Journal of Inequalities in Pure and
Applied Mathematics
SOME NEW DISCRETE NONLINEAR DELAY INEQUALITIES AND
APPLICATION TO DISCRETE DELAY EQUATIONS
volume 7, issue 4, article 122,
2006.
WING-SUM CHEUNG AND SHIOJENN TSENG
Department of Mathematics
University of Hong Kong
Hong Kong
EMail: wscheung@hku.hk
Department of Mathematics
Tamkang University
Tamsui, Taiwan 25137
EMail: tseng@math.tku.edu.tw
Received 07 September, 2005;
accepted 27 January, 2006.
Communicated by: S.S. Dragomir
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
267-05
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Abstract
In this paper, some new discrete Gronwall-Bellman-Ou-Iang-type inequalities
are established. These on the one hand generalize some existing results and
on the other hand provide a handy tool for the study of qualitative as well as
quantitative properties of solutions of difference equations.
2000 Mathematics Subject Classification: 26D10, 26D15, 39A10, 39A70.
Key words: Gronwall-Bellman-Ou-Iang-type Inequalities, Discrete inequalities, Difference equations.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Discrete Inequalities with Delay . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Immediate Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4
Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
References
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
Wing-Sum Cheung and
Shiojenn Tseng
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
1.
Introduction
It is widely recognized that integral inequalities in general provide an effective
tool for the study of qualitative as well as quantitative properties of solutions of
integral and differential equations. While most integral inequalities only give
the ‘global behavior’ of the unknown functions (in the sense that bounds are
only obtained for integrals of certain functions of the unknown functions), the
Gronwall-Bellman type (see, e.g. [3] – [8], [10] – [12], [15] – [18]) is particularly useful as they provide explicit pointwise bounds of the unknown functions.
A specific branch of this type of inequalities is originated by Ou-Iang. In his
paper [13], in order to study the boundedness behavior of the solutions of some
2nd order differential equations, Ou-Iang established the following beautiful
inequality.
Theorem 1.1 (Ou-Iang [13]). If u and f are non-negative functions on [0, ∞)
satisfying
Z x
2
2
u (x) ≤ c + 2
f (s)u(s)ds, x ∈ [0, ∞),
0
for some constant c ≥ 0, then
Z
u(x) ≤ c +
x
f (s)ds,
x ∈ [0, ∞).
0
While Ou-Iang’s inequality is interesting in its own right, it also has numerous important applications in the study of differential equations (see, e.g.,
[2, 3, 9, 11, 12]). Over the years, various extensions of Ou-Iang’s inequality
have been established. These include, among others, works of Agarwal [1],
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
Wing-Sum Cheung and
Shiojenn Tseng
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
Ma-Yang [10], Pachpatte [14] – [18], Tsamatos-Ntouyas [19], and Yang [20].
Among such extensions, the discretization is of particular interest because analogous to the continuous case, discrete versions of integral inequalities should,
in our opinion, play an important role in the study of qualitative as well as quantitative properties of solutions of difference equations.
It is the purpose of this paper to establish some new discrete GronwallBellman-Ou-Iang-type inequalities giving explicit bounds to unknown discrete
functions. These on the one hand generalize some existing results in the literature and on the other hand give a handy tool to the study of difference equations.
An application to a discrete delay equation is given at the end of the paper.
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
Wing-Sum Cheung and
Shiojenn Tseng
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
2.
Discrete Inequalities with Delay
Throughout this paper, R+ = (0, ∞) ⊂ R, Z+ = R+ ∩ Z, and for any a, b ∈ R,
Ra = [a, ∞), Za = Ra ∩Z, Z[a,b] = Z∩[a, b]. If X and Y are sets, the collection
of functions of X into Y , the collection of continuous functions of X into Y ,
and that of continuously differentiable functions of X into Y are denoted by
F(X, Y ), C(X, Y ), and C 1 (X, Y ), respectively. As usual, if u is a real-valued
function on Z[a,b] , the difference operator ∆ on u is defined as
∆u(n) = u(n + 1) − u(n) ,
n ∈ Z[a,b−1] .
In the sequel, summations over empty sets are, as usual, defined to be zero.
The basic assumptions and initial conditions used in this paper are the following:
Assumptions
(A1) f, g, h, k, p ∈ F(Z0 , R0 ) with p non-decreasing;
(A2) w ∈ C(R0 , R0 ) is non-decreasing with w(r) > 0 for r > 0;
(A3) σ ∈ F(Z0 , Z) with σ(s) ≤ s for all s ∈ Z0 and −∞ < a := inf{σ(s) :
s ∈ Z0 } ≤ 0;
(A4) ψ ∈ F(Z[a,0] , R0 ); and
(A5) φ ∈ C 1 (R0 , R0 ) with φ0 non-decreasing and φ0 (r) > 0 for r > 0.
Initial Conditions
(I1) x(s) = ψ(s) for all s ∈ Z[a,0] ;
(I2) ψ (σ(s)) ≤ φ−1 (p(s)) for all s ∈ Z0 with σ(s) ≤ 0.
