Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 538247, 21 pages
doi:10.1155/2012/538247
Research Article
Some Delay Integral Inequalities on
Time Scales and Their Applications in
the Theory of Dynamic Equations
Qinghua Feng,1, 2 Fanwei Meng,2 Yaoming Zhang,1
Jinchuan Zhou,1 and Bin Zheng1
1
School of Science, Shandong University of Technology, Shandong,
Zibo 255049, China
2
School of Mathematical Sciences, Qufu Normal University, Shandong,
Qufu 273165, China
Correspondence should be addressed to Qinghua Feng, fqhua@sina.com
Received 29 July 2011; Accepted 1 December 2011
Academic Editor: Agacik Zafer
Copyright q 2012 Qinghua Feng 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.
We establish some delay integral inequalities on time scales, which on one hand provide a handy
tool in the study of qualitative as well as quantitative properties of solutions of certain delay
dynamic equations on time scales and on the other hand unify some known continuous and discrete results in the literature.
1. Introduction
During the past decades, with the development of the theory of differential and integral
equations as well as difference equations, a lot of integral and difference inequalities have
been discovered e.g., see 1–13 and the references therein, which play an important role in
the research of boundedness, global existence, stability of solutions of differential and integral
equations as well as difference equations. On the other hand, Hilger 14 initiated 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 scales including various inequalities on time scales e.g., see
15–24, and the references therein. However, delay integral inequalities on time scales have
been paid little attention so far. Recent results in this direction include the works of Li 25
and Ma and Pečarić 26 to our best knowledge.
2
Abstract and Applied Analysis
In this paper, we will establish some new delay integral inequalities on time scales,
which unify some known continuous and discrete results in the literature. New explicit
bounds for unknown functions concerned are obtained due to the presented inequalities.
Some applications will be presented for the established results.
Throughout this paper, R denotes the set of real numbers and R 0, ∞, while Z
denotes the set of integers. For two given sets G, H, we denote the set of maps from G to H
by G, H.
2. Some Preliminaries on Time Scales
A time scale is an arbitrary nonempty closed subset of the real numbers. In this paper, T
denotes an arbitrary time scale. On T we define the forward and backward jump operators
σ ∈ T, T and ρ ∈ T, T such that σt inf{s ∈ T, s > t}, ρt sup{s ∈ T, s < t}.
Definition 2.1. The graininess μ ∈ T, R is defined by μt σt − t.
Remark 2.2. Obviously, μt 0 if T R, while μt 1 if T Z.
Definition 2.3. A point t ∈ T is said to be left-dense if ρt t and t /
inf T, right-dense if
σt t and t /
sup T, left-scattered if ρt < t, and right-scattered if σt > t.
Definition 2.4. The set Tκ is defined to be T if T does not have a left-scattered maximum,
otherwise it is T without the left-scattered maximum.
Definition 2.5. A function f ∈ T, R is called rd-continuous if it is continuous in right-dense
points and if the left-sided limits exist in left-dense points, while f is called regressive if
1 μtft /
0. Crd denotes the set of rd-continuous functions, while denotes the set of all
regressive and rd-continuous functions, and {f|f ∈ , 1 μtft > 0, for all t ∈ T}.
Definition 2.6. For some t ∈ Tκ , and a function f ∈ T, R, the delta derivative of f at t is
denoted by f Δ t provided it exists with the property such that for every ε > 0 there exists
a neighborhood U of t satisfying
fσt − fs − f Δ tσt − s ≤ ε|σt − s|
∀s ∈ U.
2.1
Remark 2.7. If T R, then f Δ t becomes the usual derivative f t, while f Δ t ft1−ft
if T Z, which represents the forward difference.
Definition 2.8. For a, b ∈ T and a function f ∈ T, R, the Cauchy integral of f is defined by
b
ftΔt Fb − Fa,
2.2
a
where F Δ t ft, t ∈ Tκ .
The following two theorems include some important properties for delta derivative and
the Cauchy integral on time scales.
Abstract and Applied Analysis
3
Theorem 2.9 see 27. If f, g ∈ T, R, and t ∈ Tκ , then
i
⎧
fσt − ft
⎪
⎪
⎨
μt
f Δ t ft − fs
⎪
⎪
⎩lim
s→t
t−s
if
μt /
0,
if
μt 0.
2.3
ii If f, g are delta differentials at t, then fg is also delta differential at t, and
Δ
fg t f Δ tgt fσtg Δ t.
