Hindawi Publishing Corporation

Abstract and Applied Analysis

Volume 2014, Article ID 623726, 4 pages
http://dx.doi.org/10.1155/2014/623726

Research Article
A Note on Gronwall’s Inequality on Time Scales

Xueru Lin
College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350002, China

Correspondence should be addressed to Xueru Lin; xuerulin2014@163.com

Received 29 May 2014; Accepted 6 June 2014; Published 1 July 2014

Academic Editor: Yonghui Xia

Copyright © 2014 Xueru Lin. 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.

This paper gives a new version of Gronwall’s inequality on time scales. The method used in the proof is much different from that in
the literature. Finally, an application is presented to show the feasibility of the obtained Gronwall’s inequality.

1. Introduction and Motivation Taking 𝑓(𝑡) ≡ 𝛼, a classical version of Gronwall’s inequality

follows (see [2, Corollary 6.7, pp 257]).
Recently, an interesting field of research is to study the
dynamic equations on time scales, which have been exten- Theorem B. Let 𝑝 ∈ R+ (T, R), 𝑝(𝑡) ≥ 0, 𝑦 ∈ C𝑟𝑑 R(T, R),
sively studied. For example, one can see [1–17] and references and 𝛼 ∈ R. Then
cited therein. A time scale T is an arbitrary nonempty closed
𝑡
subset of the real numbers R. The forward and backward jump 𝑦 (𝑡) ≤ 𝛼 + ∫ 𝑦 (𝑠) 𝑝 (𝑠) Δ𝑠, ∀𝑡 ∈ T, (3)
operators are defined by 𝜎(𝑡) := inf{𝑠 ∈ T : 𝑠 > 𝑡}, 𝜌(𝑡) := 𝑡0
sup{𝑠 ∈ T : 𝑠 > 𝑡}. A point 𝑡 ∈ T, 𝑡 > inf T, is said to be left
dense if 𝜌(𝑡) = 𝑡 and right dense if 𝑡 < inf T and 𝜎(𝑡) = 𝑡. The implies
mapping 𝜇 : T → R+ defined by 𝜇(𝑡) = 𝜎(𝑡) − 𝑡 is called
graininess. A function 𝑔 : T → R is said to be rd-continuous 𝑦 (𝑡) ≤ 𝛼𝑒𝑝 (𝑡, 𝑡0 ) , ∀𝑡 ∈ T. (4)
provided 𝑔 is continuous at right-dense points. The set of
all such rd-continuous functions is denoted by Crd (T, R). A This paper presents a new version of Gronwall’s inequality as
function 𝑝 : T → R is regressive provided 1+𝜇(𝑡)𝑝(𝑡) ≠ 0 for follows.
𝑡 ∈ T. Denote R+ (T, R) := {𝑝 ∈ Crd (T, R) : 1+𝜇(𝑡)𝑝(𝑡) > 0}.
Theorem 1. Let −𝑝 ∈ R+ (T, R) and 𝑦 ∈ C𝑟𝑑 R(T, R).
One of important topics is the differential inequalities on
Suppose that 𝑝(𝑡) ≥ 0, 𝑦(𝑡) ≥ 0, and 𝛼 > 0. Then
time scales. A nonlinear version of Gronwall’s inequality is
presented in [2, Theorem 6.4, pp 256]. This version is stated 󵄨󵄨 𝑡 󵄨󵄨
󵄨 󵄨
as follows. 𝑦 (𝑡) ≤ 𝛼 + 󵄨󵄨󵄨󵄨∫ 𝑦 (𝑠) 𝑝 (𝑠) Δ𝑠󵄨󵄨󵄨󵄨 , ∀𝑡 ∈ T, (5)
󵄨󵄨 𝑡0 󵄨󵄨
Theorem A. Let 𝑦, 𝑓 ∈ C𝑟𝑑 R(T, R), 𝑝(𝑡) ∈ R+ (T, R), and
implies
𝑝(𝑡) ≥ 0. Then

𝑡 𝛼𝑒 (𝑡, 𝑡0 ) , for 𝑡 ∈ [ 𝑡0 , +∞)T ,

𝑦 (𝑡) ≤ { 𝑝 (6)
𝑦 (𝑡) ≤ 𝑓 (𝑡) + ∫ 𝑦 (𝑠) 𝑝 (𝑠) Δ𝑠, ∀𝑡 ∈ T, (1) 𝛼𝑒−𝑝 (𝑡, 𝑡0 ) , for 𝑡 ∈ ( −∞, 𝑡0 ]T .
𝑡0