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
Wing-Sum Cheung and
Shiojenn Tseng
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
Theorem 2.1. Under Assumptions (A1) – (A5), if x ∈ F(Za , R0 ) is a function
satisfying the nonlinear delay inequality
(2.1) φ (x(n))
≤ p(n) +
n−1
X
φ0 (x (σ(s))) {f (s) + g(s)x (σ(s)) + h(s)w (x (σ(s)))}
s=0
for all n ∈ Z0 with initial conditions (I1) – (I2), then
!
!#
( "
n−1
n−1
X
X
exp
g(s)
φ−1 (p(n)) +
f (s)
(2.2) x(n) ≤ Φ−1 Φ
s=0
+ exp
s=0
n−1
X
! n−1
)
X
g(s)
h(t)
s=0
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and α ≥ 0 is chosen such that the RHS of (2.2) is well-defined, that is,
"
!
!#
n−1
n−1
X
X
Φ
exp
g(s)
φ−1 (p(n)) +
f (s)
s=0
+ exp
n−1
X
s=0
Wing-Sum Cheung and
Shiojenn Tseng
t=0
for all n ∈ Z[0,α] , where Φ ∈ C(R0 , R) is defined by
Z r
ds
Φ(r) :=
, r > 0,
1 w(s)
s=0
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
! n−1
X
g(s)
h(t) ∈ Im Φ
t=0
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for all n ∈ Z[0,α] .
J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
Proof. Fix ε > 0 and N ∈ Z[0,α] . Define u : Z[0,N ] → R0 by
(
(2.3) u(n) := φ−1 ε + p(N )
+
n−1
X
)
φ0 (x (σ(t))) [f (t) + g(t)x (σ(t)) + h(t)w (x (σ(t)))]
.
t=0
By (A5), u is non-decreasing on Z[0,N ] . For any n ∈ Z[0,N ] , by (A5) again,
u(n) ≥ φ−1 (ε + p(N )) > 0 .
(2.4)
As φ (u(n)) > φ (x(n)), we have
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
Wing-Sum Cheung and
Shiojenn Tseng
u(n) > x(n) .
(2.5)
Next, observe that if σ(n) ≥ 0, then by (A3), σ(n) ∈ Z[0,N ] and so
x (σ(n)) < u (σ(n)) ≤ u(n) .
On the other hand, if σ(n) ≤ 0, then by (A3) again, σ(n) ∈ Z[a,0] and so by
(I1), (I2), (A1), (A5) and (2.4),
x (σ(n)) = ψ (σ(n)) ≤ φ−1 (p(n)) ≤ φ−1 (p(N )) ≤ φ−1 (p(N ) + ε) ≤ u(n) .
Hence we always have
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(2.6)
x (σ(n)) ≤ u(n) for all n ∈ Z[0,N ] .
J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
Therefore, for any s ∈ Z[0,N −1] , by (2.3) and (2.6),
∆(φ ◦ u)(s) = φ (u(s + 1)) − φ (u(s))
= φ0 (x (σ(s))) {f (s) + g(s)x (σ(s)) + h(s)w (x (σ(s)))}
≤ φ0 (u(s)) {f (s) + g(s)u(s) + h(s)w (u(s))} .
On the other hand, by the Mean Value Theorem, we obtain
∆(φ ◦ u)(s) = φ (u(s + 1)) − φ (u(s))
= φ0 (ξ)∆u(s)
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
for some ξ ∈ [u(s), u(s + 1)]. Observe that by (2.4) and (A5), φ0 (ξ) > 0. Thus
by the monotonicity of φ0 , for any s ∈ Z[0,N −1] ,
φ0 (u(s))
{f (s) + g(s)u(s) + h(s)w (u(s))}
φ0 (ξ)
≤ f (s) + g(s)u(s) + h(s)w (u(s)) .
∆u(s) ≤
u(n) − u(0) =
≤
s=0
n−1
X
s=0
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Summing up, we have
n−1
X
Wing-Sum Cheung and
Shiojenn Tseng
Go Back
∆u(s)
f (s) +
II
I
n−1
X
s=0
h(s)w (u(s)) +
n−1
X
s=0
Close
g(s)u(s) ,
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Page 8 of 36
J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
or
"
−1
u(n) ≤ φ
(ε + p(N )) +
n−1
X
f (s) +
s=0
n−1
X
#
h(s)w (u(s)) +
s=0
n−1
X
g(s)u(s)
s=0
for all n ∈ Z[0,N ] . Hence by the discrete version of the Gronwall-Bellman
inequality (see, e.g., [16, Corollary 1.2.5]),
"
#
n−1
n−1
n−1
X
X
X
−1
u(n) ≤ φ (ε + p(N )) +
f (s) +
h(s)w (u(s)) exp
g(s)
"
(2.7)
≤ φ−1 (ε + p(N )) +
s=0
N
−1
X
s=0
f (s) +
s=0
n−1
X
#
h(s)w (u(s)) exp
s=0
s=0
N
−1
X
g(s)
s=0
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
Wing-Sum Cheung and
Shiojenn Tseng
for all n ∈ Z[0,N ] . Denote by v(n) the RHS of (2.7). Then v is non-decreasing
and for all n ∈ Z[0,N ] ,
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u(n) ≤ v(n) .
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(2.8)
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Therefore, for any t ∈ Z[0,N −1] ,
∆v(t) = v(t + 1) − v(t)
= h(t)w (u(t)) exp
≤ h(t)w (v(t)) exp
N
−1
X
s=0
N
−1
X
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g(s)
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g(s) .