2.4
Theorem 2.10 see 27. If a, b, c ∈ T, α ∈ R, and f, g ∈ Crd , then
b
b
b
i a ft gtΔt a ftΔt a gtΔt,
b
b
ii a αftΔt α a ftΔt,
b
a
iii a ftΔt − b ftΔt,
b
c
b
iv a ftΔt a ftΔt c ftΔt,
a
v a ftΔt 0,
b
vi if ft ≥ 0 for all a ≤ t ≤ b, then a ftΔt ≥ 0.
Definition 2.11. The cylinder transformation ξh : Ch → Zh is defined by
⎧
1
⎪
⎨ Log1 hz , if h /
0 for z /
−
,
h
h
ξh z ⎪
⎩z,
if h 0,
2.5
where Log is the principal logarithm function.
Definition 2.12. For p ∈ and s, t ∈ T, the exponential function is defined by
t
ep t, s exp
ξμτ pτ Δτ .
2.6
s
t
Remark 2.13. If T R, then for s, t ∈ R, ep t, s exp s pτdτ. If T Z, then for s, t ∈ Z
and s < t, ep t, s t−1
τs 1 pτ.
The following two theorems include some known properties on the exponential
function.
Theorem 2.14 see 28. If p ∈ , then the following conclusions hold:
i ep t, t ≡ 1, and e0 t, s ≡ 1,
4
Abstract and Applied Analysis
ii ep σt, s 1 μtptep t, s,
iii ep t, s 1/ep s, t ep s, t,
where p −p/1 μp.
Theorem 2.15 see 28. If p ∈ , and fix t0 ∈ T, then the exponential function ep t, t0 is the
unique solution of the following initial value problem:
yΔ t ptyt,
yt0 1.
2.7
For more details about the calculus of time scales, we advise to refer to 29.
3. Main Results
In the rest of this paper, for the sake of convenience, we denote T0 t0 , ∞ ∩ T, where t0 ∈ T,
and always assume T0 ⊂ Tκ .
Lemma 3.1 see 30, Theorem 2.2. Let t0 ∈ Tκ and ω : T × Tκ → R be continuous at t, t,
where t ≥ t0 , t ∈ Tκ with t > t0 . Assume that ωΔ t, · is rd-continuous on t0 , σt. If for any ε > 0,
there exists a neighborhood U of t, independent of τ ∈ t0 , σt, such that
ωσt, τ − ωs, τ − ωΔ t, τσt − s ≤ ε|σt − s| ∀s ∈ U,
3.1
where ωΔ denotes the derivative of ω with respect to the first variable, then
t
ωt, τΔτ
3.2
ωΔ t, τΔτ ωσt, t.
3.3
gt :
t0
implies
Δ
g t t
t0
Theorem 3.2. Suppose u ∈ Crd T0 , R , ω ∈ R , R with ωu > 0 for u > 0, and ω is
nondecreasing. τi ∈ T0 , T with τi t ≤ t, i 1, 2, and −∞ < α inf{min{τi t, i 1, 2}, t ∈
T0 } ≤ t0 . f, g ∈ Crd α, ∞ ∩ T, R . p, q, C are constants, and p > q > 0, C ≥ 0. If for t ∈ T0 , ut
satisfies the following inequality:
up t ≤ Cp/p−q p
p−q
t
t0
fτ1 suq τ1 sωuτ1 s gτ2 suq τ2 s Δs
3.4
Abstract and Applied Analysis
5
with the initial condition
ut φt,
φτi t ≤ C1/p−q ,
t ∈ α, t0 ∩ T,
3.5
∀t ∈ T0 , τi t ≤ t0 , i 1, 2,
where φ ∈ Crd α, t0 ∩ T, R , then,
ut ≤
−1
G
G C
t
gτ2 sΔs
t0
where Gx x
1
t
1/p−q
fτ1 sΔs
,
t ∈ T0 ,
3.6
t0
1/ωr 1/p−q dr, x > 0 with G∞ ∞, and G−1 is the inverse of G.
Proof. Assume C > 0. Denote the right side of 3.4 by zt. Then
ut ≤ z1/p t,
3.7
t ∈ T0 .
If τi t ≥ t0 , for t ∈ T0 , since τi t ≤ t, then τi t ∈ T0 , and from 3.7 we have
uτi t ≤ z1/p τi t ≤ z1/p t,
i 1, 2.
3.8
If τi t ≤ t0 , from 3.5 we obtain
uτi t φτi t ≤ C1/p−q ≤ z1/p t,
i 1, 2.
3.9
So from 3.8 and 3.9 we always have
uτi t ≤ z1/p t,
i 1, 2, t ∈ T0 .