Remark 2. Note that, for 𝑡 ∈ (−∞, 𝑡0 ]T , inequality (5) reduces

implies to
𝑡 𝑡
𝑦 (𝑡) ≤ 𝑓 (𝑡) + ∫ 𝑒𝑝 (𝑡, 𝜎 (𝑠)) 𝑓 (𝑠) 𝑝 (𝑠) Δ𝑠, ∀𝑡 ∈ T. (2) 𝑦 (𝑡) ≤ 𝛼 − ∫ 𝑦 (𝑠) 𝑝 (𝑠) Δ𝑠, (7)
𝑡0 𝑡0
2 Abstract and Applied Analysis

which is different from inequality (3) in Theorem B. Since Lemma 4. Suppose that 𝑔 : T → R+ is positive delta differen-
Theorem B requires 𝑝(𝑡) ≥ 0, we see that Theorem B tiable on T and 𝑔Δ (𝑡)/𝑔(𝑡) is regressive. Then 𝜉𝜇(𝑡) (𝑔Δ (𝑡)/𝑔(𝑡))
cannot be applied to (7). Moreover, the method used to is a preantiderivative of function Log[𝑔(𝑡)], where 𝜉ℎ (𝑧) =
prove Theorem A cannot be used to prove Theorem 1. To (1/ℎ)Log(1 + 𝑧ℎ) and Log is the principal logarithm function.
explain this, recall the proof of Theorem A in [2]. Let 𝑧(𝑡) =
𝑡
∫𝑡 𝑦(𝑠)𝑝(𝑠)Δ𝑠. Then 𝑧(𝑡0 ) = 0 and Proof. Let 𝑓(𝑥) = Log 𝑥. Obviously, 𝑓 : R+ → R is
0
continuous on R+ . To prove Lemma 4, it suffices to show that
𝑧Δ = 𝑦 (𝑡) 𝑝 (𝑡) [Log[𝑔(𝑡)]]Δ = 𝜉𝜇(𝑡) (𝑔Δ (𝑡)/𝑔(𝑡)). In fact, by using Lemma 3,
(8) we have
≤ [𝑓 (𝑡) + 𝑧 (𝑡)] 𝑝 (𝑡) = 𝑝 (𝑡) 𝑧 (𝑡) + 𝑝 (𝑡) 𝑓 (𝑡) . Δ
[Log[𝑔(𝑡)]]
By comparing theorem and variation of constants formula,
1
Δ
𝑡 = (𝑓 ∘ 𝑔) (𝑡) = {∫ 𝑓󸀠 (𝑔 (𝑡) + ℎ𝜇 (𝑡) 𝑔Δ (𝑡)) 𝑑ℎ} 𝑔Δ (𝑡)
0
𝑧 (𝑡) ≤ ∫ 𝑒𝑝 (𝑡, 𝜎 (𝑠)) 𝑓 (𝑠) 𝑝 (𝑠) Δ𝑠, (9)
𝑡0 1
1
= {∫ 𝑑ℎ} 𝑔Δ (𝑡)
and hence Theorem A follows in view of 𝑦(𝑡) ≤ 𝑓(𝑡) + 𝑧(𝑡). 0 𝑔 (𝑡) + ℎ𝜇 (𝑡) 𝑔Δ (𝑡)
Now we try to adopt the same idea used in [2] to estimate 󵄨󵄨ℎ=1
𝑡 1 Δ 󵄨󵄨 Δ
inequality (7). Let 𝑧(𝑡) = ∫𝑡 𝑦(𝑠)𝑝(𝑠)Δ𝑠. Then 𝑧(𝑡0 ) = 0 and ={ Log [𝑔 (𝑡) + ℎ𝜇 (𝑡) 𝑔 (𝑡)] 󵄨󵄨 } 𝑔 (𝑡)
0 𝜇 (𝑡) 𝑔Δ (𝑡) 󵄨󵄨ℎ=0

