Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 546423, 11 pages
doi:10.1155/2010/546423
Research Article
Volterra Discrete Inequalities of Bernoulli Type
Sung Kyu Choi and Namjip Koo
Department of Mathematics, Chungnam National University,
Daejeon 305-764, Republic of Korea
Correspondence should be addressed to Namjip Koo, njkoo@cnu.ac.kr
Received 27 April 2010; Revised 18 October 2010; Accepted 12 December 2010
Academic Editor: P. J. Y. Wong
Copyright q 2010 S. K. Choi and N. Koo. 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 obtain the discrete versions of integral inequalities of Bernoulli type obtained in Choi 2007
and give an application to study the boundedness of solutions of nonlinear Volterra difference
equations.
1. Introduction
Integral inequalities of Gronwall type have been very useful in the study of ordinary
differential equations. Sugiyama 1 proved the discrete analogue of the well-known
Gronwall-Bellman inequality 2–5 which find numerous applications in the theory of
finite difference equations. See 6–11 for differential inequalities and difference inequalities.
Willett and Wong 12 established some discrete generalizations of the results of
Gronwall 5. The discrete analogue of the result of Bihari 13 was partially given by
Hull and Luxemburg 14 and was used by them for the numerical treatment of ordinary
differential equations. Pachpatte 15 obtained some general versions of Gronwall-Bellman
inequality. Oguntuase 16 established some generalizations of the inequalities obtained
in 15. However, there were some defects in the proofs of Theorems 2.1 and 2.7 in 16.
Choi et al. 17 improved the results of 16 and gave an application to boundedness of the
solutions of nonlinear integrodifferential equations.
In this paper, we establish the discrete analogues of integral inequalities of Bernoulli
type in 17 and give an application to study the boundedness of solutions of nonlinear
Volterra difference equations.
2
Journal of Inequalities and Applications
2. Main Results
Pachpatte 11 proved the following useful discrete inequality which can be used in the proof
of various discrete inequalities. Let Æ n0 {n0 , n0 1, . . . , n0 k, . . .} and Æ n0 , l {n0 , n0 1, . . . , n0 k, . . . , l} for fixed nonnegative integers n0 and l.
Lemma 2.1 see 11, Theorem 2.3.4. Let un be a positive sequence defined on Æ n0 , and let
bn and kn be nonnegative sequences defined on Æ n0 . Suppose that
Δun ≤ bnun knup n,
n ∈ Æ n0 ,
2.1
where p ≥ 0, p /
1 is a constant. Then, one has
1/1−p
n−1
1
1−p
1−p
un ≤
u n0 1 − p
kse s 1
,
en
sn0
where en n−1
sn0 1
n ∈ Æ n0 , β ,
2.2
bs−1 and
β sup n ∈ Æ n0 : u
1−p
n−1
1−p
kse s 1 > 0 .
n0 1 − p
2.3
sn0
If we set bn 0 in Lemma 2.1, then we can obtain the following corollary.
Corollary 2.2. Suppose that
n ∈ Æ n0 , p ≥ 0, p /
1.
Δun ≤ knup n,
2.4
Then,
un ≤ u
1−p
1/1−p
n−1
ks
,
n0 1 − p
n ∈ Æ n0 , β ,
2.5
sn0
where
β sup n ∈ Æ n0 : u
1−p
n−1
ks > 0 .
n0 1 − p
2.6
sn0
Willet and Wong 12, Theorem 4 proved the nonlinear difference inequality by using
the mean value theorem. We obtain the following result which is slightly different from Willet
and Wong’s Theorem 4.
Journal of Inequalities and Applications
3
Theorem 2.3. Let un, bn, kn be nonnegative sequences defined on Æ n0 and bn < 1 for each
n ∈ Æ n0 . Suppose that
un ≤ u0 n−1
bsus sn0
n−1
n ∈ Æ n0 ,
ksup s,
2.7
sn0
where p ≥ 0, p /
1 and u0 is a positive constant. Then one has
1
un ≤
e−b n
1−p
u0
1/1−p
n−1
1−p
1−p
kse−b s
,
n ∈ Æ n0 , β ,
2.8
sn0
where
e−b n n−1
β sup n ∈ Æ n0 :
1 − bs,
sn0
1−p
u0
n−1
1−p
1−p
kse−b s > 0 .
2.9
sn0
Proof. Let the right hand of 2.7 denote by
wn u0 n−1
bsus sn0
n−1
ksup s.
