Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 320794, 12 pages
doi:10.1155/2011/320794
Research Article
Nonlinear Dynamical Integral Inequalities in Two
Independent Variables and Their Applications
Yuangong Sun
School of Mathematics, University of Jinan, Jinan, Shandong 250022, China
Correspondence should be addressed to Yuangong Sun, sunyuangong@yahoo.cn
Received 25 May 2011; Accepted 17 July 2011
Academic Editor: Hassan A. El-Morshedy
Copyright q 2011 Yuangong Sun. 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.
In this paper, we investigate some nonlinear dynamical integral inequalities involving the forward
jump operator in two independent variables. These inequalities provide explicit bounds on
unknown functions, which can be used as handy tools to study the qualitative properties of
solutions of certain partial dynamical systems on time scales pairs.
1. Introduction
Theory of dynamical equations on time scales, which goes back to Hilger’s landmark paper
1, has received considerable attention in recent years. For example, see the monographs 2,
3 and the references cited therein. Since dynamical integral inequalities usually can be used
as handy tools to study the qualitative theory of dynamical equations on time scales, many
researchers devoted to the study of different types of integral inequalities on time scales. We
refer the readers to 4–19.
To the best of our knowledge, the theory of partial dynamic equations on time scales
has received less attention 20–24. The main purpose of this paper is to investigate several
nonlinear integral inequalities in two independent variables on time scale pairs, which can
be used to estimate explicit bounds of solutions of certain partial dynamical equations on
time scales. Unlike some existing results in the literature e.g., 12, the integral inequalities
considered in this paper involve the forward jump operator σt and σs on a pair of
which results in difficulties in the estimation on the explicit bounds
time scales T and T,
As an application, we study the qualitative
of unknown functions ut, s for t ∈ T and s ∈ T.
property of certain partial dynamical equations on time scales.
Throughout this paper, a knowledge and understanding of time scales and time scale
are two unbounded time scales, t0 ∈ T and
notations is assumed. In what follows, T and T
2
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Crd T, T
is the set of right-dense continuous functions on T × T.
For an excellent
s0 ∈ T.
introduction to the calculus on time scales, we refer the reader to monographs 2, 3.
2. Problem Statements
Before establishing the main results of this paper, we first present two useful lemmas as
follows.
Lemma 2.1. Let c ≥ 0, x ≥ 0, and 0 < λ < 1. Then, for any k > 0,
2.1
cxλ ≤ kx θc, k, λ
holds, where θc, k, λ 1 − λλλ/1−λ c1/1−λ kλ/λ−1 .
Proof. Set Fx cxλ − kx. It is not difficult to see that Fx obtains its maximum at x λc/k1/1−λ and
2.2
Fmax 1 − λλλ/1−λ c1/1−λ kλ/1−λ .
This completes the proof of Lemma 2.1.
Lemma 2.2. Let y, p, q, r ∈ Crd T with pt, qt ≥ 0 for t ∈ T. Then
yΔ t ≤ ptyt qt
yσt rt,
1 μtqt
t ∈ T,
2.3
implies
yt ≤ yt0 ep⊕q t, t0 t
ep⊕q t, σs 1 μsqs rsΔs,
t ∈ T,
2.4
t0
where p ⊕ q p q μpq and μt σt − t.
Proof. Note that yσt yt μtyΔ t, we have
yΔ t ≤ ptyt qt
yt μtyΔ t rt
1 μtqt
2.5
that is,
yΔ t ≤ p ⊕ q tyt 1 μtqt rt.
By Theorem 6.1 2, page 255, we get that Lemma 2.2 holds.
2.6
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3
Consider the following nonlinear integral inequalities in two independent variables
on time scales T × T:
ut, s ≤ at, s bt, s
t s g τ, η u τ, η h1 τ, η uλ1 στ, η ΔηΔτ,
t0
ut, s ≤ at, s bt, s
t s g τ, η u τ, η h1 τ, η uλ1 στ, η
t0
s0
h2 τ, η u
ut, s ≤ at, s bt, s
2.7
s0
τ, σ η ΔηΔτ,
λ2
2.8
t s g τ, η u τ, η h1 τ, η uλ1 στ, η
t0
s0
h2 τ, η uλ2 τ, σ η h3 τ, η uλ3 στ, σ η ΔηΔτ,
2.9
where ut, s, at, s, bt, s, gt, s, and hi t, s i 1, 2, 3 are nonnegative right-dense
0 < λi < 1 i 1, 2, 3 are constants.
