔt. 2.12

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Then

u x, y

≤v^1/p x, y

≤v^1/p X, y

, ∀x∈x0, X

T, y∈T0. 2.13

Ifτ1x≥x0andτ2y≥y0, thenτ1x∈x0, XT, τ2y∈T0, and

u

τ₁x, τ2

y

≤v^1/p

τ₁x, τ2

y

≤v^1/p x, y

. 2.14

Ifτ₁x≤x₀orτ₂y≤y₀, then from2.9we have

u

τ₁x, τ2

y φ

τ₁x, τ2

y

≤a^1/p x, y

≤v^1/p x, y

. 2.15

From2.14and2.15we always have

u

τ1x, τ2

y

≤v^1/p x, y

, x∈x0, X

T, y∈T0. 2.16

Moreover

v X, y

a X, y

b X, y

y y0

_X

x0

fs, tuτ1s, τ2t ΔsΔt

≤a X, y

b X, y

y y0

_X

x0

fs, tv^1/ps, tΔsΔt.

2.17

FromLemma 2.3, we have

v^1/ps, t≤ 1

pK^1−p/pvs, t p−1

p K^1/p, ∀K >0. 2.18

So

v X, y

≤a X, y

b X, y

y y0

_X

x0

fs, t 1

pK^1−p/pvs, t p−1 p K^1/p

ΔsΔt

≤a X, y

b X, y

y y0

_X

x0

fs, tp−1

p K^1/pΔsΔt b

X, y

y y0

_X

x0

fs, t1

pK^1−p/pΔs vX, tΔt H1

X, y b

X, y

y y0

H2X, tvX, tΔt.

2.19

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Then applyingLemma 2.1to2.19, we obtain

v X, y

≤H1

X, y b

X, y

y y0

e_H₂ y, σt

H2X, tH1X, tΔt. 2.20

So

u x, y

≤v^1/p X, y

≤

H₁ X, y

b X,y

y y0

e_H

2

y, σt

H₂X, tH1X, tΔt

1/p

,

x∈x0, X

T, y∈T0.

2.21

SettingxXin2.21, it follows that

u X, y

≤

H1

X, y b

X, y

y y0

e_H₂ y, σt

H2X, tH1X, tΔt

1/p

. 2.22

ReplacingXwithxin2.22, we obtain the desired inequality.

Remark 2.6. Theorem 2.5is the 2D extension of27, Theorem 1. For its special caseT R, the established bound forux, yin2.10is a new bound compared with the result in12, Theorem 2.2.

Remark 2.7. Assumebx, y≡1 inTheorem 2.5. If we applyLemma 2.2instead ofLemma 2.1 to2.19in the proof ofTheorem 2.5, then we obtain another bound forux, yas follows:

u x, y

≤ H1

x, y eH2

y, y0

_1/p

,

x, y

∈T0×T0. 2.23

Now we will establish a more general inequality than that inTheorem 2.5.

Theorem 2.8. Supposeu, a, b, f, φ, τ1, τ2, α, β are the same as inTheorem 2.5, andg ∈ CrdT0 × T0,R.p, q, rare constants, andp≥q≥0, p≥r≥0, ρ /0. If forx, y∈T0×T0,ux, ysatisfies the following inequality:

u^p x, y

≤a x, y

b x, y

y y0

_x

x0

fs, tu^qτ1s, τ2t gs, tu^rτ1s, τ2t ΔsΔt,

2.24 with the initial condition2.9, then

u x, y

≤

H1

x, y b

x, y

y y0

e

H2

y, σtH2x, tH1x, tΔt

1/p

,

x, y

∈T0×T0, 2.25

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where

H1

x, y a

x, y b

x, y

y y0

_x

x0

fs, tp−q

p K^q/pgs, tp−r p K^r/p

ΔsΔt, H₂

x, y

_x

x0

f

s, yq

pK^q−p/pg s, yr

pK^r−p/p

Δs, H₂

x, y b

x, yH₂ x, y

.

