doi:10.1155/2011/283926
Research Article
Nonlinear Integral Inequalities in Two Independent Variables on Time Scales
Wei Nian Li
Department of Mathematics, Binzhou University, Shandong 256603, China
Correspondence should be addressed to Wei Nian Li,[email protected] Received 7 December 2010; Accepted 18 February 2011
Academic Editor: Jianshe Yu
Copyrightq2011 Wei Nian Li. 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.
We investigate some nonlinear integral inequalities in two independent variables on time scales.
Our results unify and extend some integral inequalities and their corresponding discrete analogues which established by Pachpatte. The inequalities given here can be used as handy tools to study the properties of certain partial dynamic equations on time scales.
1. Introduction
The theory of dynamic equations on time scales unifies existing results in differential and finite difference equations and provides powerful new tools for exploring connections between the traditionally separated fields. During the last few years, more and more scholars have studied this theory. For example, we refer the reader to1,2and the references cited therein. At the same time, some integral inequalities used in dynamic equations on time scales have been extended by many authors3–11.
On the other hand, a few authors have focused on the theory of partial dynamic equations on time scales12–17. However, only10,11have studied integral inequalities useful in the theory of partial dynamic equations on time scales, as far as we know. In this paper, we investigate some nonlinear integral inequalities in two independent variables on time scales, which can be used as handy tools to study the properties of certain partial dynamic equations on time scales.
Throughout this paper, a knowledge and understanding of time scales and time scale notation is assumed. For an excellent introduction to the calculus on time scales, we refer the reader to1,2.
2. Main Results
In what follows,Tis an arbitrary time scale, Crddenotes the set of rd-continuous functions,R denotes the set of all regressive and rd-continuous functions,R{p∈ R: 1μtpt>0 for allt ∈T},Rdenotes the set of real numbers,R 0,∞, andN0 {0,1,2, . . .}denotes the set of nonnegative integers. We use the usual conventions that empty sums and products are taken to be 0 and 1, respectively. Throughout this paper, we always assume thatT1 andT2
are time scales,t0∈T1,s0∈T2,t≥t0,s≥s0,Ω T1×T2, and we writexΔtt, sfor the partial delta derivatives ofxt, swith respect tot, andxΔtΔst, sfor the partial delta derivatives of xΔtt, swith respect tos.
The following two lemmas are useful in our main results.
Lemma 2.1see18. Ifx, y∈R, and 1/p1/q1 withp >1, then
x1/py1/q≤ x p y
q, 2.1
with equality holding if and only ifxy.
Lemma 2.2Comparison Theorem1. Supposeu, b∈Crd,a∈ R. Then,
uΔt≤atut bt, t∈T 2.2
implies
ut≤ut0eat, t0
t
t0
eat, στbτΔτ, t∈T. 2.3
Next, we establish our main results.
Theorem 2.3. Assume that ut, s, at, s,bt, s,gt, s, and ht, s are nonnegative functions defined fort, s ∈ Ωthat are right-dense continuous for t, s ∈ Ω, andp > 1 is a real constant.