2.10
sn0
Then, we have
Δwn bnun knup n
2.11
≤ bnwn knwp n
≤ bnwn 1 knwp n,
n ∈ Æ n0 ,
since wn is nondecreasing. Multiplying 2.11 by the factor e−b n, we obtain
Δe−b nwn e−b nΔwn − bne−b nwn 1
≤ kne−b nwp n
1−p
kne−b ne−b nwnp ,
2.12
n ∈ Æ n0 ,
since e−b n is a positive sequence on Æ n0 . From Corollary 2.2, we obtain
e−b nwn ≤
1−p
u0
1/1−p
n−1
1−p
1−p
kse−b s
,
sn0
n ∈ Æ n0 , β ,
2.13
4
Journal of Inequalities and Applications
where
β sup n ∈ Æ n0 :
n−1
1−p
1−p
kse−b s > 0 .
1−p
u0
2.14
sn0
Since un ≤ wn and e−b n > 0, this implies that our inequality holds.
Remark 2.4. Note that 2.7 with p 1 in Theorem 2.3 implies
un ≤ u0
n−1
1 bs ks
sn0
n−1
≤ u0 exp
bs ks ,
2.15
n ∈ Æ n0 .
sn0
Hence, we can obtain a comparison result for linear difference inequalities.
The following theorem can be regarded as an extension of the inequality given by
Willett and Wong in 12 which is the discrete analogue of the inequality given by Choi et al.
in 17, Theorem 2.7.
Theorem 2.5. Let un and bn < 1 be nonnegative sequences defined on Æ n0 , and let kn, m be
a nonnegative function for n, m ∈ Æ n0 with n ≥ m. Suppose that
un ≤ c n−1
s−1
bs us sn0
ks, τu τ ,
p
n ∈ Æ n0 ,
2.16
τn0
where c is a positive constant and p > 0, p /
1 is a constant. Then, one has
1/q
n−1
s−1
τ−1
bs
q
q
un ≤ c e−b τ kτ 1, τ ,
c q
|Δτ kτ, σ|
sn0 e−b s
τn0
σn0
where q 1 − p, e−b n n−1
sn0 1
2.17
− bs, Δn kn, m kn 1, m − kn, m, and
β sup n ∈ Æ n0 : c q
q
n ∈ Æ n0 , β ,
n−1
sn0
q
e−b s
ks 1, s s−1
|Δs ks, τ| > 0 .
2.18
τn0
Proof. Define vn by the right member of 2.16. Then,
Δvn bnun bn
n−1
kn, sup s
sn0
≤ bnvn bn
n−1
sn0
2.19
kn, svp s,
Journal of Inequalities and Applications
5
1. Letting
by un ≤ vn and up n ≤ vp n for 0 ≤ p /
n−1
wn vn wn0 vn0 c,
kn, svp s,
2.20
sn0
we obtain
Δwn Δvn kn 1, nvp n n−1
Δn kn, svp s
sn0
≤ bnwn kn 1, nwp n n−1
|Δn kn, s|wp s
2.21
sn0
≤ bnwn kn 1, n n−1
|Δn kn, s| wp n,
sn0
for each n ∈ Æ n0 . By Theorem 2.3, we have
1/1−p
n−1
s−1
1
1−p
1−p
wn ≤
e−b s ks 1, s ,
c 1−p
|Δs ks, τ|
e−b n
sn0
τn0
where e−b n n−1
sn0 1
n ∈ Æ n0 , β ,
2.22
− bs and
β sup n ∈ Æ n0 : c q
q
n−1
sn0
1−p
e−b s
ks 1, s s−1
|Δs ks, τ| > 0 .
2.23
τn0
Substituting 2.22 into 2.19 and then summing it from n0 to n, we have
1/q
n−1
s−1
τ−1
bs
q
q
vn ≤ c e−b τ kτ 1, τ , n ∈ Æ n0 , β ,
c q
|Δτ kτ, σ|
sn0 e−b s
τn0
σn0
2.24
where q 1 − p. Hence, the proof is complete.
Remark 2.6. We suppose further that Δn kn, m is a nonnegative function for n, m ∈
with n ≥ m in Theorem 2.5. Then, we have
1/q
n−1
s−1
τ−1
bs
q
q
un ≤ c e−b τ kτ 1, τ Δτ kτ, σ
,
c q
sn0 e−b s
τn0
σn0
Æ n0 n ∈ Æ n0 , β ,
2.25
6
Journal of Inequalities and Applications
where
β sup n ∈ Æ n0 : c q
q
n−1
sn0
q
e−b s
ks 1, s s−1
Δs ks, τ > 0 .