3.10
Furthermore,
p fτ1 tuq τ1 tωuτ1 t gτ2 tuq τ2 t
p−q
p ≤
fτ1 tzq/p tω z1/p t gτ2 tzq/p t ,
p−q
zΔ t 3.11
that is,
p zΔ t
1/p
gτ
,
fτ
≤
z
tω
t
t
1
2
zq/p t p − q
3.12
6
Abstract and Applied Analysis
According to 29, Theorem 1.90, considering zΔ t ≥ 0, we have
p p−q/p
z
t
p−q
Δ
Δ
z t
1 zt hμtzΔ t
−q/p
dh
0
zΔ t
q/p
z t
≤
−q/p
1 zΔ t
1 hμt
dh
zt
0
3.13
zΔ t
.
zq/p t
Combining 3.12 and 3.13, we obtain
zp−q/p t
Δ
≤ fτ1 tω z1/p t gτ2 t.
3.14
Setting t s in 3.14, an integration with respect to s from t0 to t yields
p−q/p
z
p−q/p
t − z
t fτ1 sω z1/p s gτ2 s Δs.
t0 ≤
3.15
t0
Since zt0 Cp/p−q , then 3.15 implies
zt ≤
t C
fτ1 sω z
1/p
p/p−q
s gτ2 s Δs
3.16
.
t0
Fix T ∈ T0 , and let t ∈ t0 , T ∩ T. Then,
zt ≤
C
T
gτ2 sΔs t0
Denote vt by C T
t0
t
fτ1 sω z
1/p
p/p−q
s Δs
.
3.17
t0
gτ2 sΔs t
t0
fτ1 sωz1/p sΔs. Then,
zt ≤ vp/p−q t,
t ∈ t0 , T ∩ T,
vΔ t fτ1 tω z1/p t ≤ fτ1 tω v1/p−q t ,
3.18
3.19
that is,
vΔ t
ω v1/p−q t
≤ fτ1 t.
3.20
Abstract and Applied Analysis
7
On the other hand, for t ∈ t0 , T ∩ T, if σt > t, then
1
Gvσt − Gvt
σt − t
σt − t
GvtΔ vσt
vt
1
dr
ω r 1/p−q
1
vΔ t
vσt − vt
≤
1/p−q 1/p−q .
σt − t
ω v
ω v
t
t
3.21
If σt t, then
Gvt − Gvs
1
lim
s→t
s→tt − s
t−s
GvtΔ lim
vt
vs
ω
1
r 1/p−q
dr
1
vt − vs
vΔ t
lim
1/p−q 1/p−q ,
s→t
t−s
ω ξ
ω v
t
3.22
where ξ lies between vs and vt. So from 3.21 and 3.22 we always have
GvtΔ ≤
vΔ t
3.23
.
ω v1/p−q t
Combining 3.20 and 3.23, we deduce
GvtΔ ≤ fτ1 t.
3.24
Setting t s in 3.24, an integration with respect to s from t0 to t yields
Gvt − Gvt0 ≤
t
3.25
fτ1 sΔs.
t0
Considering vt0 C T
t0
gτ2 sΔs, and G is increasing, then we obtain
−1
vt ≤ G
G C
T
gτ2 sΔs
t0
t
fτ1 sΔs ,
t ∈ t0 , T ∩ T.
3.26
t0
Combining 3.7, 3.18, and 3.26, we have
ut ≤
−1
G
G C
T
t0
gτ2 sΔs
t
1/p−q
fτ1 sΔs
,
t ∈ t0 , T ∩ T.
3.27
t0
Setting t T in 3.27, considering that T ∈ T0 is selected arbitrarily, after substituting T with
t, we get the desired result.
If C 0, then we carry out the process above with C replaced by ε, where ε > 0, and
after letting ε → 0, we also get the desired result. So the proof is complete.
8
Abstract and Applied Analysis
Remark 3.3. If we take T R, t0 0, then Theorem 3.2 reduces to 31, Theorem 2.1. If T R,
t0 0, p 2, q 1, then Theorem 3.2 reduces to 32, Theorem 1. If T R, t0 0, p 2, q 1,
τ1 t τ2 t t, then Theorem 3.2 reduces to 33, Theorem 3a6. If we take T Z, t0 0,
p 2, q 1, τ1 t τ2 t t, then Theorem 3.2 reduces to 33, Theorem 6b6.