𝑧Δ = 𝑦 (𝑡) 𝑝 (𝑡) ≤ [𝑓 (𝑡) − 𝑧 (𝑡)] 𝑝 (𝑡) 1

{
{ {Log [𝑔 (𝑡) + 𝜇 (𝑡) 𝑔Δ (𝑡)] − Log [𝑔 (𝑡)]}
{
{ 𝜇 (𝑡)
= −𝑝 (𝑡) 𝑧 (𝑡) + 𝑝 (𝑡) 𝑓 (𝑡) (10) {
{
{
{
{
{ if 𝜇 (𝑡) ≠ 0,
{
= −𝑝 (𝑡) 𝑧𝜎 + (1 + 𝜇 (𝑡) 𝑝 (𝑡)) 𝑝 (𝑡) 𝑓 (𝑡) . ={
{
{ 𝑔Δ (𝑡)
{
{
By comparing theorem and variation of constants formula, {
{
{ 𝑔 (𝑡)
{
we have {
{
𝑡 { if 𝜇 (𝑡) = 0
𝑧 (𝑡) ≤ ∫ 𝑒⊖𝑝 (𝑡, 𝜎 (𝑠)) 𝑓 (𝑠) 𝑝 (𝑠) Δ𝑠, (11)
𝑡0 { 1 𝑔 (𝑡) + 𝜇 (𝑡) 𝑔Δ (𝑡)
{
{ {Log } if 𝜇 (𝑡) ≠ 0,
{ 𝜇 (𝑡) 𝑔 (𝑡)
which implies ={ Δ
{
{ 𝑔 (𝑡)
{
if 𝜇 (𝑡) = 0
𝑡
{ 𝑔 (𝑡)
−𝑧 (𝑡) ≥ − ∫ 𝑒⊖𝑝 (𝑡, 𝜎 (𝑠)) 𝑓 (𝑠) 𝑝 (𝑠) Δ𝑠. (12)
𝑡0
1 𝑔Δ (𝑡)
= Log {1 + 𝜇 (𝑡) }
If we were to use the same idea as in [2], we should combine 𝜇 (𝑡) 𝑔 (𝑡)
(12) with
𝑔Δ (𝑡)
= 𝜉𝜇(𝑡) ( ).
𝑦 (𝑡) ≤ 𝑓 (𝑡) − 𝑧 (𝑡) . (13) 𝑔 (𝑡)
(15)
However, on one side, 𝑦(𝑡) ≤ ⋅ ⋅ ⋅ ; on the other side, 𝑓(𝑡) −
𝑧(𝑡) ≥ ⋅ ⋅ ⋅ . These two inequalities cannot lead us anywhere.
Therefore, some novel proof is employed to prove
Theorem 1. One can see the detailed proof in the next section.
Proof of Theorem 1. To prove Theorem 1, we divide it into two
cases.
2. Proof of Main Result
Case 1. For 𝑡 ∈ [𝑡0 , +∞)T , in this case, we have
Before our proof of Theorem 1, we need some lemmas.
󵄨󵄨 𝑡 󵄨󵄨 𝑡
󵄨 󵄨
Lemma 3 (chain rule [2]). Assume 𝑔 : T → X is delta 𝑦 (𝑡) ≤ 𝛼 + 󵄨󵄨󵄨󵄨∫ 𝑦 (𝑠) 𝑝 (𝑠) Δ𝑠󵄨󵄨󵄨󵄨 = 𝛼 + ∫ 𝑦 (𝑠) 𝑝 (𝑠) Δ𝑠,
differentiable on T. Assume further that 𝑓 : X → X is 󵄨󵄨 𝑡0 󵄨󵄨 𝑡0
continuously differentiable. Then 𝑓 ∘ 𝑔 : T → X is delta
differentiable and satisfies for 𝑡 ∈ [𝑡0 , +∞)T .
(16)
1
Δ 󸀠 Δ Δ
(𝑓 ∘ 𝑔) (𝑡) = {∫ 𝑓 (𝑔 (𝑡) + ℎ𝜇 (𝑡) 𝑔 (𝑡)) 𝑑ℎ} 𝑔 (𝑡) .
0 Hence, it is easy to conclude that 𝑦(𝑡) ≤ 𝛼𝑒𝑝 (𝑡, 𝑡0 ) for 𝑡 ∈
(14) [𝑡0 , +∞)T .
Abstract and Applied Analysis 3