2.26
τn0
If we set kn, m cndm in Theorem 2.5, then we obtain the following corollary
from Theorem 2.5. This is an analogue of the nonlinear difference inequality in 17,
Corollary 2.8.
Corollary 2.7. Let un, bn < 1, cn, dn be nonnegative sequences defined on Æ n0 , and let
u0 un0 be a positive constant. Suppose that
un ≤ u0 n−1
bs us cs
sn0
s−1
n ∈ Æ n0 ,
dτu τ ,
p
2.27
τn0
where p ≥ 0, p /
1 is a constant. Then, one has
1/q
n−1
s−1
τ−1
bs
q
q
un ≤ u0 e−b τ cτ 1dτ |Δcτ|
dσ
,
u0 q
sn0 e−b s
τn0
σn0
where q 1 − p, e−b n n−1
sn0 1
β sup n ∈ Æ n0 :
q
u0
q
n ∈ Æ n0 , β ,
2.28
− bs, and
n−1
sn0
q
e−b s
cs 1ds |Δcs|
s−1
dτ
>0 .
2.29
τn0
If we use Lemma 2.1 in the proof of Theorem 2.5, then we obtain the following bound
of un which contains double fold summations.
Corollary 2.8. Let un and bn be nonnegative sequences defined on Æ n0 , and let kn, m be a
nonnegative function for n, m ∈ Æ n0 with n ≥ m. Suppose that
un ≤ c n−1
sn0
bs us s−1
τn0
ks, τu τ ,
p
n ∈ Æ n0 ,
2.30
Journal of Inequalities and Applications
7
where c is a positive constant, and p > 0, p /
1 is a constant. Then, one has
1/1−p
n−1
s−1
1
1−p
1−p
un ≤
bs
c 1−p
ks, τ e s 1
,
en
sn0
τn0
where en n−1
sn0 1
n ∈ Æ n0 , β ,
2.31
bs−1 and
β sup n ∈ Æ n0 : c
1−p
1−p
n−1
bs
sn0
s−1
ks, τ e
1−p
s 1 > 0 .
2.32
τn0
Proof. Define vn by the right member of 2.30. Then, we have
Δvn bnun bn
n−1
kn, sup s
sn0
≤ bnvn bn
kn, svp s
sn0
≤ bnvn n−1
bn
n−1
2.33
kn, s vp n,
sn0
since un ≤ vn and vp n is nondecreasing in n. From Lemma 2.1, we obtain
1/1−p
n−1
s−1
1
1−p
1−p
vn ≤
c 1−p
ks, τ e s 1
,
bs
en
sn0
τn0
where en n−1
sn0 1
n ∈ Æ n0 , β ,
2.34
bs−1 and
β sup n ∈ Æ n0 : c
1−p
1−p
n−1
bs
sn0
s−1
ks, τ e
1−p
s 1 > 0 .
2.35
τn0
Since un ≤ vn, the proof is complete.
If we set p 1 in Theorem 2.5, then we can obtain the following discrete analogue of
Theorem 2.2 in 17 which improve in 16, Theorem 2.1.
Corollary 2.9. Let un and bn be nonnegative sequences defined on
nonnegative function for each n, m ∈ Æ n0 with n ≥ m. Suppose that
un ≤ c n−1
sn0
bs us s−1
τn0
Æ n0 and kn, m be a
ks, τuτ ,
n ∈ Æ n0 ,
2.36
8
Journal of Inequalities and Applications
where c is a positive constant. Then, one has
un ≤ c 1 n−1
bs
sn0
≤c 1
n−1
s−1 τn0
bs exp
sn0
where pn bn kn 1, n 1 pτ
n−1
s−1
2.37
,
pτ
n ∈ Æ n0 ,
τn0
sn0
|Δn kn, s|.
The proof of this corollary follows by the similar argument as in the proof of
Theorem 2.5. We omit the details.
3. An Application
In this section, we present an application of nonlinear difference inequalities established
in Theorem 2.5 to study the boundedness of the solutions of nonlinear Volterra difference
equations.
Consider the difference equation of Volterra type
Δxn Anxn n−1
Kn, sxs Fn,
xn0 x0 ,
3.1
sn0
where An and Kn, s are d × d matrices for each n, s ∈ Æ n0 and F : Æ n0 →
Êd .