Theorem 3.4. Suppose u, a ∈ Crd T0 , R , and a is nondecreasing. α, φ, τi , i 1, 2 are defined
Δ
Δ
2
as in Theorem 3.2. f, h, ftΔ , hΔ
t ∈ Crd T0 × α, ∞ ∩ T, R , g, d, gt , dt ∈ Crd T0 , R , where
Δ
ftΔ , ht , gtΔ , dtΔ denote the delta derivative of f, h, g, d with respect to the first variable. If for t ∈ T0 ,
ut satisfies the following inequality:
ut ≤ at t
ft, τ1 suτ1 s gt, sus Δs
t0
t
ht, τ2 suτ2 sΔs
3.28
t
t0
dt, susΔs
t0
with the initial condition
ut φt,
φτi t ≤ at,
t ∈ α, t0 ∩ T,
3.29
∀t ∈ T0 , τi t ≤ t0 , i 1, 2,
then
ut ≤
provided that 1 at
F1 t t
t
t0
ate−F1 t, t0 ,
t
1 at t0 e−F1 σs, t0 F2 sΔs
t ∈ T0
3.30
e−F1 σs, t0 F2 sΔs > 0 and 1 − μtF1 t > 0 for ∀t ∈ T0 , where
Δ
ft, τ1 s gt, s Δs
,
F2 t t
t0
ht, τ2 sΔs
t0
Δ
t
dt, sΔs
.
t0
3.31
Proof. Assume a0 > 0. Fix T ∈ T0 , and let t ∈ t0 , T ∩T. If the right side of 3.28 is atzt,
then
ut ≤ at zt,
3.32
and similar to the process of 3.8–3.10, we have
uτi t ≤ at zt,
i 1, 2.
3.33
Abstract and Applied Analysis
9
Furthermore, by Lemma 3.1 and Theorem 2.9ii
zΔ t t ftΔ t, τ1 suτ1 s gtΔ t, sus Δs fσt, τ1 tuτ1 t gσt, tut
t0
t
t0
t
hΔ
t t, τ2 suτ2 sΔs
hσt, τ2 tuτ2 t
ht, τ2 suτ2 sΔs
t0
≤ at zt
t
t0
2
at zt
dtΔ t, susΔs
dσt, t
ftΔ t, τ1 s gtΔ t, s Δs fσt, τ1 t gσt, t
t
hΔ
t t, τ2 sΔs
t0
t
t
σt
hσt, τ2 t
dσt, sΔs
t0
t
ht, τ2 sΔs
t0
≤ aT zt
dσt, susΔs
t0
t
t0
σt
t0
dtΔ t, sΔs
dσt, t
Δ
ft, τ1 s gt, s Δs
t0
2
aT zt
t
ht, τ2 sΔs
Δ
t
t0
dt, sΔs
,
t0
3.34
that is,
zΔ t
1
≤
2
aT zt
aT zt
t
t
t0
ht, τ2 sΔs
t0
≤
Δ
ft, τ1 s gt, s Δs
Δ
t
dt, sΔs
3.35
t0
1
F1 t F2 t,
aT zt
where F1 , F2 are defined in 3.31.
Considering zΔ t ≥ 0, from 3.35, we deduce
zΔ t
1
zΔ t
F1 t F2 t.
≤
≤
2
aT zt
aT ztaT zσt aT zt
3.36
10
Abstract and Applied Analysis
Let vt 1/aT zt. Then, by Theorem 2.9ii, vΔ t −zΔ t/aT ztaT zσt, and 3.36 implies
vΔ t vtF1 t ≥ F2 t.
3.37
On the other hand, since 1 − μtF1 t > 0, then 1 μt−F1 t 1/1 − μtF1 t > 0.
So −F1 ∈ , and e−F1 t, t0 > 0, ∀t ∈ T0 . By Theorem 2.14i, we have e−F1 t0 , t0 1.
Furthermore, by a combination of Theorem 2.9ii, Theorems 2.15, and 2.14, we obtain
Δ Δ
vte−F1 t, t0 e−F1 t, t0 vt e−F1 σt, t0 vΔ t
−F1 te−F1 t, t0 vt e−F1 σt, t0 vΔ t
−F1 t
Δ
vt v t
e−F1 σt, t0 1 μt−F1 t
e−F1 σt, t0 vΔ t F1 tvt .
3.38
Combining 3.37 and 3.38, we deduce
Δ
vte−F1 t, t0 ≥ e−F1 σt, t0 F2 t.
3.39
Setting t s in 3.39, an integration with respect to s from t0 to t yields
vte−F1 t, t0 − vt0 e−F1 t0 , t0 ≥
t
e−F1 σs, t0 F2 sΔs.
3.40
t0
Considering vt0 1/aT , it is then followed by
vt ≥
1 aT t
t0
e−F1 σs, t0 F2 sΔs
aT e−F1 t, t0 ,
3.41
and furthermore
aT zt ≤
aT e−F1 t, t0 .
t
1 aT t0 e−F1 σs, t0 F2 sΔs
3.42
Combining 3.32 and 3.42, we obtain
ut ≤
aT e−F1 t, t0 ,
t
1 aT t0 e−F1 σs, t0 F2 sΔs
t ∈ t0 , T ∩ T.