Page 9 of 36
s=0
J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
On the other hand, by the Mean Value Theorem, we have
∆(Φ ◦ v)(t) = Φ (v(t + 1)) − Φ (v(t))
= Φ0 (η)∆v(t)
1
=
∆v(t)
w(η)
for some η ∈ [v(t), v(t + 1)]. Observe that by (2.4), (2.8), and (A2), w(η) > 0.
Therefore, as w is non-decreasing,
N −1
X
1
∆(Φ ◦ v)(t) ≤
h(t)w (v(t)) exp
g(s)
w(η)
s=0
≤ h(t) exp
N
−1
X
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for all t ∈ Z[0,N −1] . Summing up, we have
t=0
∆(Φ ◦ v)(t) ≤
n−1
X
t=0
Wing-Sum Cheung and
Shiojenn Tseng
g(s)
s=0
n−1
X
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
h(t) exp
Contents
N
−1
X
s=0
g(s) .
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On the other hand,
n−1
X
t=0
∆(Φ ◦ v)(t) = Φ (v(n)) − Φ (v(0))
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
"
= Φ (v(n)) − Φ
exp
N
−1
X
!
φ−1 (ε + p(N )) +
g(s)
s=0
N
−1
X
!#
f (s)
,
s=0
therefore,
"
Φ (v(n)) ≤ Φ
exp
!
N
−1
X
−1
g(s)
φ
(ε + p(N )) +
s=0
+
N
−1
X
s=0
n−1
X
!#
f (s)
h(t) exp
N
−1
X
t=0
g(s)
s=0
for all n ∈ Z[0,N ] . In particular, taking n = N we have
"
Φ (v(N )) ≤ Φ
exp
N
−1
X
!
g(s)
−1
φ
(ε + p(N )) +
N
−1
X
s=0
+
exp
Wing-Sum Cheung and
Shiojenn Tseng
!#
f (s)
s=0
N
−1
X
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! N −1
X
g(s)
h(t).
s=0
t=0
Since N ∈ Z[0,α] is arbitrary,
"
Φ (v(n)) ≤ Φ
exp
n−1
X
!
g(s)
−1
φ
(ε + p(n)) +
n−1
X
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
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!#
f (s)
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s=0
s=0
+
exp
n−1
X
s=0
! n−1
X
g(s)
h(t)
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t=0
J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
for all n ∈ Z[0,α] . Hence
( "
v(n) ≤ Φ−1
Φ
exp
n−1
X
!
φ−1 (ε + p(n)) +
g(s)
s=0
+ exp
n−1
X
!#
f (s)
s=0
n−1
X
! n−1
)
X
g(s)
h(t)
s=0
t=0
and so by (2.5) and (2.8),
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
x(n) < u(n) ≤ v(n)
( "
≤ Φ−1
Φ
exp
n−1
X
!
g(s)
φ−1 (ε + p(n)) +
s=0
+ exp
n−1
X
!#
f (s)
s=0
n−1
X
! n−1
)
X
g(s)
h(t)
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t=0
Contents
s=0
for all n ∈ Z[0,α] . Finally, letting ε → 0+ , we conclude that
( "
x(n) ≤ Φ−1
Φ
exp
n−1
X
s=0
!
g(s)
φ−1 (p(n)) +
n−1
X
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J
!#
f (s)
s=0
n−1
X
+ exp
s=0
Wing-Sum Cheung and
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! n−1
)
X
g(s)
h(t)
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t=0
Page 12 of 36
for all n ∈ Z[0,α] .
J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
R ∞ ds
Remark 1. In many cases the non-decreasing function w satisfies 1 w(s)
=
√
∞. For example, w = constant > 0, w(s) = s, etc., are such functions. In
such cases Φ(∞) = ∞ and so we may take α → ∞, that is, (2.2) is valid for
all n ∈ Z0 .
Theorem 2.2. Under Assumptions (A1) – (A5), if x ∈ F(Za , R0 ) is a function
satisfying the nonlinear delay inequality
(
n−1
X
φ (x(n)) ≤ p(n) +
φ0 (x (σ(s))) f (s) + g(s)x (σ(s))
s=0
+ h(s)
s−1
X
)
k(t)w (x (σ(t)))
t=0
for all n ∈ Z0 with initial conditions (I1) – (I2), then
( "
!
!#
n−1
n−1
X
X
(2.9) x(n) ≤ Φ−1 Φ
exp
g(s)
φ−1 (p(n)) +
f (s)
s=0
s=0
+ exp
n−1
X
s=0
! n−1 s−1
)
XX
g(s)
h(s)k(t)
Wing-Sum Cheung and
Shiojenn Tseng
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JJ
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s=0 t=0
for all n ∈ Z[0,β] , where Φ ∈ C(R0 , R) is as defined in Theorem 2.1, and β ≥ 0
is chosen such that the RHS of (2.9) is well-defined, that is,
"
!
!#
n−1
n−1
X
X
−1
Φ
exp
g(s)
φ (p(n)) +
f (s)
s=0
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
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s=0
J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
+
exp
n−1
X
! n−1 s−1
XX
g(s)
h(s)k(t) ∈ Im Φ
s=0
s=0 t=0
for all n ∈ Z[0,β] .