continuous functions on T × T,
The reason for studying inequalities of type 2.7–2.9 is that sometimes we may need
to estimate the solutions of the following partial dynamical equation in the form
uΔt Δs t, s ft, s, ut, s, uσt, s, ut, σs, uσt, σs
2.10
with boundary conditions ut, s0 αt, ut0 , s βs, and ut0 , s0 u0 , where f : T ×
× R3 → R is right-dense continuous, R −∞, ∞, and u0 is a constant. Integrating 2.10
T
yields
ut, s αt βs − u0
t s
f τ, η, u τ, η , u στ, η , u τ, σ η , u στ, σ η ΔηΔτ.
t0
2.11
s0
Therefore, the study on the integral inequalities of type 2.7–2.9 can provide explicit
bounds of solutions of system 2.10 in some cases.
3. Main Results
Now, let us present the main results of this paper.
such that
Theorem 3.1. If there exists a positive function k1 t, s ∈ Crd T, T,
μtbσt, sk1 t, s < 1,
t, s ∈ T × T,
3.1
4
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then inequality 2.7 implies
ut, s ≤ at, s bt, s
t
ep1 ⊕q1 ·,s t, στ 1 μτq1 τ, s r1 τ, sΔτ,
3.2
t0
where
p1 t, s bt, s
s
g t, η Δη,
s0
bσt, sk1 t, s
,
1 − μtbσt, sk1 t, s
s
s
r1 t, s aσt, sk1 t, s at, s
g t, η Δη θ
h1 t, η Δη, k1 t, s, λ1 .
q1 t, s s0
3.3
s0
Proof. Define a function vt, s by
vt, s t s g τ, η u τ, η h1 τ, η uλ1 στ, η ΔηΔτ.
t0
3.4
s0
vt, s is nondecreasing with respect to t and s, and
Then, vt, s ≥ 0 for t, s ∈ T × T,
t, s ∈ T × T.
ut, s ≤ at, s bt, svt, s,
3.5
A delta derivative of vt, s with respect to t yields
Δt
v t, s s
g t, η u t, η h1 t, η uλ1 σt, η Δη
s0
≤ ut, s u σt, s
λ1
s
3.6
h1 t, η Δη.
s0
By Lemma 2.1, we have
u σt, s
λ1
s
s0
h1 t, η Δη ≤ k1 t, suσt, s θ
s
s0
h1 t, η Δη, k1 t, s, λ1 .
3.7
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5
It follows from 3.5, 3.6, and 3.7 that
Δt
v t, s ≤ at, s bt, svt, s
s
g t, η Δη
s0
aσt, s bσt, svσt, sk1 t, s
s
θ
h1 t, η Δη, k1 t, s, λ1 .
3.8
s0
Notice the definitions of p1 t, s, q1 t, s, and r1 t, s, we have
q1 t, s
r1 t, s,
1 μtq1 t, s
t, s ∈ T × T.
3.9
ep1 ⊕q1 ·,s t, στ 1 μτq1 τ, s r1 τ, sΔτ,
t, s ∈ T × T.
3.10
vΔt t, s ≤ p1 t, svt, s Since vt0 , s 0, by Lemma 2.2 we get
vt, s ≤
t
t0
Then, 3.5 and 3.10 imply 3.2.
such that k2 t, s is ΔTheorem 3.2. If there exist positive functions k1 t, s, k2 t, s ∈ Crd T, T,
Δs
differentiable with respect to s, k2 t, s ∈ Crd T, T, and
μtbσt, sk1 t, s < 1,
t, s ∈ T × T,
3.11
then inequality 2.8 implies
ut, s ≤ at, s bt, s
t
ep2 ⊕q2 ·,s t, στ 1 μτq2 τ, s r2 τ, sΔτ,
3.12
t0
where
s k2 t, σ η b t, σ η
Δη k 2 t, η p2 t, s p1 t, s k2 t, s Δη,
1 μ η b t, σ η
s0
Δs
q2 t, s q1 t, s,
k2 t, s max 0, −k2Δs t, s ,
s θ h2 t, η ,
r2 t, s r1 t, s s0
k2 t, σ η a t, σ η
k2 t, σ η
, λ2 Δη.
1 μ η b t, σ η
1 μ η b t, σ η
3.13
6
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Proof. Set
t s g τ, η u τ, η h1 τ, η uλ1 στ, η h2 τ, η uλ2 τ, σ η ΔηΔτ.
zt, s t0
3.14
s0
and
Then, zt, s is nonnegative and nondecreasing with respect to t and s on T × T,
ut, s ≤ at, s bt, szt, s,
t, s ∈ T × T.