3.43
Setting t T in 3.43, since T ∈ T0 is selected arbitrarily, after substituting T with t, we get
the desired result.
Abstract and Applied Analysis
11
If a0 0, then we carry out the process above with at replaced by at ε, and after
letting ε → 0, we also get the desired result. So the proof is complete.
Remark 3.5. If we take T R, t0 0, then Theorem 3.4 reduces to 34, Corollary 2.5. If T R,
t0 0, τ1 t τ2 t t, at ≡ C, where C is a nonnegative constant, and ft, s, gt, s, ht, s
are replaced by fs, gs, hs, then Theorem 3.4 reduces to 35, Theorem 1. If we take
T Z, t0 0, τ1 t τ2 t t, t ≡ C, where C is a nonnegative constant, and ft, s, t, s,
ht, s are replaced by fs, gs, hs, then Theorem 3.4 reduces to 35, Theorem 5.
Next we will study the delay integral inequality on time scales with the following form
u t ≤ C
p
p/p−q
p
p−q
p
p−q
t t
fτ1 sup τ1 sΔs
t0
fτ1 su τ1 s
q
t0
s
gτ1 ξu
p−q
τ1 ξΔξ hτ2 su τ2 s Δs,
q
t0
3.44
where u, f, g, p, q, C, α, φ, τi , i 1, 2 are defined as in Theorem 3.2, and h ∈ Crd α, ∞ ∩ T, R .
Lemma 3.6. Suppose u, f, g, τ1 are defined as in Theorem 3.2, and f gτ1 · ∈ , then for
t ∈ T0 ,
ut ≤ 1 t
t s
fτ1 s
fτ1 susΔs gτ1 ξuξΔξ Δs
t0
t0
3.45
t0
implies
ut ≤ 1 t
fτ1 sefgτ1 · s, t0 Δs.
3.46
t0
Proof. Denote the right side of 3.46 by zt. Then, ut ≤ zt, t ∈ T0 , and
zΔ t fτ1 tut fτ1 t
t
gτ1 ξuξΔξ
t0
≤ fτ1 tzt fτ1 t
t
3.47
gτ1 ξzξΔξ.
t0
Let mt zt t
t0
gτ1 ξzξΔξ. Then, zt ≤ mt, and zΔ t ≤ fτ1 tmt. Furthermore,
mΔ t zΔ t gτ1 tzt ≤ fτ1 tmt gτ1 tmt.
3.48
Since f gτ1 · ∈ , by 28, Theorem 5.4, we have mt ≤ efgτ1 · t, t0 . So,
zΔ t ≤ fτ1 tefgτ1 · t, t0 .
3.49
12
Abstract and Applied Analysis
Using zt0 1, an integration for 3.49 from t0 to t yields
zt ≤ 1 t
fτ1 sefgτ1 · s, t0 Δs,
3.50
t0
which confirms the desired inequality. So the proof is complete.
Theorem 3.7. If for t ∈ T0 , ut satisfies the inequality 3.44 with the initial condition 3.5, then
ut ≤
Jt 1 t
1/p−q
fτ1 sefgτ1 · s, t0 Δs
,
t ∈ T0
3.51
t0
provided f gτ1 · ∈ , where
Jt C t
hτ2 sΔs.
3.52
t0
Proof. Let
zt C εp/p−q
t t
p
p
fτ1 sup τ1 sΔs fτ1 suq τ1 s
p − q t0
p − q t0
s
3.53
gτ1 ξup−q τ1 ξΔξ hτ2 suq τ2 s Δs,
×
t0
where ε > 0 is an arbitrary small constant. Then,
ut ≤ z1/p t,
t ∈ T0 ,
3.54
and similar to 3.8–3.10
uτi t ≤ z1/p t,
i 1, 2, t ∈ T0 .
3.55
Abstract and Applied Analysis
13
Furthermore,
zΔ t ≤
p
fτ1 tup τ1 t
p−q
t
p
q
p−q
q
fτ1 tu τ1 t
gτ1 ξu τ1 ξΔξ hτ2 tu τ2 t
p−q
t0
p
fτ1 tzt
p−q
t
p
q/p
p−q/p
q/p
fτ1 tz t
gτ1 ξz
ξΔξ hτ2 tz t .
p−q
t0
3.56
Δ
Using 3.13, we obtain p/p − qzp−q/p t ≤ zΔ t/zq/p t. So 3.56 implies
t
Δ
zp−q/p t ≤ fτ1 tzp−q/p t fτ1 t
gτ1 ξzp−q/p ξΔξ hτ2 t.