𝑡
Case 2. For 𝑡 ∈ (−∞, 𝑡0 ]T , let 𝑧(𝑡) = ∫𝑡 𝑦(𝑠)𝑝(𝑠)Δ𝑠. For any 3. An Application
0
𝑠 ∈ [𝑡, 𝑡0 ]T , we have
Inequality (5) has many potential applications. For instance,
󵄨󵄨 𝑠 󵄨󵄨 it can be used to study the property of the solutions to the
󵄨 󵄨
𝑦 (𝑠) ≤ 𝛼 + 󵄨󵄨󵄨󵄨∫ 𝑦 (𝜏) 𝑝 (𝜏) Δ𝜏󵄨󵄨󵄨󵄨 dynamic systems. Consider the following linear system:
󵄨󵄨 𝑡0 󵄨󵄨
(17)
𝑠 𝑥Δ = 𝐴 (𝑡) 𝑥. (26)
= 𝛼 − ∫ 𝑦 (𝜏) 𝑝 (𝜏) Δ𝜏 = 𝛼 − 𝑧 (𝑠) .
𝑡0
Let 𝑋(𝑡, 𝑡0 , 𝑥0 ) and 𝑋(𝑡, 𝑡0 , 𝑥̃0 ) be two solutions of (26)
Noting that 𝑦 ≥ 0, 𝑝 ≥ 0, 𝛼 > 0, we have 𝛼 − 𝑧(𝑠) > 0. Thus, satisfying the initial conditions 𝑋(𝑡0 ) = 𝑥0 and 𝑋(𝑡0 ) = 𝑥̃0 ,
we have respectively.

𝑦 (𝑠) Theorem 5. Suppose that 𝐴(𝑡) is bounded on T. Then one has

≤ 1. (18)
𝛼 − 𝑧 (𝑠) 󵄩󵄩 󵄩
󵄩󵄩𝑋 (𝑡, 𝑡0 , 𝑥0 ) − 𝑋 (𝑡, 𝑡0 , 𝑥̃0 )󵄩󵄩󵄩
Multiplied by −𝑝(𝑠) on both sides of the above inequality, it 󵄩󵄩 󵄩
󵄩𝑥 − 𝑥̃0 󵄩󵄩󵄩 𝑒𝑝1 (𝑡, 𝑡0 ) , for 𝑡 ∈ [𝑡0 , +∞)T , (27)
follows that ≤ {󵄩󵄩󵄩 0 󵄩
󵄩󵄩𝑥0 − 𝑥̃0 󵄩󵄩󵄩 𝑒−𝑝1 (𝑡, 𝑡0 ) , for 𝑡 ∈ (−∞, 𝑡0 ]T ,
−𝑦 (𝑠) 𝑝 (𝑠)
≥ −𝑝 (𝑠) , (19)
𝛼 − 𝑧 (𝑠) where 𝑝1 (𝑡) ≡ 𝑀.
or Proof. Integrating (7) over [𝑡0 , 𝑡], we have
[𝛼 − 𝑧(𝑠)]Δ
≥ −𝑝 (𝑠) . (20) 𝑋 (𝑡, 𝑡0 , 𝑥0 )
𝛼 − 𝑧 (𝑠)
𝑡
Since −𝑝 ∈ R+ , −𝑝 ≤ [𝛼 − 𝑧(𝑠)]Δ /(𝛼 − 𝑧(𝑠)) ∈ R+ . Using = 𝑥0 + ∫ [𝐴 (𝑠) 𝑋 (𝑠, 𝑡0 , 𝑥0 ) + 𝑓 (𝑠, 𝑋 (𝑠, 𝑡0 , 𝑥0 ))] Δ𝑠.
the fact that 𝜉𝜇(𝑡) (𝑧) is nondecreasing with respect to 𝑧 for 𝑧 ∈ 𝑡0
(28)
R+ , we have