2.26

Proof. FixX∈T0, andx∈x0, XT, y∈T0. Let

v x, y

a x, y

b x, y

y y0

_x

x0

fs, tu^qτ1s, τ2t gs, tu^rτ1s, τ2t ΔsΔt.

2.27

Then

u x, y

≤v^1/p x, y

≤v^1/p X, y

, ∀x∈x0, X

T, y∈T0. 2.28

Similar to2.14–2.16, we obtain

u

τ₁x, τ2

y

≤v^1/p x, y

, x∈x0, X

T, y∈T0. 2.29

So

v X, y

a X, y

b X, y

y y0

_X

x0

fs, tu^qτ1s, τ2t gs, tu^rτ1s, τ2t ΔsΔt

≤a X, y

b X, y

y y0

_X

x0

fs, tv^q/ps, t gs, tv^r/ps, t ΔsΔt.

2.30

FromLemma 2.3, we have

v^q/ps, t≤ q

pK^q−p/pvs, t p−q

p K^q/p, ∀K >0, v^r/ps, t≤ r

pK^r−p/pvs, t p−r

p K^r/p, ∀K >0.

2.31

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Combining2.30and2.31we get that

v X, y

≤a X, y

b X, y

y y0

_X

x0

fs, t

q

pK^q−p/pvs, t p−q p K^q/p

gs, t r

pK^r−p/pvs, t p−r p K^r/p

ΔsΔt

≤a X, y

b X, y

y y0

_X

x0

fs, tp−q

ΔsΔt b

X, y

y y0

_X

x0

fs, tq

pK^q−p/pgs, tr

pK^r−p/p

Δs

vX, tΔt

H₁ X, y

b X, y

y y0

H₂X, tvX, tΔt.

2.32

ApplyingLemma 2.1to2.32yields

v X, y

≤H₁ X, y

b X, y

y y0

e

H2

y, σtH₂X, tH₁X, tΔt. 2.33

Then

u x, y

≤v^1/p X, y

≤

H1

X, y b

X, y

y y0

e_H

2

y, σtH2X, tH1X, tΔt

1/p

,

x∈x0, X

T, y∈T0.

2.34

SettingxXin2.34yields

u X, y

≤

H1

X, y b

X, y

y y0

e_H

2

y, σtH2X, tH1X, tΔt

1/p

. 2.35

Considering X ∈ T0 is arbitrary and replacing X with x in2.35, we obtain the desired inequality.

Remark 2.9. Assumebx, y≡1 inTheorem 2.8. If we applyLemma 2.2instead ofLemma 2.1 to2.32in the proof ofTheorem 2.8, then we obtain another bound forux, yas follows:

u x, y

≤ H1

x, y e_H

2

y, y0

^1/p

,

x, y

∈T0×T0. 2.36

Remark 2.10. Theorem 2.8is the 2D extension of27, Theorem 3.

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Theorem 2.11. Supposeu, f, α, β, φ, τ1, τ2are the same as inTheorem 2.5, andC >0 is a constant.

If forx, y∈T0×T0,ux, ysatisfies the following inequality:

u² x, y

≤C _y

y0

_x

x0

fs, tuτ1s, τ2t uτ1s, στ2tΔsΔt 2.37

x, y φ

x, y

, ifx∈α, x0

Tor y∈

β, y0 T, φ

τ1x, τ2

y

≤C^1/2, ∀ x, y

∈T0×T0, ifτ1x≤x0 orτ2

y

≤y0,

2.38

then

u x, y

≤√ C

_y

y0

_x

x0

fs, tΔsΔt, x, y

∈ T0×T0

. 2.39

Proof. Let the right side of2.37bev²x, y. Then

u x, y

≤v x, y

, ∀ x, y

∈ T0×T0

. 2.40

Forx, y∈T0×T0, ifτ1x≥x0 and τ2y≥y0, thenτ1x∈T0 andτ2y∈T0, and from 2.40we have

u

τ1x, τ2

y

≤v

τ1x, τ2

y

≤v x, y

. 2.41

Ifτ1x≤x0orτ2y≤y0, from2.38we have u

τ₁x, τ2

y φ

τ₁x, τ2

y

≤a^1/2

τ₁x, τ2

y

≤a^1/2 x, y

≤v x, y

. 2.42

So from2.41and2.42, we always have

u

τ1x, τ2

y

≤v x, y

, ∀ x, y

∈ T0×T0

. 2.43

Similarly, whenτ1x≥ x0andστ2y≥ y0, thenτ1x ∈T0 andστ2y ∈T0, and from 2.40we have

u

τ1x, σ τ2

y

≤v

τ1x, σ τ2

y

≤v x, σ

y

. 2.44

Whenτ1x ≤ x0orστ2y ≤ y0, consideringστ2y ≥ τ2y ≥ β, from2.38it follows that

u

τ1x, σ τ2

y φ

τ1x, σ τ2

y

≤C^1/2≤v x, y

≤v x, σ

y

. 2.45

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Combining2.44and2.45, we always have u

τ1x, σ τ2

y

≤v x, σ

y , ∀

x, y

∈ T0×T0

. 2.46

By2.43and2.46, we obtain

v² x, y

≤C _y

y0

_x

x0

fs, tvs, t vs, σtΔsΔt, x∈T0, y∈T0. 2.47

Let the right side of2.47bez²x, y. Then v

x, y

≤z x, y

, ∀ x, y

∈ T0×T0

, 2.48

z²

x, y^Δ

y _x

x0

f s, y

v s, y

v s, σ

y

Δs

≤ _x

x0

f s, y

Δs

v x, y

v x, σ

y

≤ _x

x0

f s, y

Δs

z x, y

z x, σ

y

.

2.49

Considering zx, y zx, σy ≥ zx0, y₀ C > 0, and z²x, y^Δ_y zx, y zx, σyzx, y^Δ_y, from2.49it follows that

z x, y_Δ

y ≤ _x

x0

f s, y

Δs. 2.50 An integration of 2.50 with respect to y from y₀ to y yields zx, y − zx, y0 ≤ _y

y0

_x

x0fs, tΔsΔt.

Consideringzx, y0

√C, it follows that

z x, y

≤√ C

_y

y0

_x

x0

fs, tΔsΔt. 2.51

Then combining2.40,2.48, and2.51, we obtain

u x, y

≤v x, y

≤z x, y

≤√ C

_y

y0

_x

x0

fs, tΔsΔt, 2.52

and the proof is complete.

Remark 2.12. If we takeT R, thenTheorem 2.11becomes the extension of the known Ou- Iang’s inequality13to the 2D case.

The following theorem provides a more general result thanTheorem 2.11.

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Theorem 2.13. Supposepis a positive integer, andp≥2. Under the conditions ofTheorem 2.11, if ux, ysatisfies

u^p x, y

≤C _y

y0

_x

x0

fs, t^p−1

l0

u^lτ1s, τ2tu^p−1−lτ1s, στ2t ΔsΔt, x, y

∈ T0×T0

,

2.53

x, y φ

x, y

, ifx∈α, x0

Tory∈

β, y₀ T, φ

τ1x, τ2

y

≤C^1/p, ∀ x, y

∈T0×T0, ifτ1x≤x0 orτ2

y

≤y0, 2.54

then

u x, y

≤C^1/p _y

y0

_x

x0

fs, tΔsΔt, x∈T0, y∈T0. 2.55

The proof ofTheorem 2.13is similar toTheorem 2.11. As long as we notice a delta d iﬀerentiable functionzx, y, the following formula26, Equation6.2holds:

z^p x, y_Δ

y

z x, y_Δ

y

p−1 l0

z^l

x, y z^p−1−l

x, σ

y

. 2.56

Then following a similar manner as inTheorem 2.11, we can deduce the desired result.