Then,
upt, s≤at, sbt, s t
t0
s
s0
g τ, η
up τ, η
h τ, η
u τ, η
ΔηΔτ, t, s∈Ω 2.4
implies
ut, s≤
at, s bt, smt, sey·,st, t01/p
, t, s∈Ω, 2.5
where
mt, s t
t0
s
s0
a τ, η
g τ, η
p−1
p a
τ, η p
h
τ, η
ΔηΔτ, 2.6
yt, s s
s0
g t, η
h t, η p
b
t, η
Δη, t, s∈Ω. 2.7
Proof. Define a functionzt, sby
zt, s t
t0
s
s0
g τ, η
up τ, η
h τ, η
u τ, η
ΔηΔτ, t, s∈Ω. 2.8
Then,2.4can be written as
upt, s≤at, s bt, szt, s, t, s∈Ω. 2.9
From2.9, byLemma 2.1, we have
ut, s≤at, s bt, szt, s1/p1p−1/p
≤ at, s
p bt, szt, s
p p−1
p , t, s∈Ω. 2.10
It follows from2.8–2.10that
zt, s≤ t
t0
s
s0
g
τ, η a
τ, η b
τ, η z
τ, η
h
τ, η p−1a τ, η
p b
τ, η z
τ, η p
ΔηΔτ
mt, s
t
t0
s
s0
g τ, η
h τ, η
p
b τ, η
z τ, η
ΔηΔτ, t, s∈Ω,
2.11
where mt, s is defined by 2.6. It is easy to see that mt, s is nonnegative, right-dense continuous, and nondecreasing fort, s∈Ω. Letε >0 be given, and from2.11, we obtain
zt, s
mt, s ε ≤1 t
t0
s
s0
g τ, η
h τ, η
p
b
τ, η z τ, η m
τ, η
εΔηΔτ, t, s∈Ω. 2.12
Define a functionvt, sby
vt, s 1 t
t0
s
s0
g τ, η
h τ, η
p
b
τ, η z τ, η m
τ, η
εΔηΔτ, t, s∈Ω. 2.13
It follows from2.12and2.13that
zt, s≤mt, s εvt, s, t, s∈Ω. 2.14
From2.13, a delta derivative with respect totyields
vΔtt, s s
s0
g t, η
h t, η p
b
t, η z t, η m
t, η εΔη
≤ s
s0
g t, η
h t, η p
b
t, η v
t, η Δη
≤ s
s0
g t, η
h t, η p
b
t, η Δη
vt, s yt, svt, s, t, s∈Ω,
2.15
whereyt, sis defined by2.7. Noting thatvt0, s 1,yt, s ≥ 0, and usingLemma 2.2, from2.15, we obtain
vt, s≤ey·,st, t0, t, s∈Ω. 2.16
It follows from2.9,2.14, and2.16that
ut, s≤
at, s bt, smt, s εey·,st, t01/p
, t, s ∈Ω. 2.17
Lettingε → 0 in2.17, we immediately obtain the required2.5. The proof ofTheorem 2.3 is complete.
Remark 2.4. LettingT1 T2RandT1 T2N0, respectively, we easily see thatTheorem 2.3reduces to Theorem 2.3.3c1and Theorem 5.2.4d1in19.
Theorem 2.5. Assume that all assumptions ofTheorem 2.3hold. Ifat, s>0 andat, sis nonde- creasing fort, s∈Ω, then
upt, s≤apt, sbt, s t
t0
s
s0
g τ, η
up τ, η
h τ, η
u τ, η
ΔηΔτ, t, s∈Ω 2.18
implies
ut, s≤at, s
1bt, snt, sew·,st, t01/p
, t, s∈Ω, 2.19
where
nt, s t
t0
s
s0
g
τ, η h
τ, η a1−p
τ, η ΔηΔτ,
wt, s s
s0
g t, η
h t, η
a1−p τ, η p
b
t, η
Δη, t, s∈Ω.
2.20
Proof. Noting thatat, s>0 andat, sis nondecreasing fort, s∈Ω, from2.18, we have ut, s
at, s p
≤1bt, s t
t0
s
s0
g τ, η
u τ, η a
τ, η p
h τ, η
a1−p τ, ηu
τ, η a
τ, η
ΔηΔτ, t, s∈Ω.
2.21
ByTheorem 2.3, from2.21, we easily obtain the desired2.19. This completes the proof of Theorem 2.5.
Remark 2.6. IfT1T2RinTheorem 2.5, then we easily obtain Theorem 2.3.3c2in19.