Proof. Fix ε > 0 and M ∈ Z[0,β] . Define u : Z[0,M ] → R0 by
(
(2.10) u(n) := φ−1
ε + p(M ) +
n−1
X
"
φ0 (x (σ(δ))) · f (δ) + g(δ)x (σ(δ))
δ=0
+h(δ)
δ−1
X
#)
k(t)w (x (σ(t)))
.
t=0
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
Wing-Sum Cheung and
Shiojenn Tseng
By (A5), u is non-decreasing on Z[0,M ] . For any n ∈ Z[0,M ] , by (A5) again,
(2.11)
u(n) ≥ φ−1 (ε + p(M )) > 0 .
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As φ (u(n)) > φ (x(n)), we have
(2.12)
u(n) > x(n) .
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Using the same arguments as in the derivation of (2.6) in the proof of Theorem
2.1, we have
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x (σ(n)) ≤ u(n) for all n ∈ Z[0,M ] .
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(2.13)
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
Hence for any s ∈ Z[0,M −1] , by (2.10) and (2.13),
∆(φ ◦ u)(s) = φ (u(s + 1)) − φ (u(s))
(
= φ0 (x (σ(s))) f (s) + g(s)x (σ(s)) + h(s)
s−1
X
)
k(t)w (x (σ(t)))
t=0
(
0
≤ φ (u(s)) f (s) + g(s)u(s) + h(s)
s−1
X
)
k(t)w (u(t))
.
t=0
On the other hand, by the Mean Value Theorem,
∆(φ ◦ u)(s) = φ (u(s + 1)) − φ (u(s))
= φ0 (ξ)∆u(s)
for some ξ ∈ [u(s), u(s + 1)]. Observe that by (2.12) and (A5), φ0 (ξ) > 0.
Thus by the monotonicity of φ0 , for any s ∈ Z[0,M −1] ,
(
)
s−1
X
φ0 (u(s))
f (s) + g(s)u(s) + h(s)
k(t)w (u(t))
∆u(s) ≤
φ0 (ξ)
t=0
≤ f (s) + g(s)u(s) + h(s)
s−1
X
t=0
u(n) − u(0) =
Wing-Sum Cheung and
Shiojenn Tseng
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Summing up, we have
n−1
X
k(t)w (u(t)) .
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
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∆u(s)
Page 15 of 36
s=0
J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
≤
n−1
X
f (s) +
n−1
X
s=0
h(s)
s−1
X
s=0
k(t)w (u(t)) +
n−1
X
t=0
g(s)u(s) ,
s=0
or
"
−1
u(n) ≤ φ
(ε + p(M )) +
n−1
X
f (s) +
s=0
n−1
X
h(s)
s=0
s−1
X
#
n−1
X
k(t)w (u(t)) +
g(s)u(s)
t=0
s=0
for all n ∈ Z[0,M ] . Hence by the discrete version of the Gronwall-Bellman
inequality (see, e.g., [16, Corollary 1.2.5]),
"
n−1
X
−1
u(n) ≤ φ (ε + p(M )) +
f (s)
s=0
+
n−1
X
h(s)
s=0
s−1
X
#
k(t)w (u(t)) exp
t=0
≤ φ
g(s)
(ε + p(M )) +
M
−1
X
(2.14)
+
s=0
h(s)
s−1
X
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f (s)
JJ
J
s=0
n−1
X
Wing-Sum Cheung and
Shiojenn Tseng
s=0
"
−1
n−1
X
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
#
k(t)w (u(t)) exp
t=0
M
−1
X
g(s)
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s=0
for all n ∈ Z[0,M ] . Denote by v(n) the RHS of (2.14). Then v is non-decreasing
and for all n ∈ Z[0,M ] ,
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Page 16 of 36
(2.15)
u(n) ≤ v(n) .
J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
Therefore, for any δ ∈ Z[0,M −1] ,
∆v(δ) = v(δ + 1) − v(δ)
= h(δ)
δ−1
X
!
k(t)w (u(t)) exp
t=0
≤ h(δ)
δ−1
X
!
k(t)w (v(t)) exp
t=0
≤ h(δ)w (v(δ))
δ−1
X
!
k(t) exp
t=0
M
−1
X
s=0
M
−1
X
s=0
M
−1
X
g(s)
g(s)
g(s) .
s=0
On the other hand, by the Mean Value Theorem,
∆(Φ ◦ v)(δ) = Φ (v(δ + 1)) − Φ (v(δ))
1
∆v(δ)
= Φ0 (η)∆v(δ) =
w(η)
for some η ∈ [v(δ), v(δ + 1)]. Observe that by (2.11), (2.14), and (A2), w(η) >
0. Therefore, as w is non-decreasing,
!
δ−1
M
−1
X
X
1
∆(Φ ◦ v)(δ) ≤
h(δ)w (v(δ))
k(t) exp
g(s)
w(η)
t=0
s=0
!