Theorem 2.14. Suppose u, f, τ₁, τ₂, φ, α, β are the same as in Theorem 2.5, ω ∈ CR,R is nondecreasing, andp, Care constants with p ≥ 1, C > 0. Furthermore, define a bijective function Gsuch thatGzx, y^Δ_y zx, y^Δ_y/ωz^1/px, y. If forx, y∈T0×T0,ux, ysatisfies the following inequality:

u^p x, y

≤C _y

y0

_x

x0

fs, tωuτ1s, τ2t

ΔsΔt, 2.57

x, y φ

x, y

, ifx∈α, x0

Tory∈

β, y₀ T, φ

τ1x, τ2

y

≤C^1/p, ∀ x, y

∈T0×T0, ifτ1x≤x0or τ2

y

≤y0, 2.58

then

u x, y

≤

G⁻¹

GC _y

y0

η₁x, tΔt

1/p

,

x, y

∈ T0×T0

, 2.59

whereη₁x, y _x

x0fs, yΔs.

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T, y∈T0. Let

v x, y

C _y

y0

_x

x0

fs, t

ΔsΔt, x∈x0, X

T, y∈T0. 2.60

Then

u x, y

≤v^1/p x, y

≤v^1/p X, y

, ∀x∈x0, X

T, y∈T0. 2.61

Similar to2.14–2.16, we obtain u

τ1x, τ2

y

≤v^1/p x, y

, x∈x0, X

T, y∈T0. 2.62

Moreover,

v^Δ_y X, y

_X

x0

f s, y

ω u

τ1s, τ2

y Δs

≤ _X

x0

f

s, y ω

v^1/p s, y

Δs

≤ _X

x0

f s, y

Δs

ω v^1/p

X, y η1

X, y ω

v^1/p X, y

.

2.63

LetvX, ybe the solution of the following problem:

v^Δ_y X, y

η₁ X, y

ω v^1/p

X, y , v

X, y₀

C. 2.64

ConsideringvX, y0 Candωis nondecreasing and continuous, then from2.63,2.64, andLemma 2.4, we have

v X, y

≤v X, y

, y∈T0. 2.65

On the other hand, from the definition ofGwe haveGvX, y^Δ_y v^Δ_yX, y/ωvX, y η₁X, y. Then an integration with respect toyfromy₀toyyields

G v

X, y

−G v

X, y0

_y

y0

η1X, tΔt, 2.66

that is,

v X, y

≤G⁻¹

GC _y

y0

η1X, tΔt , y∈T0. 2.67

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Combining2.61,2.65, and2.67, we have

u x, y

≤

G⁻¹

GC _y

y0

η₁X, tΔt

1/p

, x∈x0, X

T, y∈T0. 2.68

SettingxXin2.68, we get the desired result.

Remark 2.15. If we take T R, then Theorem 2.14 reduces to 14, Theorem 2.1, while Theorem 2.14reduces to15, Theorem 2.1if we takeTZ.

Theorem 2.16. Supposeu, f, τ₁, τ₂, φ, α, βare the same as inTheorem 2.5, and furthermore,uis delta diﬀerential onT0with respect toy,g∈CrdT0×T0,R.ω∈CR,Ris nondecreasing, andωis submultiplicative, that is,ωαβ≤ωαωβ, for all α, β∈R.C >0 is a constant.Gis a bijective function such thatGzx, y ^Δ_y zx, y^Δ_y /ωzx, y. If forx, y∈T0×T0,ux, ysatisfies the following inequality:

u x, y

≤C _y

y0

_x

x0

fs, tωuτ1s, τ2t gs, tuτ1s, τ2t

ΔsΔt, 2.69

x, y φ

x, y

, ifx∈α, x0

Tory∈

β, y₀ T, φ

τ₁x, τ2

y

≤C, ∀ x, y

∈ T0×T0

, ifτ₁x≤x₀ or τ₂ y

≤y₀, 2.70

then

u x, y

≤G⁻¹

GC

_y

y0

η2x, tΔt eB1

y, y0

, x, y

∈ T0×T0

, 2.71

whereB1x, y _x

x0gs, yΔs, η2x, y ωeB1y, y0_x

x0fs, yΔsandeB1y, y0is the unique solution of the following equation:

z^Δ_y x, y

B₁ x, y

z x, y

, z x, y₀

1. 2.72

T, y∈T0. Let

v x, y

C _y

y0

_x

x0

fs, tωuτ1s, τ2t gs, tuτ1s, τ2t

ΔsΔt. 2.73

Then

u x, y

≤v x, y

≤v X, y

, ∀x∈x0, X

T, y∈T0. 2.74

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Similar to2.14–2.16, we can obtain u

τ1x, τ2

y

≤v x, y

, x∈x0, X

T, y∈T0. 2.75

Furthermore we have

v X, y

C _y

y0

_X

x0

fs, tωuτ1s, τ2t gs, tuτ1s, τ2t ΔsΔt

≤C _y

y0

_X

x0

fs, tωvs, t gs, tvs, t ΔsΔt

≤C _y

y0

_X

x0

fs, tωvs, tΔsΔt _y

y0

_X

x0

gs, tΔs

vX, tΔt, y∈T0. 2.76

LetB2X, y C_y

y0

_X

x0fs, tωvs, tΔsΔt. Then from2.76it follows that

v X, y

≤B2

X, y

_y

y0

B1X, tvX, tΔt, y∈T0. 2.77

ConsideringB₂X, yis nondecreasing iny, by applyingLemma 2.2to2.77, we obtain v

X, y

≤B2

X, y eB1

y, y0

, y∈T0. 2.78

On the other hand, B2

X, y_Δ

y _X

x0

f s, y

ω v

s, y Δs≤

_X

x0

f s, y

Δs ω v

X, y

≤ _X

x0

f s, y

Δs ω B2

X, y eB1

y, y0

≤ _X

x0

f s, y

Δs ω B2

X, y ω

eB1

y, y0

ω

B2

X, y η2

X, y .

2.79

LetvX, ybe the solution of the following equation:

v^Δ_y X, y

η₂ X, y

ω v

X, y , v

X, y₀

C. 2.80

ConsideringB₂X, y0 Candωis nondecreasing and continuous, then from2.79,2.80, andLemma 2.4, we have

B2

X, y

≤v X, y

, y∈T0. 2.81

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From the definition ofGand2.80, we haveGvX, y ^Δ_y v^Δ_yX, y/ωvX, y η2X, y.

Then similar to2.66and2.67, we obtain

B₂ X, y

≤v X, y

≤G⁻¹

GC

_y

y0

η₂X, tΔt , y∈T0. 2.82

Combining2.74,2.78, and2.82, we have

u x, y

≤G⁻¹

GC

_y

y0

η₂X, tΔt e_B₁ y, y₀

, x∈x0, X

T, y∈T0. 2.83

SettingxXin2.83, we obtain

u X, y

≤G⁻¹

GC

_y

y0

η₂X, tΔt e_B₁ y, y₀

, y∈T0. 2.84

ReplacingXwithxin2.84yields the desired inequality2.71.

Theorem 2.17. Under the conditions ofTheorem 2.16, ifp, Care constants withp >0, C >0, and forx, y∈T0×T0,ux, ysatisfies the following inequality:

u^p x, y

≤C _y

y0

_x

x0

fs, tωuτ1s, τ2t gs, tu^pτ1s, τ2t

ΔsΔt, 2.85

with the initial condition2.58, then

u x, y

≤

G⁻¹

GC _y

y0

η₃x, tΔt e_J₁

y, y₀_1/p

,

x, y

∈T0×T0, 2.86

where G is defined as in Theorem 2.14, J₁x, y _x

x0gs, yΔs, η3x, y ωeJ1y, y0^1/p_x

x0fs, yΔs, ande_J₁y, y0is the unique solution of the following equation:

z^Δ_y x, y

J₁ x, y

z x, y

, z x, y₀

1. 2.87

The proof ofTheorem 2.17is similar to that ofTheorem 2.16, and we omit it here.