3.15
By Lemma 2.1, we have
Δt
z t, s ≤ ut, s
s
s
g t, η Δη u σt, s
λ1
s0
h1 t, η Δη
s0
h2 t, η uλ2 t, σ η Δη
s0
≤ ut, s
s
g t, η Δη k1 t, suσt, s
s0
θ
s
s
h1 t, η Δη, k1 t, s, λ1
3.16
s0
k2 t, σ η
u t, σ η Δη
s0 1 μ η b t, σ η
s k2 t, σ η
θ h2 t, η ,
, λ2 Δη.
1 μ η b t, σ η
s0
s
Substituting 3.15 into 3.16, we get
zΔt t, s ≤ at, s bt, szt, s
s
g t, η Δη
s0
aσt, s bσt, szσt, sk1 t, s
s
θ
h1 t, η Δη, k1 t, s, λ1
s0
k2 t, σ η
a t, σ η b t, σ η z t, σ η Δη
s0 1 μ η b t, σ η
s k2 t, σ η
θ h2 t, η ,
, λ2 Δη.
1 μ η b t, σ η
s0
s
3.17
Discrete Dynamics in Nature and Society
7
Note that
z t, σ η z t, η μ η zΔη t, η .
3.18
Integrating by parts, we have
s
s0
k2 t, σ η
a t, σ η b t, σ η z t, σ η Δη
1 μ η b t, σ η
≤
s
s0
s k2 t, σ η a t, σ η
k2 t, σ η b t, σ η
Δη z t, η Δη
1 μ η b t, σ η
s0 1 μ η b t, σ η
s
k2 t, σ η zΔη t, η Δη
3.19
s0
≤
s
s0
s k2 t, σ η a t, σ η
k2 t, σ η b t, σ η
Δη zt, s
Δη
1 μ η b t, σ η
s0 1 μ η b t, σ η
zt, s k2 t, s s
s0
Δη k 2 t, η Δη
.
Therefore, it follows from 3.17 and 3.19 that
zΔt t, s ≤ p2 t, szt, s q2 t, s
zσt, s r2 t, s,
1 μtq2 t, s
t, s ∈ T × T.
3.20
This together with Lemma 2.2 and 3.15 yields 3.12.
such that
Theorem 3.3. If there exist positive functions k1 t, s, k2 t, s, k3 t, s ∈ Crd T, T,
Δs
Δs
and
k2 t, s, k3 t, s are Δ-differentiable with respect to s, k2 t, s, k3 t, s ∈ Crd T, T,
μtΛt, s < 1,
t, s ∈ T × T,
3.21
then inequality 2.9 implies
ut, s ≤ at, s bt, s
t
t0
ep3 ⊕q3 ·,s t, στ 1 μτq3 τ, s r3 τ, sΔτ,
3.22
8
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where
Λt, s bσt, sk1 t, s k3 t, s s s0
Δη k 3 t, η
k3 t, σ η b σt, σ η
Δη,
1 μ η b σt, σ η
p3 t, s p2 t, s,
Λt, s
,
1 − μtΛt, s
s k3 t, σ η a σt, σ η
r3 t, s r2 t, s Δη
s0 1 μ η b σt, σ η
s k3 t, σ η
θ3 h3 t, η ,
, λ3 Δη,
1 μ η b σt, σ η
s0
q3 t, s 3.23
Δs
and k3 t, s max{0, −k3Δs t, s}.
Proof. Let the nonnegative and nondecreasing function wt, s be defined by
wt, s t s g τ, η u τ, η h1 τ, η uλ1 στ, η
t0
s0
h2 τ, η u
λ2
τ, σ η h3 τ, η uλ3 στ, σ η ΔηΔτ.
3.24
Then,
ut, s ≤ at, s bt, swt, s,
t, s ∈ T × T.