δ−1
M
−1
X
X
≤ h(δ)
k(t) exp
g(s)
t=0
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
Wing-Sum Cheung and
Shiojenn Tseng
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s=0
J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
for all δ ∈ Z[0,M −1] . Summing up, we have
n−1
X
∆(Φ ◦ v)(δ) ≤
δ=0
n−1
X
h(δ)
δ−1
X
δ=0
t=0
n−1
X
δ−1
X
!
k(t) exp
M
−1
X
g(s) ,
s=0
or
Φ (v(n)) ≤ Φ (v(0)) +
h(δ)
φ
(ε + p(M )) +
M
−1
X
!
f (s) exp
s=0
+
n−1
X
h(δ)
k(t) exp
M
−1
X
t=0
δ=0
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
M
−1
X
#
g(s)
s=0
!
δ−1
X
g(s)
s=0
M
−1
X
"
=Φ
k(t) exp
t=0
δ=0
−1
!
Wing-Sum Cheung and
Shiojenn Tseng
g(s)
s=0
Title Page
for all n ∈ Z[0,M ] . In particular, taking n = M this yields
"
Φ (v(M )) ≤ Φ
−1
φ
(ε + p(M )) +
M
−1
X
+
!
f (s) exp
s=0
M
−1
X
δ=0
M
−1
X
Contents
JJ
J
#
g(s)
s=0
h(δ)
δ−1
X
t=0
!
k(t) exp
M
−1
X
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g(s) .
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
Since M ∈ Z[0,β] is arbitrary,
"
φ−1 (ε + p(n)) +
Φ (v(n)) ≤ Φ
n−1
X
!
f (s) exp
s=0
+
n−1
X
h(δ)
n−1
X
#
g(s)
s=0
δ−1
X
!
k(t) exp
t=0
δ=0
n−1
X
g(s)
s=0
for all n ∈ Z[0,β] . Hence
( "
v(n) ≤ Φ−1
φ−1 (ε + p(n)) +
Φ
n−1
X
+
!
f (s) exp
s=0
n−1
X
n−1
X
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
#
g(s)
Wing-Sum Cheung and
Shiojenn Tseng
s=0
h(δ)
δ−1
X
!
k(t) exp
t=0
δ=0
n−1
X
)
g(s)
Title Page
s=0
Contents
and so by (2.12) and (2.15),
x(n) < u(n) ≤ v(n)
( "
≤ Φ−1
Φ
φ−1 (ε + p(n)) +
+
n−1
X
!
f (s) exp
s=0
n−1
X
δ−1
X
δ=0
t=0
h(δ)
n−1
X
JJ
J
#
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g(s)
s=0
!
k(t) exp
n−1
X
s=0
II
I
)
g(s)
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
for all n ∈ Z[0,β] . Finally, letting ε → 0+ , we conclude that
( "
!
!#
n−1
n−1
X
X
x(n) ≤ Φ−1 Φ
exp
g(s)
φ−1 (p(n)) +
f (s)
s=0
s=0
+
exp
n−1
X
! n−1 δ−1
)
XX
g(s)
h(δ)k(t)
s=0
δ=0 t=0
for all n ∈ Z[0,β] .
Remark 2. Similar to the previous remark, in case Φ(∞) = ∞, (2.9) holds for
all n ∈ Z0 .
Theorem 2.3. Under Assumptions (A1), (A3) and (A4), if x ∈ F(Za , R0 ) is a
function satisfying the nonlinear delay inequality
r
r
x (n) ≤ c +
n−1
X
xr (σ(s)) {f (s) + g(s)xr (σ(s))} ,
n ∈ Z0 ,
s=0
Wing-Sum Cheung and
Shiojenn Tseng
Title Page
Contents
with initial conditions (I1) and
(I3)
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
ψ (σ(s)) ≤ c
for all s ∈ Z0 with σ(s) ≤ 0 ,
where r, c > 0 are constants, then
"
#− r1
n−1
n
n−1
Y
X
Y
(2.16)
x(n) ≤ c−r
(1 − f (s)) −
g(s)
(1 − f (t))
s=0
s=1
t=s
for all n ∈ Z[0,γ] , where γ ≥ 0 is chosen such that the RHS of (2.16) is welldefined.
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
Proof. Define u ∈ F(Z0 , R0 ) by
(2.17)
ur (n) := cr +
n−1
X
xr (σ(s)) {f (s) + g(s)xr (σ(s))} ,
n ∈ Z0 .
s=0
Clearly, u ≥ 0 is non-decreasing and
x(n) ≤ u(n) for all n ∈ Z0 .
(2.18)
Similar to the derivation of (2.6) in the proof of Theorem 2.1, we easily establish
x (σ(n)) ≤ u(n) for all n ∈ Z0 .
By (2.17), for any n ∈ Z0 ,
r
r
r
∆u (n) = u (n + 1) − u (n)
= xr (σ(n)) {f (n) + g(n)xr (σ(n))}
≤ ur (n) {f (n) + g(n)ur (n)}
≤ ur (n + 1) {f (n) + g(n)ur (n)} .