3. Some Simple Applications

In this section, we will present some examples to illustrate the validity of our results in deriving explicit bounds of solutions of certain delay dynamic equations on time scales.

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Example 3.1. Consider the following delay dynamic integral equation:

u^p x, y

C _y

y0

_x

x0

Ms, t, uτ1s, τ2tΔsΔt, x, y

∈T0×T0, 3.1

with the initial condition u

x, y φ

x, y

, ifx∈α, x0

Tory∈

β, y₀ T, φ

τ1x, τ2

y

≤ |C|^1/p, ∀ x, y

∈ T0,T0

, ifτ1x≤x0 orτ2

y

≤y0, 3.2

whereu∈CrdT0×T0,R,φ, α, β, τ1, τ2are the same as inTheorem 2.8, andM∈CrdT0×T0× R,R. Furthermore, assume|Ms, t, u| ≤fs, t|u|^qgs, t|u|^r, wheref, g∈C_rdT0×T0,R, andp, q, rare the same as inTheorem 2.8.

From3.1we have u^p

x, y≤ |C|

_y

y0

_x

x0

|Ms, t, uτ1s, τ2t|ΔsΔt

≤ |C|

_y

y0

_x

x0

fs, t|uτ1s, τ2t|^qgs, t|uτ1s, τ2t|^r ΔsΔt.

3.3

Then according toTheorem 2.8, we can obtain the following estimate:

u

x, y≤

H1

x, y

_y

y0

e_H

2

y, σtH2x, tH1x, tΔt

1/p

,

x, y

∈T0×T0, 3.4

where H1

x, y |C|

_y

y0

_x

x0

fs, tp−q

ΔsΔt, ∀K >0, H₂

x, y

_x

x0

f

s, yq

pK^q−p/pg s, yr

pK^r−p/p

Δs, ∀K >0.

3.5

Example 3.2. Considering the following delay dynamic integral equation:

u³ x, y

C _y

y0

_x

x0

Ns, t, uτ1s, τ2t, uτ1s, στ2tΔsΔt, x, y

∈T0×T0, 3.6

with the initial condition u

x, y φ

x, y

, ifx∈α, x0

Tory∈

β, y₀ T;

φ

τ₁x, τ2

y

≤ |C|^1/3, ∀x∈T0, y∈T0, ifτ₁x≤x₀ orτ₂ y

≤y₀, 3.7

whereu∈C_rdT0×T0,R, andN∈C_rdT0×T0×R²,R.

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Assume|Nx, y, u, v| ≤fx, y|u|²|v|², wheref∈CrdT0×T0,R, then from3.6 we have

u³

x, y≤ |C|

_y

y0

_x

x0

|Ns, t, uτ1s, τ2t, uτ1s, στ2t|ΔsΔt

≤ |C|

_y

y0

_x

x0

fs, t

|uτ1s, τ2t|²|uτ1s, στ2t|² ΔsΔt

≤ |C|

_y

y0

_x

x0

fs, t

|uτ1s, τ2t|²|uτ1s, στ2t|²

|uτ1s, τ2t||uτ1s, στ2t|ΔsΔt.

3.8

According toTheorem 2.13p3, we can reach the following estimate:

u

x, y≤ |C|^1/3 _y

y0

_x

x0

fs, tΔsΔt, x, y

∈T0×T0. 3.9

4. Conclusions

In this paper, we established some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales. As one can see, the presented results provide a handy tool for deriving bounds for solutions of certain delay dynamic equations on time scales. Furthermore, the process of constructing Theorems2.5,2.8,2.14,2.16and2.17can be applied to the situation withnindependent variables.

Acknowledgments

This work is supported by the Natural Science Foundation of Shandong Province ZR2010AZ003 China. The authors thank the referees very much for their careful comments and valuable suggestions on this paper.

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