3.57
t0
Considering zt0 C εp/p−q , an integration for 3.57 from t0 to t yields
zp−q/p t ≤ C ε t
fτ1 szp−q/p sΔs
t0
t s
p−q/p
fτ1 s
gτ1 ξz
ξΔξ hτ2 s Δs
t0
Jε t t0
t
3.58
p−q/p
fτ1 sz
sΔs
t0
t s
p−q/p
fτ1 s
gτ1 ξz
ξΔξ Δs,
t0
t0
where Jε t Jt ε, and Jt is defined in 3.52. Then,
zp−q/p t
≤1
Jε t
t
zp−q/p s
Δs fτ1 s
Jε s
t0
t s
zp−q/p ξ
fτ1 s
Δξ Δs.
gτ1 ξ
Jε ξ
t0
t0
3.59
Denote zp−q/p t/Jε t vt. Then,
vt ≤ 1 t
t0
t s
fτ1 s
fτ1 svsΔs gτ1 ξvξΔξ Δs.
t0
t0
3.60
14
Abstract and Applied Analysis
A suitable application of Lemma 3.6 to 3.60 yields
vt ≤ 1 t
3.61
fτ1 sefgτ1 · s, t0 Δs.
t0
Combining 3.54 and 3.61, we obtain
ut ≤
Jε t 1 t
1/p−q
fτ1 sefgτ1 · s, t0 Δs
.
3.62
t0
After letting ε → 0, we get the desired result. So the proof is complete.
Remark 3.8. If we take T R, t0 0, p 2, q 1, τ1 t τ2 t t, then Theorem 3.7 reduces to
33, Theorem 1a2. If we take T Z, t0 0, p 2, q 1, τ1 t τ2 t t, then Theorem 3.7
reduces to 33, Theorem 4b2.
Now we present a more general inequality than in Theorem 3.7.
Theorem 3.9. Suppose u, f, g, ω, p, q, C, α, φ, τi , i 1, 2 are defined as in Theorem 3.2, and h ∈
Crd α, ∞ ∩ T, R . If for t ∈ T0 , ut satisfies the following inequality:
up t ≤ Cp/p−q ×
s
p
p−q
t
fτ1 sup τ1 sΔs t0
p
p−q
t
fτ1 suq τ1 s
t0
3.63
gτ1 ξ ωuτ1 ξΔξ hτ2 suq τ2 sΔs
t0
with the initial condition 3.5, then for t ∈ T0 ,
ut ≤
Jt t
1/p−q
s
−1 fτ1 sG
,
fτ1 ξ gτ1 ξ Δξ Δs
GJs t0
3.64
t0
where Jt is defined as in Theorem 3.7, and
Gx
x
1
1
dr,
r ω r 1/p−q
−1 is the inverse of G.
with G∞
∞, and G
x>0
3.65
Abstract and Applied Analysis
15
Proof. Let the right side of 3.63 is zt, ε > 0 is an arbitrary small constant, and Jε t is
defined as in Theorem 3.7. Then, ut ≤ z1/p t, t ∈ T0 , and similar to the process of 3.53–
3.58, we obtain
zp−q/p t ≤ Jε t t t
fτ1 szp−q/p s Δs
t0
fτ1 s
s
t0
gτ1 ξω z
1/p
3.66
ξ Δξ Δs.
t0
Fix T ∈ T0 , and let t ∈ t0 , T ∩ T. Considering Jε t is nondecreasing, we deduce that
zp−q/p t ≤ Jε T t t
fτ1 szp−q/p sΔs
t0
fτ1 s
s
t0
gτ1 ξω z
1/p
3.67
ξ Δξ Δs.
t0
Denote the right side of 3.66 by vt. Then, zp−q/p t ≤ vt, t ∈ t0 , T ∩ T, and
Δ
p−q/p
v t fτ1 tz
t fτ1 t
t
gτ1 ξω z1/p ξ Δξ
t0
≤ fτ1 tvt fτ1 t
t
gτ1 ξω v1/p−q ξ Δξ.
3.68
t0
Denote mt vt t
t0
gτ1 ξωv1/p−q ξΔξ. Then, vt ≤ mt, and
vΔ t ≤ fτ1 tmt.
3.69
Furthermore,
mΔ t vΔ t gτ1 tω v1/p−q t ≤ fτ1 tmt gτ1 tω m1/p−q t
≤ fτ1 t gτ1 t mt ω m1/p−q t ,
3.70
that is,
mΔ t
≤ fτ1 t gτ1 t.
mt ω m1/p−q t
3.71
On the other hand, similar to 3.21–3.23, we have
Gmt
Δ
≤
mΔ t
.
mt ω m1/p−q t
3.72
16
Abstract and Applied Analysis
So combining 3.71 and 3.72, we deduce
Δ
≤ fτ1 t gτ1 t.