(𝛼 − 𝑧 (𝑠))Δ Denoting 𝑀 = sup𝑡∈T ‖𝐴(𝑡)‖, simple computation leads us to

𝜉𝜇(𝑠) [ ] ≥ 𝜉𝜇(𝑠) [−𝑝 (𝑠)] . (21)
𝛼 − 𝑧 (𝑠) 󵄩󵄩 󵄩
󵄩󵄩𝑋 (𝑡, 𝑡0 , 𝑥0 ) − 𝑋 (𝑡, 𝑡0 , 𝑥̃0 )󵄩󵄩󵄩
An integration of the above inequality over [𝑡, 𝑡0 ]T leads to 󵄨󵄨 𝑡 󵄨
󵄩 󵄩 󵄨 󵄩 󵄩 󵄨󵄨
≤ 󵄩󵄩󵄩𝑥0 − 𝑥̃0 󵄩󵄩󵄩 + 𝑀 󵄨󵄨󵄨󵄨∫ 󵄩󵄩󵄩𝑋 (𝑠, 𝑡0 , 𝑥0 ) − 𝑋 (𝑠, 𝑡0 , 𝑥̃0 )󵄩󵄩󵄩 Δ𝑠󵄨󵄨󵄨󵄨 .
󵄨󵄨 𝑡0 󵄨󵄨
𝑡0
(𝛼 − 𝑧 (𝑠))Δ 𝑡0
∫ 𝜉𝜇(𝑠) [ ] Δ𝑠 ≥ ∫ 𝜉𝜇(𝑠) [−𝑝 (𝑠)] Δ𝑠. (22) (29)
𝑡 𝛼 − 𝑧 (𝑠) 𝑡

By Theorem 1, it follows from (29) that

It follows from Lemma 4 that
󵄩󵄩 󵄩
󵄨𝑡 0 𝑡 󵄩󵄩𝑋 (𝑡, 𝑡0 , 𝑥0 ) − 𝑋 (𝑡, 𝑡0 , 𝑥̃0 )󵄩󵄩󵄩
Log[𝛼 − 𝑧(𝑠)]󵄨󵄨󵄨𝑡0 ≥ ∫ 𝜉𝜇(𝑠) [−𝑝 (𝑠)] Δ𝑠, (23)
𝑡
󵄩󵄩 󵄩 (30)
󵄩𝑥 − 𝑥̃0 󵄩󵄩󵄩 𝑒𝑝1 (𝑡, 𝑡0 ) , for 𝑡 ∈ [𝑡0 , +∞)T ,
or ≤ {󵄩󵄩󵄩 0 󵄩
󵄩󵄩𝑥0 − 𝑥̃0 󵄩󵄩󵄩 𝑒−𝑝1 (𝑡, 𝑡0 ) , for 𝑡 ∈ (−∞, 𝑡0 ]T .
𝑡0
Log 𝛼 − Log [𝛼 − 𝑧 (𝑡)] ≥ ∫ 𝜉𝜇(𝑡) [−𝑝 (𝑠)] Δ𝑠, (24)
𝑡

which leads to Remark 6. One can see that, for the case 𝑡 ∈ (−∞, 𝑡0 ]T , (29)
reduces to
𝑡0
𝛼 − 𝑧 (𝑡) ≤ 𝛼 exp (− ∫ 𝜉𝜇(𝑠) [−𝑝 (𝑠)] Δ𝑠) 󵄩󵄩 󵄩
𝑡
󵄩󵄩𝑋 (𝑡, 𝑡0 , 𝑥0 ) − 𝑋 (𝑡, 𝑡0 , 𝑥̃0 )󵄩󵄩󵄩
(25) (31)
𝑡 𝑡
󵄩 󵄩 󵄩 󵄩
= 𝛼 exp (∫ 𝜉𝜇(𝑠) [−𝑝 (𝑠)] Δ𝑠) = 𝛼𝑒−𝑝 (𝑡, 𝑡0 ) . ≤ 󵄩󵄩󵄩𝑥0 − 𝑥̃0 󵄩󵄩󵄩 − 𝑀 ∫ [󵄩󵄩󵄩𝑋 (𝑠, 𝑡0 , 𝑥0 ) − 𝑋 (𝑠, 𝑡0 , 𝑥̃0 )󵄩󵄩󵄩] Δ𝑠.
𝑡0 𝑡 0

Therefore, 𝑦(𝑡) ≤ 𝛼 − 𝑧(𝑡) ≤ 𝑒−𝑝 (𝑡, 𝑡0 ) for 𝑡 ∈ (−∞, 𝑡0 ]T . This As you see, Theorem B cannot be used to (31) because the
completes the proof of Theorem 1. essential condition in Theorem B is 𝑝(𝑡) ≥ 0.
4 Abstract and Applied Analysis