Theorem 2.7. Assume thatut, s,at, s, andbt, sare nonnegative functions defined fort, s∈ Ωthat are right-dense continuous fort, s∈Ω, andp >1 is a real constant. Iff:Ω×R → Ris right-dense continuous onΩand continuous onRsuch that
0≤ft, s, x−f t, s, y
≤φ t, s, y
x−y
, 2.22
fort, s∈Ω,x≥y≥0, whereφ :Ω×R → Ris right-dense continuous onΩand continuous onR, then
upt, s≤at, s bt, s t
t0
s
s0
f τ, η, u
τ, η
ΔηΔτ, t, s∈Ω 2.23
implies
ut, s≤
at, s bt, smt, se w·,s t, t01/p
, t, s∈Ω, 2.24
where
mt, s
t
t0
s
s0
f
τ, η,p−1a τ, η p
ΔηΔτ, 2.25
wt, s
s
s0
φ
t, η,p−1a t, η p
b t, η
p Δη, t, s∈Ω. 2.26
Proof. Define a functionzt, sby
zt, s t
t0
s
s0
f τ, η, u
τ, η
ΔηΔτ, t, s∈Ω. 2.27
As in the proof ofTheorem 2.3, from2.23, we easily see that 2.9and 2.10hold. Com- bining2.10,2.27and noting the assumptions onf, we have
zt, s≤ t
t0
s
s0
f
τ, η,p−1a τ, η
p b
τ, η z
τ, η p
−f
τ, η,p−1a τ, η p
f
τ, η,p−1a τ, η p
ΔηΔτ
≤mt, s t
t0
s
s0
φ
τ, η,p−1a τ, η p
b τ, η
p z
τ, η ΔηΔτ,
2.28
where mt, s is defined by 2.25. It is easy to see thatmt, s is nonnegative, right-dense continuous, and nondecreasing fort, s∈Ω. The remainder of the proof is similar to that of Theorem 2.3and we omit it.
Remark 2.8. Letting T1 T2 R and T1 T2 N0 in Theorem 2.7, respectively, we can obtain Theorem 2.3.4d1and Theorem 5.2.4d2in19.
Theorem 2.9. Assume thatut, s,at, s, andbt, sare nonnegative functions defined fort, s∈ Ωthat are right-dense continuous fort, s∈Ω, andp >1 is a real constant. Iff:Ω×R → Ris right-dense continuous onΩand continuous onR, andΨ∈CR,Rsuch that
0≤ft, s, x−f t, s, y
≤φ t, s, y
Ψ−1 x−y
, 2.29
fort, s∈Ω,x≥y≥0, whereφ :Ω×R → Ris right-dense continuous onΩand continuous onR,Ψ−1is the inverse function ofΨ, and
Ψ−1 xy
≤Ψ−1xΨ−1 y
, x, y∈R, 2.30
then
upt, s≤at, s bt, sΨ t
t0
s
s0
f τ, η, u
τ, η ΔηΔτ
, t, s∈Ω 2.31
implies
ut, s≤
at, s bt, sΨ
mt, sew·,st, t01/p
, t, s∈Ω, 2.32
wheremt, s is defined by2.25, and
wt, s s
s0
φ
t, η,p−1a t, η p
Ψ−1
b t, η
p
Δη, t, s∈Ω. 2.33
Proof. Define a functionzt, sby2.27. Similar to the proof ofTheorem 2.3, we have
upt, s≤at, s bt, sΦzt, s, 2.34
ut, s≤ p−1at, s
p bt, s
p Φzt, s, t, s∈Ω. 2.35
From2.27,2.35and the assumptions onfandΨ, we obtain
zt, s≤ t
t0
s
s0
f
τ, η,p−1a τ, η
p b
τ, η Ψ
z τ, η p
−f
τ, η,p−1a τ, η p
f
τ, η,p−1a τ, η p
ΔηΔτ
≤mt, s t
t0
s
s0
φ
τ, η,p−1a τ, η p
Ψ−1
b τ, η
p
z τ, η
ΔηΔτ,
2.36
wheremt, s is defined by2.25. Obviously,mt, s is nonnegative, right-dense continuous, and nondecreasing fort, s∈Ω. The remainder of the proof is similar to that ofTheorem 2.3, and we omit it here. This completes the proof ofTheorem 2.9.
Remark 2.10. We note that whenT1 T2 R,Theorem 2.9reduces to Theorem 2.3.4d2in 19.
Remark 2.11. Using our main results, we can obtain many integral inequalities for some peculiar time scales. For example, letting T1 R, T2 N0, fromTheorem 2.3, we easily obtain the following result.
Corollary 2.12. Assume that ut, s,at, s,bt, s,gt, sand ht, sare nonnegative functions defined fort∈R,s∈N0that are continuous fort∈R, andp >1 is a real constant. Then,
upt, s≤at, s bt, s t
0
⎧⎨
⎩ s−1 η0
g τ, η
up τ, η
h τ, η
u
τ, η⎫
⎬
⎭dτ, t∈R, s∈N0
2.37
implies
ut, s≤
⎧⎨
⎩at, s bt, sm∗t, s×exp
⎛
⎝t
0
⎡
⎣s−1
η0
g
τ, η h
τ, η p
b
τ, η⎤
⎦dτ
⎞
⎠
⎫⎬
⎭
1/p
,
t∈R, s∈N0, 2.38
where
m∗t, s t
0
⎧⎨
⎩ s−1 η0
a τ, η
g τ, η
p−1
p a
τ, η p
h
τ, η⎫⎬
⎭dτ. 2.39
3. Some Applications
In this section, we present two applications of our main results.