Gmt
3.73
Considering mt0 zt0 Jε T , an integration for 3.73 from t0 to t yields
ε T Gmt
≤ GJ
t
fτ1 ξ gτ1 ξ Δξ.
3.74
t0
is increasing, then
Since G
t
−1 mt ≤ G
fτ1 ξ gτ1 ξ Δξ .
GJε T 3.75
t0
Combining 3.69 and 3.75, we obtain
t
−1 v t ≤ fτ1 tG
fτ1 ξ gτ1 ξ Δξ .
GJε T Δ
3.76
t0
Setting t T in 3.76, since T is selected from T0 arbitrarily, then in fact 3.76 holds for all
t ∈ T0 , that is,
−1
Δ
v t ≤ fτ1 tG
ε t GJ
t
fτ1 ξ gτ1 ξ Δξ ,
∀t ∈ T0 .
3.77
t0
Considering vt0 Jε T , an integration for 3.77 from t0 to t yields
vt ≤ Jε T t
−1
fτ1 sG
t0
ε s GJ
s
fτ1 ξ gτ1 ξ Δξ Δs,
3.78
t0
which implies
ut ≤
Jε T t
−1
fτ1 sG
t0
ε s × GJ
s
1/p−q
,
fτ1 ξ gτ1 ξ Δξ Δs
3.79
t ∈ t0 , T ∩ T.
t0
Setting t T in the above inequality, considering T is selected from T0 arbitrarily, after
replacing T with t and letting ε → 0, we get the desired result. So the proof is complete.
Abstract and Applied Analysis
17
Remark 3.10. If we take T R, t0 0, p 2, q 1, τ1 t τ2 t t, then Theorem 3.9
reduces to 33, Theorem 3a7. If we take T Z, t0 0, p 2, q 1, τ1 t τ2 t t, then
Theorem 3.9 reduces to 33, Theorem 6b7.
Following in a similar manner as the proof of Theorem 3.7 and 3.9, then we present
two more theorems as follows, the special cases of which p 2, q 1, t0 0, τ1 t τ2 t t unify the continuous 33, Theorems 1a3, 3a8 and discrete inequalities 33,
Theorems 4b3, 6b8.
Theorem 3.11. Suppose u, f, g, h, p, q, C, α, φ, τi , i 1, 2 are defined as in Theorem 3.9. If for t ∈ T0 ,
ut satisfies the following inequality:
u t ≤ C
p
p/p−q
p
p−q
t fτ1 suq τ1 s
t0
×
s
3.80
gτ1 ξup−q τ1 ξΔξ hτ2 suq τ2 s Δs
t0
with the initial condition 3.5, then
ut ≤ JteH t, t0 1/p−q ,
provided H ∈ , where Ht fτ1 t
t
t0
t ∈ T0
3.81
gτ1 ξΔξ, and Jt is defined as in Theorem 3.7.
Theorem 3.12. Suppose u, f, g, h, ω, p, q, C, α, φ, τi , i 1, 2 are defined as in Theorem 3.9. If for
t ∈ T0 , ut satisfies the following inequality:
u t ≤ C
p
p/p−q
p
p−q
t fτ1 suq τ1 s
t0
×
s
3.82
gτ1 ξωuτ1 ξΔξ hτ2 suq τ2 s Δs
t0
with the initial condition 3.5, then
ut ≤
−1
G
GJt t
t0
fτ1 s
s
1/p−q
gτ1 ξΔξΔs
,
t ∈ T0 , t ∈ T0 ,
3.83
t0
where G is defined as in Theorem 3.2, and Jt is defined as in Theorem 3.7.
4. Some Applications
In this section, we will present some applications for the results which we have established
above and apply them to qualitative and quantitative analysis of solutions of certain delay
dynamic equations on time scales.
18
Abstract and Applied Analysis
Example 4.1. Consider the following delay dynamic integral equation on time scales:
u t C p
t
Fτs, uτsΔs,
t ∈ T0
4.1
t0
with the initial condition
ut φt, t ∈ α, t0 ∩ T,
φτt ≤ |C|1/p , ∀t ∈ T0 , τt ≤ t0 ,
4.2
where u ∈ Crd T0 , R, τ ∈ T0 , T with τt ≤ t, −∞ < α inf{τt, t ∈ T0 } ≤ t0 , φ ∈
Crd α, t0 ∩ T, R, F ∈ Crd α, ∞ ∩ T × R, R, and p is a constant with p > 1.