Conflict of Interests Abstract and Applied Analysis, vol. 2014, Article ID 632109, 11
pages, 2014.
The author declares that there is no conflict of interests [16] Y. Gao, Y. Xia, X. Yuan, and P. Wong, “Linearization of
regarding the publication of this paper. nonautonomous impulsive system with nonuniform exponen-
tial dichotomy,” Abstract and Applied Analysis, vol. 2014, Article
Acknowledgment ID 860378, 7 pages, 2014.
[17] Y. Xia, X. Yuan, K. I. Kou, and P. J. Y. Wong, “Existence and
This work was supported by JB12254. uniqueness of solution for perturbed nonautonomous systems
with nonuniform exponential dichotomy,” Abstract and Applied
Analysis, vol. 2014, Article ID 725098, 10 pages, 2014.
References
[1] M. Bohner, G. Guseinov, and A. Peterson, Introduction to the
Time Scales Calculus, Advances in Dynamic Equations on Time
Scales, Birkhäauser, Boston, Mass, USA, 2003.
[2] M. Bohner and A. Peterson, Dynamic Equations on Time Scales,
An Introduction with Applications, Birkhäuser, Boston, Mass,
USA, 2001.
[3] R. P. Agarwal, M. Bohner, and D. O’Regan, “Dynamic equations
on time scales: a survey,” Journal of Computational and Applied
Mathematics, vol. 141, no. 1-2, pp. 1–26, 2002.
[4] R. P. Agarwal, M. Bohner, and D. O’Regan, “Time scale
boundary value problems on infinite intervals,” Journal of
Computational and Applied Mathematics, vol. 141, no. 1-2, pp.
27–34, 2002.
[5] L. Erbe, A. Peterson, and P. Řeháka, “Comparison theorems for
linear dynamic equations on time scales,” Journal of Mathemat-
ical Analysis and Applications, vol. 275, no. 1, pp. 418–438, 2002.
[6] L. Erbe and A. Peterson, “Green functions and comparison
theorems for di ff erential equations on measure chains,”
Dynamics of Continuous, Discrete and Impulsive Systems, vol. 6,
no. 1, pp. 121–137, 1999.
[7] W. N. Li, “Some integral inequalities useful in the theory of
certain partial dynamic equations on time scales,” Computers
and Mathematics with Applications, vol. 61, no. 7, pp. 1754–1759,
2011.
[8] W. N. Li, “Some delay integral inequalities on time scales,”
Computers & Mathematics with Applications, vol. 59, no. 6, pp.
1929–1936, 2010.
[9] Y. H. Xia, J. Cao, and M. Han, “A new analytical method for the
linearization of dynamic equation on measure chains,” Journal
of Differential Equations, vol. 235, no. 2, pp. 527–543, 2007.
[10] Y. Xia, X. Chen, and V. G. Romanovski, “On the linearization
theorem of Fenner and Pinto,” Journal of Mathematical Analysis
and Applications, vol. 400, no. 2, pp. 439–451, 2013.
[11] Y.-H. Xia, J. Li, and P. J. Y. Wong, “On the topological
classification of dynamic equations on time scales,” Nonlinear
Analysis: Real World Applications, vol. 14, no. 6, pp. 2231–2248,
2013.
[12] Y. Xia, “Global analysis of an impulsive delayed Lotka-Volterra
competition system,” Communications in Nonlinear Science and
Numerical Simulation, vol. 16, no. 3, pp. 1597–1616, 2011.
[13] Y. H. Xia, “Global asymptotic stability of an almost periodic
nonlinear ecological model,” Communications in Nonlinear
Science and Numerical Simulation, vol. 16, no. 11, pp. 4451–4478,
2011.
[14] Y. Xia and M. Han, “New conditions on the existence and
stability of periodic solution in Lotka-Volterra’s population
system,” SIAM Journal on Applied Mathematics, vol. 69, no. 6,
pp. 1580–1597, 2009.
[15] Y. Gao, X. Yuan, Y. Xia, and P. J. Y. Wong, “Linearization
of impulsive differential equations with ordinary dichotomy,”
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