As u(0) = c, by elementary analysis, we infer from (2.17) that
(2.19)
u(n) ≤ y(n) for all n ∈ Z[0,ρ]
where Z[0,ρ] is the maximal lattice on which the unique solution y(n) to the
discrete Bernoulli equation

 ∆y r (n) = y r (n + 1) {f (n) + g(n)y r (n)} , n ∈ Z0
(2.20)

y(0) = c
Some New Discrete Nonlinear
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Shiojenn Tseng
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
is defined. Now the unique solution for (2.20) is (see, e.g., [1])
"
(2.21)
y(n) = c−r
n−1
Y
s=0
(1 − f (s)) −
n
X
s=1
g(s)
n−1
Y
#− r1
(1 − f (t))
t=s
for all n ∈ Z[0,γ] . The assertion now follows from (2.18), (2.19) and (2.21).
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
Wing-Sum Cheung and
Shiojenn Tseng
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
3.
Immediate Consequences
Direct application of the results in Section 2 yields the following consequences
immediately.
Corollary 3.1. Under Assumptions (A1) – (A4), if x ∈ F(Za , R0 ) is a function
satisfying the nonlinear delay inequality
(3.1) xα (n)
≤ p(n) +
n−1
X
x
α−1
(σ(s)) {f (s) + g(s)x (σ(s)) + h(s)w (x (σ(s)))}
s=0
for all n ∈ Z0 with initial conditions (I1) and
(I4)
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
Wing-Sum Cheung and
Shiojenn Tseng
1
ψ (σ(s)) ≤ p α (s) for all s ∈ Z0 with σ(s) ≤ 0 ,
Title Page
where α ≥ 1 is a constant, then
( "
(3.2) x(n) ≤ Φ−1
Φ
!
!#
n−1
n−1
1
1X
1X
g(α)
p α (n) +
f (s)
exp
α s=0
α s=0
! n−1
)
n−1
1X
1X
g(α)
h(t)
+ exp
α s=0
α t=0
for all n ∈ Z[0,µ] , where µ ≥ 0 is chosen such that the RHS of (3.2) is welldefined for all n ∈ Z[0,µ] , and Φ is defined as in Theorem 2.1.
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
Proof. Let φ : R0 → R0 be defined by φ(r) = rα , r ∈ R0 . Then φ satisfies
Assumption (A5). By (3.1) we have
n−1
X
f (s) g(s)
h(s)
0
φ (x(n)) ≤ p(n)+
φ (x (σ(s)))
+
x (σ(s)) +
w (x (σ(s))) .
α
α
α
s=0
Furthermore, it is easy to see that
1
φ (x(s)) ≤ p α (s) = φ−1 (p(s))
for all s ∈ Z0 with σ(s) ≤ 0 .
Thus Theorem 2.1 applies and the assertion follows.
Remark 3.
(i) In Corollary 3.1, if we set α = 2, p(n) ≡ c2 , g(n) ≡ 0, we have
2
2
x (n) ≤ c +
n−1
X
x (σ(s)) {f (s) + h(s)w (x (σ(s)))} ,
n ∈ Z0
s=0
( "
x(n) ≤ Φ−1
Wing-Sum Cheung and
Shiojenn Tseng
Title Page
Contents
JJ
J
implies
#
)
n−1
n−1
1X
1X
Φ c+
f (s) +
h(s) ,
2 s=0
2 s=0
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
n ∈ Z[0,µ] .
II
I
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Close
This is the discrete analogue of a result of Pachpatte in [14]. Furthermore,
if σ = id, this reduces to a result of Pachpatte in [18].
(ii) In case Φ(∞) = ∞, (3.2) holds for all n ∈ Z0 .
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
Corollary 3.2. Under Assumptions (A1) – (A4) with p ∈ F(Z0 , R+ ), if x ∈
F(Za , R1 ) satisfies the nonlinear delay inequality
(3.3) xα (n)
≤ p(n) +
n−1
X
xα (σ(s)) {f (s) + g(s) ln x (σ(s)) + h(s)w (ln x (σ(s)))}
s=0
for all n ∈ Z0 with initial conditions (I1) and
(I5)
ψ (σ(s)) ≤
1
ln (p(s))
α
for all s ∈ Z0 with σ(s) ≤ 0 ,
Wing-Sum Cheung and
Shiojenn Tseng
where α > 0 is a constant, then
(
"
(3.4) x(n) ≤ exp Φ−1 Φ
×
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
!
n−1
1X
exp
g(s)
α s=0
!!
n−1
1
1X
ln p(n) +
f (s)
α
α s=0
! n−1
#)
n−1
1X
1X
g(s)
h(t)
+ exp
α s=0
α t=0
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for all n ∈ Z[0,ν] , where ν ≥ 0 is chosen such that the RHS of (3.4) is welldefined for all n ∈ Z[0,ν] , and Φ is defined as in Theorem 2.1.
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
Proof. Letting y(n) = ln x(n), (3.3) becomes
(3.5)
exp (αy(n))
≤ p(n) +
n−1
X
exp (αy (σ(s))) {f (s) + g(s)y (σ(s)) + h(s)w (y (σ(s)))} .
s=0
Let φ : R0 → R0 be defined by φ(r) = exp(αr), r ∈ R0 . Then φ satisfies
Assumption (A5). Hence from (3.5), we have
n−1
X
f (s) g(s)
h(s)
0
φ (y(n)) ≤ p(n)+
φ (y (σ(s)))
+
y (σ(s)) +
w (y (σ(s))) .