Lemma 3.1 see 18, Theorem 2.9.1. Assume that there exists a d × d matrix Ln, s defined on
Æ n0 × Æ n0 and satisfying
Kn, s Δs Ln, s Ln, s 1As n−1
Ln, τ 1Kτ, s 0,
3.2
τs1
where Δs Ln, s Ln, s 1 − Ln, s.
Then, 3.1 is equivalent to the ordinary linear difference equation
Δyn Bnyn Ln, n0 x0 Hn,
yn0 x0 ,
3.3
where Bn An − Ln, n and
Hn Fn n−1
sn0
Ln, s 1Fs.
3.4
Journal of Inequalities and Applications
9
Consider the linear nonhomogeneous difference equation
Δxn Bnxn P n,
where Bn is a d×d matrix over Ê and P : Æ n0 →
formula of difference equations.
3.5
xn0 x0 ,
Êd . We present the variation of constants
Lemma 3.2. The solution xn xn, n0 , x0 of 3.5 is given by the variation of constants formula
xn Φn, n0 x0 n−1
Φn, s 1P s,
3.6
sn0
where Φn, n0 is a fundamental matrix solution of the difference equation Δxn Bnxn such
that Φn0 , n0 is the identity matrix.
Now, we give an application of our results. We consider the perturbation of linear
Volterra difference equation 3.1 with Fn 0
Δxn Anxn n−1
Kn, sxs Gn, xn,
3.7
sn0
with initial condition xn0 x0 , where G : Æ n0 × Êd →
Êd .
Theorem 3.3. Suppose that the following conditions hold for n ≥ n0 , 0 < α < 1:
i |Φn, n0 | ≤ M1 ,
ii |Ln, s| ≤ M2 αn−s ,
iii |Gs, x| ≤ bs|x|, n ≥ s ≥ n0 ,
where M1 and M2 are some positive constants and bn is a nonnegative sequence defined on Æ n0 −s
with ∞
sn0 α bs < ∞. Then, all solutions of 3.7 are bounded in Æ n0 .
Proof. By Lemma 3.1, 3.7 is equivalent to
Δxn Bnxn Ln, n0 x0 Gn, xn n−1
Ln, s 1Gs, xs,
3.8
sn0
with xn0 x0 , where Bn An − Ln, n and Ln, s is a solution of 3.2. It follow from
Lemma 3.2 that the solution xn, n0 , x0 of 3.8 is given by
xn Φn, n0 x0 n−1
sn0
Φn, s1 Ls, n0 x0 Gs, xs
s−1
τn0
Ls, τ 1Gτ, xτ ,
n ≥ n0 .
3.9
10
Journal of Inequalities and Applications
By using the conditions i–iii, we obtain
|xn| |Φn, n0 ||x0 | n−1
|Φn, s 1||Ls, n0 ||x0 |
sn0
n−1
|Φn, s 1| |Gs, xs| sn0
s−1
|Ls, τ 1||Gτ, xτ|
τn0
n−1
≤ M1 |x0 | M1
s−n0
M2 α
|x0 | M1
sn0
n−1
bs |xs| sn0
s−1
|Ls, τ 1|bτ|xτ|
τn0
n−1
s−1
1 M2 − α
M1 |x0 | ≤
M1 bs |xs| |Ls, τ 1|bτ|xτ| ,
1−α
sn0
τn0
n ≥ n0 .
3.10
Letting un |xn|, c 1 M2 − α/1 − αM1 |x0 | and
|Ls, τ 1|bτ ≤ M2 αs−τ−1 bτ ks, τ,
3.11
and employing the above estimate by Corollary 2.9, then we have
un ≤ c 1 n−1
sn0
where M c1 ∞
sn0
M1 bs exp
s−1
pτ
≤ M,
n ∈ Æ n0 ,
3.12
τn0
M1 bs exp ∞
τn0 pτ, because
pτ bτ kτ 1, τ τ−1
|Δτ kτ, σ|
σn0
bτ M2 αbτ M2
τ−1
1 − α τ α
α−σ bσ
α
σn0
≤ bτ1 M2 α ατ M2
3.13
∞
1 − α α−σ bσ,
α σn0
and ∞
τn0 pτ < ∞. Hence, the solutions xn of 3.7 are bounded in Æ n0 , and the proof is
complete.
Acknowledgments
This work was supported by Basic Science Research Program through the National Research
Foundation of Korea NRF funded by the Ministry of Education, Science, and Technology
NRF-2010-0008835. The authors are thankful to the referee for giving valuable comments
for the improvement of this paper.
Journal of Inequalities and Applications
11
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