3.25
Based on the same arguments as in Theorem 3.2, we have
wΔt t, s ≤ p2 t, szt, s bσt, sk1 t, swσt, s r2 t, s
s k3 t, σ η a σt, σ η
Δη
s0 1 μ η b σt, σ η
s k3 t, σ η
θ3 h3 t, η ,
, λ3 Δη
1 μ η b σt, σ η
s0
s k3 t, σ η b σt, σ η
z σt, σ η Δη.
s0 1 μ η b σt, σ η
3.26
Discrete Dynamics in Nature and Society
9
Notice that
k3 t, σ η b σt, σ η
z σt, σ η Δη
1
μ
η
b
σt,
σ
η
s0
s k3 t, σ η b σt, σ η z σt, η μ η zΔη σt, η Δη
s0 1 μ η b σt, σ η
s s
k3 t, σ η b σt, σ η
≤ zσt, s
k3 t, σ η zΔη σt, η Δη
Δη s0 1 μ η b σt, σ η
s0
s k3 t, σ η b σt, σ η
Δη Δη .
k3 t, η ≤ zσt, s k3 t, s 1 μ η b σt, σ η
s0
s
3.27
By 3.26 and 3.27, we have
wΔt t, s ≤ p3 t, swt, s Λt, swσt, s r3 t, s,
t, s ∈ T × T.
3.28
Using the fact Λt, s q3 t, s/1 μtq3 t, s, Lemma 2.2 and 3.25, we get that 3.22
holds.
It is worthy to mention that although some additional assumptions such as
μtbσt, sk1 t, s < 1 and μtΛt, s < 1 are imposed in Theorems 3.1–3.3, they are easy to
be satisfied by choosing appropriate adjusting functions k1 t, s and k3 t, s.
4. Applications
We now consider some applications of the main results in the partial dynamical system 2.10
under the boundary condition
ut, s0 αt,
ut0 , s βs,
ut0 , s0 u0 .
4.1
Denote at, s |αt| |βs| |u0 |. We have the following corollaries.
R 0, ∞, and
Corollary 4.1. Let T Z {0, 1, 2, . . .}, T
ft, s, ut, s, ut 1, s ≤ |ut, s| |ut 1, s|λ1 ,
t, s ∈ T × T.
4.2
Then, the solution of system 2.10 under the boundary condition 4.1 satisfies
t−1
1
aτ 1, s
aτ, ss θ s, , λ1 ,
|ut, s| ≤ at, s 2 2 2st−1−τ
2
2
τ0
for t, s ∈ T × T.
4.3
10
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it follows from 2.11 and 4.2 that
Proof. For t, s ∈ T × T,
|ut, s| ≤ at, s t−1 s u τ, η u τ 1, η λ1 .
τ0
4.4
0
Let k1 t, s 1/2 be a constant. A straightforward computation yields
p1 t, s s,
q1 t, s 1,
1
at 1, s
at, ss θ s, , λ1 .
r1 t, s 2
2
4.5
Since p1 ⊕ q1 t, s 2 2s, we get 4.3 by Theorem 3.1.
Z, and
Corollary 4.2. Let T T
ft, s, ut, s, ut 1, s, ut, s 1 ≤ |ut, s| |ut 1, s|λ1 |ut, s 1|λ2 .
4.6
Then, the solution of system 2.10 under the boundary condition 4.1 satisfies
⎡
⎤
s−1 aτ, η 1
t−1
1
η0
t−1−τ ⎣
⎦,
r1 τ, s sθ s, , λ2 |ut, s| ≤ at, s 2 3 3s
2
2
τ0
4.7
where r1 t, s is defined as in Corollary 4.1 for t, s ∈ T × T.
it follows from 2.11 and 4.6 that
Proof. For t, s ∈ T × T,
|ut, s| ≤ at, s t−1 s−1 u τ, η u τ 1, η λ1 u τ, η 1 λ2
4.8
τ0 η0
Let k1 t, s 1/2 and k2 t, s 1. A straightforward computation
holds for t, s ∈ T × T.
yields
p2 t, s 1 3s
,
2
1
r2 t, s r1 t, s sθ s, , λ2
2
q2 t, s 1,
s−1 η0 a t, η 1
2
Hence, p2 ⊕ q2 3 3s. By Theorem 3.2, we have that 4.7 holds.
4.9
.
Discrete Dynamics in Nature and Society
11
For the case when f satisfies
ft, s, ut, s, ut 1, s, ut, s 1, ut 1, s 1
≤ |ut, s| |ut 1, s|λ1 |ut, s 1|λ2 |ut 1, s 1|λ3
4.10
on Z × Z, the solution of system 2.10 under the boundary condition 4.1 can be similarly
estimated by Theorem 3.3. We omit it here.
Acknowledgment
This paper was supported by the Natural Science Foundations of Shandong Province
ZR2010AL002, JQ201119 and the National Natural Science Foundation of China
61174217.
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