α
α
α
s=0
Furthermore, it is easy to see that
ψ (σ(s)) ≤
1
ln (p(s)) = φ−1 (p(s))
α
for all s ∈ Z0 with σ(s) ≤ 0 .
y(n) ≤ Φ
−1
Φ
n−1
!
1X
exp
g(s)
α s=0
Wing-Sum Cheung and
Shiojenn Tseng
Title Page
Contents
Thus Theorem 2.1 applies and we have
( "
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
n−1
!#
1
1X
ln p(n) +
f (s)
α
α s=0
! n−1
)
n−1
1X
1X
+ exp
g(s)
h(t)
α s=0
α t=0
for all n ∈ Z[0,ν] , and from this the assertion follows.
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
Remark 4. In case Φ(∞) = ∞, (3.4) holds for all n ∈ Z0 .
Corollary 3.3. Under Assumptions (A1) – (A4), if x ∈ F(Za , R0 ) satisfies the
nonlinear delay inequality
(3.6) xα (n) ≤ p(n) +
n−1
X
(
xα−1 (σ(s)) f (s) + g(s)x (σ(s))
s=0
+ h(s)
s−1
X
)
k(t)w (x (σ(t)))
t=0
for all n ∈ Z0 with initial conditions (I1) and (I4), where α ≥ 1 is a constant,
then
!
!#
"
n−1
n−1
X
X
1
1
1
g(s)
p α (n) +
f (s)
(3.7) x(n) ≤ Φ−1 Φ
exp
α s=0
α s=0
!
!)
n−1
n−1
s−1
X
1X
1X
+ exp
g(s)
h(s)
k(t)
α s=0
α s=0
t=0
for all n ∈ Z[0,η] , where η ≥ 0 is chosen such that the RHS of (3.7) is welldefined for all n ∈ Z[0,η] , and Φ is defined as in Theorem 2.1.
Proof. Let φ : R0 → R0 be defined by φ(r) = rα , r ∈ R0 . Then φ satisfies
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
Wing-Sum Cheung and
Shiojenn Tseng
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
Assumption (A5). By (3.6),
φ (x(n)) ≤ p(n) +
n−1
X
(
φ0 (x (σ(s)))
s=0
f (s) g(s)
+
x (σ(s))
α
α
+
s−1
h(s) X
α
)
k(t)w (x (σ(t)))
t=0
for all n ∈ Z0 . Furthermore, it is easy to see that
1
ψ (σ(s)) ≤ p α (s) = φ−1 (p(s))
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
for all s ∈ Z0 with σ(s) ≤ 0 .
Thus Theorem 2.2 applies and we have
!
!#
( "
n−1
n−1
X
X
1
1
1
x(n) ≤ Φ−1 Φ
exp
g(s)
p α (n) +
f (s)
α s=0
α s=0
!
)
n−1
n−1 s−1
1X
1 XX
+ exp
g(s) ·
h(s)k(t)
α s=0
α s=0 t=0
for all n ∈ Z[0,η] .
Wing-Sum Cheung and
Shiojenn Tseng
Title Page
Contents
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Remark 5.
(i) In Corollary 3.3, if we put α = 2, p(n) ≡ c2 , g(n) ≡ 0, we have
(
)
n−1
s−1
X
X
2
2
x (n) ≤ c +
x (σ(s)) f (s) + h(s)
k(t)w (x (σ(t))) ,
s=0
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n ∈ Z0
Page 28 of 36
t=0
J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
implies
( "
x(n) ≤ Φ−1
#
)
n−1
n−1
s−1
X
1X
1X
Φ c+
f (s) +
h(s)
k(t) ,
2 s=0
2 s=0
t=0
n ∈ Z[0,η] .
This is the discrete analogue of a result of Pachpatte in [14]. Furthermore,
if σ = id and w = id, this reduces to a result of Pachpatte in [18].
(ii) In case Φ(∞) = ∞, (3.7) holds for all n ∈ Z0 .
Corollary 3.4. Under Assumptions (A1) – (A4) with p ∈ F(Z0 , R+ ), if x ∈
F(Za , R1 ) satisfies the nonlinear delay inequality
α
(3.8) x (n) ≤ p(n) +
n−1
X
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
Wing-Sum Cheung and
Shiojenn Tseng
(
α
x (σ(s)) f (s) + g(s) ln x (σ(s))
s=0
+ h(s)
s−1
X
)
k(t)w (ln x (σ(t)))
t=0
for all n ∈ Z0 with initial conditions (I1) and
(I6)
ψ (σ(s)) ≤
1
ln (p(s))
α
for all s ∈ Z0 with σ(s) ≤ 0 ,
where α > 0 is any constant, then
(
"
(3.9) x(n) ≤ exp Φ−1 Φ
!
n−1
1X
exp
g(s)
α s=0
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
×
!!
n−1
1
1X
ln p(n) +
f (s)
α
α s=0
!
#)
n−1
n−1
s−1
X
1X
1X
+ exp
g(s) ·
h(s)
k(t)
α s=0
α s=0
t=0
for all n ∈ Z[0,λ] , where λ ≥ 0 is chosen such that the RHS of (3.9) is welldefined for all n ∈ Z[0,λ] , and Φ is defined as in Theorem 2.1.