Theorem 4.2. Suppose ut is a solution of 4.1-4.2, and |Ft, u| ≤ ft|u|p gt|u|p−1 , where
f, g ∈ Crd α, ∞ ∩ T, R , then
ut ≤ C t
t
gτsΔs exp
t0
t ∈ T0 .
fτsΔs ,
4.3
t0
Proof. In fact, from 4.1 we have
|ut|p ≤ |C| t
|Fτs, uτs|Δs
t0
t fτs|uτs|p gτs|uτs|p−1 Δs
≤ |C| 4.4
t0
t fτs|uτs|p−1 ω|uτs| gτs|uτs|p−1 Δs,
|C| t0
where ωr r. Then a suitable application of Theorem 3.2 with τ1 τ2 τ, q p − 1 to
4.4 yields
−1
ut ≤ G
G C
t
gτsΔs
t0
t
fτsΔs ,
t ∈ T0 ,
4.5
t0
x
where Gx 1 1/r dr lnx, for all x > 0. Using the expression of G in 4.5, we obtain
the desired result, and the proof is complete.
Example 4.3. Consider the following delay dynamic differential equation on time scales
u t
3
Δ
F τt, uτt,
t
Mτξ, uτξΔξ ,
t0
t ∈ T0
4.6
Abstract and Applied Analysis
19
with the initial condition
ut0 u0 ,
ut φt,
φτt ≤ |u0 |,
4.7
t ∈ α, t0 ∩ T,
∀t ∈ T0 , τt ≤ t0 ,
where u, τ, α, φ are defined as in Example 4.1, F ∈ Crd α, ∞ ∩ T × R2 , R, M ∈ Crd α, ∞ ∩
T × R, R.
Theorem 4.4. Suppose ut is a solution of 4.6-4.7, and |Ft, u, v|
ft|u||v|, |Mt, u| ≤ gt|u|2 , where f, g ∈ Crd α, ∞ ∩ T, R , then
t
1
ut ≤ |u0 |
fτsefgτ· s, t0 Δs ,
≤
ft|u|3 t ∈ T0
4.8
t0
provided f gτ· ∈ .
Proof. The equivalent integral form of 4.6-4.7 is denoted by
u t 3
u30
t
s
F τs, uτs,
t0
Mτξ, uτξΔξ Δs.
4.9
t0
So,
s
t Mτξ, uτξΔξ Δs
|ut| ≤ |u0 | F τs, uτs,
t0
t0
t s
3
3
≤ |u0 | fτs|uτs| fτs|uτs| Mτξ, uτξΔξ Δs
t0
t0
3
3
s
3/2 t
2
2
3
≤ u0 fτs|uτs| fτs|uτs|
gτξ|uτξ| Δξ Δs.
t0
t0
4.10
Then a suitable application of Theorem 3.7 with p 3, q 1, τ1 τ2 τ, ht ≡ 0 to 4.10
yields the desired inequality 4.8, and the proof is complete.
t
Theorem 4.5. If under the conditions of Theorem 4.4, for t ∈ T0 , t0 fτ1 sefgτ1 . s, t0 Δs ≤
L, where L > 0 is a constant, then the trivial solution of 4.6-4.7 is uniformly stable.
Theorem 4.6. Assume |Ft, u1 , v1 − Ft, u2 , v2 |
≤
ft|u31 − u32 | ft |u31 − u32 ||v1 −
v2 |, |Mt, u1 − Mt, u2 | ≤ gt| |u31 − u32 |, where f, g are defined as in Theorem 4.4, then 4.64.7, have at most one solution.
20
Abstract and Applied Analysis
Proof. Suppose u1 t, u2 t are two solutions of 4.6-4.7. By 4.9 we have
s
t 3
3
Mτξ, u1 τξΔξ
u1 t − u2 t ≤ F τs, u1 τs,
t0 t0
Mτξ, u2 τξΔξ Δs
−F τs, u2 τs,
t0
s
t ≤
fτsu31 τs − u32 τs fτs u31 τs − u32 τs
t0
×
s
4.11
Mτξ, u1 τξ − Mτξ, u2 τξΔξ Δs
t0
t fτsu31 τs − u32 τs fτs u31 τs − u32 τs
≤
t0
×
s
t0
3
3
gτξ u1 τξ − u2 τξ Δξ Δs.
Treat |u31 t − u32 t| as one variable, and applying Theorem 3.7 to 4.11 with p 1, q 1/2, τ1 τ2 τ, ht ≡ 0 yields |u31 − u32 | ≤ 0, which implies u1 t ≡ u2 t, and the proof is
complete.
Acknowledgments
This work is partially supported by National Natural Science Foundation of China
11171178 and 11101248, Natural Science Foundation of Shandong Province ZR2009AM011,
ZR2010AQ026, and ZR2010AZ003 China, and Specialized Research Fund for the Doctoral
Program of Higher Education 20103705 110003China. The authors thank the referees very
much for their careful comments and valuable suggestions on this paper.
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