Example 3.1. Consider the following partial dynamic equation on time scales
upt, sΔtΔsFt, s, ut, s rt, s, t, s∈Ω, 3.1
with the initial boundary conditions
ut, s0 αt, ut0, s βs, ut0, s0 γ, 3.2
wherep >1 is a constant,F:T1×T2×R → Ris right-dense continuous onΩand continuous onR,r : T1 ×T2 → Ris right-dense continuous onΩ,α : T1 → Rand β : T2 → Rare right-dense continuous, andγ∈Ris a constant.
Assume that
|Ft, s, v| ≤gt, s|v|pht, s|v|, 3.3
wheregt, sand ht, sare nonnegative right-dense continuous functions fort, s ∈ Ω. If ut, sis a solution of3.1,3.2, thenut, ssatisfies
|ut, s| ≤
a0t, s Mt, seY·,st, t01/p
, t, s∈Ω, 3.4
where
a0t, s ##αpt βps−γp##
t
t0
s
s0
##r
τ, η##ΔηΔτ, Mt, s
t
t0
s
s0
a0
τ, η g
τ, η
p−1 p a0
τ, η p
h
τ, η ΔηΔτ,
Yt, s s
s0
g t, η
h t, η p
Δη, t, s∈Ω.
3.5
In fact, the solutionut, sof3.1,3.2satisfies
upt, s αpt βps−γp t
t0
s
s0
F τ, η, u
τ, η
ΔηΔτ t
t0
s
s0
r τ, η
ΔηΔτ, t, s∈Ω.
3.6
Therefore,
|ut, s|p≤a0t, s t
t0
s
s0
##F τ, η, u
τ, η##ΔηΔτ, t, s∈Ω. 3.7
It follows from3.3and3.7that
|ut, s|p≤a0t, s t
t0
s
s0
g
τ, η##u
τ, η##ph
τ, η##u
τ, η##ΔηΔτ, t, s∈Ω. 3.8
UsingTheorem 2.3, from3.8, we easily obtain3.4.
Example 3.2. Consider the following dynamic equation on time scales:
upt, s K t
t0
s
s0
H τ, η, u
τ, η
ΔηΔτ, t, s∈Ω, 3.9
whereK >0,p >1 are constants,H :T1×T2×R → Ris right-dense continuous onΩand continuous onR.
Assume that
|Ht, s, v| ≤ht, s|v|, t, s∈Ω, 3.10
whereht, s is a nonnegative right-dense continuous function fort, s ∈ Ω. If ut, sis a solution of3.9, then
|ut, s| ≤ K
1nt, seq·,st, t01/p
, t, s∈Ω, 3.11
where
nt, s K1−p/p t
t0
s
s0
h τ, η
ΔηΔτ,
qt, s K1−p/p p
s
s0
h t, η
Δη, t, s∈Ω.
3.12
In fact, ifut, sis a solution of3.9, then
|ut, s|p≤K t
t0
s
s0
##H τ, η, u
τ, η##ΔηΔτ, t, s∈Ω. 3.13
It follows from3.10and3.13that
|ut, s|p≤K t
t0
s
s0
h
τ, η##u
τ, η##ΔηΔτ, t, s∈Ω. 3.14
Therefore, byTheorem 2.5, from3.14, we immediately obtain3.11.
Acknowledgments
This work is supported by the National Natural Science Foundation of China10971018, the Natural Science Foundation of Shandong ProvinceZR2009AM005, China Postdoctoral Science Foundation Funded Project20080440633, Shanghai Postdoctoral Scientific Program 09R21415200, the Project of Science and Technology of the Education Department of Shandong ProvinceJ08LI52, and the Doctoral Foundation of Binzhou University2006Y01.
The author thanks the referees very much for their careful comments and valuable suggestions on this paper.
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