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
Proof. Letting y(n) = ln x(n), (3.8) becomes
(3.10)
exp (αy(n)) ≤ p(n) +
n−1
X
(
exp (αy (σ(s))) f (s) + g(s)y (σ(s))
Wing-Sum Cheung and
Shiojenn Tseng
s=0
+ h(s)
s−1
X
)
k(t)w (y (σ(t)))
Title Page
t=0
Contents
for all n ∈ Z0 . Let φ : R0 → R0 be defined by φ(r) = exp(αr), r ∈ R0 . Then
φ satisfies Assumption (A5). Hence from (3.10), we have
φ (y(n)) ≤ p(n) +
n−1
X
×
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φ0 (y (σ(s)))
s=0
(
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h(s)
f (s) g(s)
+
y (σ(s)) +
α
α
α
s−1
X
t=0
)
k(t)w (y (σ(t)))
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Page 30 of 36
J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
for all n ∈ Z0 . Furthermore, it is easy to check that
1
ψ (σ(s)) ≤ ln (p(s)) = φ−1 (p(s)) for all s ∈ Z0 with σ(s) ≤ 0 .
α
Thus Theorem 2.2 applies and we have
( "
!
!#
n−1
n−1
X
X
1
1
1
y(n) ≤ Φ−1 Φ
exp
g(s)
ln p(n) +
f (s)
α s=0
α
α s=0
!
)
n−1
n−1 s−1
1X
1 XX
+ exp
g(s) ·
h(s)k(t)
α s=0
α s=0 t=0
for all n ∈ Z[0,λ] , and from this the assertion follows.
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
Wing-Sum Cheung and
Shiojenn Tseng
Remark 6.
(i) In Corollary 3.4, if we set α = 2, p(n) ≡ c2 , g(n) ≡ 0, then
(
)
n−1
s−1
X
X
x2 (n) ≤ c2 +
x2 (σ(s)) f (s) + h(s)
k(t)w (ln x (σ(t))) ,
s=0
Title Page
n ∈ Z0
implies
(
x(n) ≤ exp Φ
"
−1
Contents
t=0
Φ
!
#)
n−1
n−1
s−1
X
1
1X
1X
ln p(n) +
f (s) +
h(s)
k(t)
2
2 s=0
2 s=0
t=0
n ∈ Z[0,λ] .
This is the discrete version of a result of Pachpatte in [14].
(ii) In case Φ(∞) = ∞, (3.9) holds for all n ∈ Z0 .
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4.
Application
Consider the discrete delay equation
(4.1)
xα (n) = F
n, x (σ(n)) ,
n−1
X
!
G (n, s, x (σ(s))) ,
n ∈ Z0
s=0
with initial conditions (I1) and (I4), where α ≥ 1 is a constant, σ, ψ satisfy
Assumptions (A3), (A4), x ∈ F(Za , R), F ∈ C(Z0 × R2 , R), and G ∈ C(Z20 ×
R, R). If F, G satisfy
|F (n, u, v)| ≤ p(n) + K|v| , n ∈ Z0 , u, v ∈ R ,
|G(n, s, v)| ≤ [f (s) + g(s)|v| + h(s)w (|v|)] |v|α−1 ,
n, s ∈ Z0 , v ∈ R ,
for some p, f, g, h, w satisfying (A1) and (A2), and some constant K > 0, then
every solution of (4.1) satisfies
!
n−1
X
|x(n)|α = F n, x (σ(n)) ,
G (n, s, x (σ(s))) s=0
n−1
X
≤ p(n) + K G (n, s, x (σ(s)))
≤ p(n) + K
≤ p(n) + K
s=0
n−1
X
|G (n, s, x (σ(s)))|
s=0
n−1
X
Some New Discrete Nonlinear
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Application to Discrete Delay
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Shiojenn Tseng
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[f (s) + g(s) |x (σ(s))| + h(s)w (|x (σ(s))|)] |x (σ(s))|α−1
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J. Ineq. Pure and Appl. Math. 7(4) Art. 122, 2006
for all n ∈ J(x) := the maximal existence lattice on which x is defined. Applying Corollary 3.1, this yields
( "
|x(n)| ≤ Φ
−1
Φ
!
!#
n−1
n−1
1
KX
KX
exp
g(α)
p α (n) +
f (s)
α s=0
α s=0
! n−1
)
n−1
KX
KX
+ exp
g(α)
h(t)
α s=0
α t=0
for all n ∈ J(x) ∩ Z[0,µ] . This gives the boundedness of solutions of (4.1).
Some New Discrete Nonlinear
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Wing-Sum Cheung and
Shiojenn Tseng
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References
[1] R.P. AGARWAL, Difference Equations and Inequalities, Marcel Dekker,
New York, 2000.
[2] D. BAINOV AND P. SIMEONOV, Integral Inequalities and Applications,
Kluwer Academic Publishers, Dordrecht, 1992.
[3] E.F. BECKENBACH
New York, 1961.
AND
R. BELLMAN, Inequalities, Springer-Verlag,
[4] R. BELLMAN, The stability of solutions of linear differential equations,
Duke Math. J., 10 (1943), 643–647.
[5] I. BIHARI, A generalization of a lemma of Bellman and its application
to uniqueness problems of differential equations, Acta Math. Acad. Sci.
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[6] W.S. CHEUNG, On some new integrodifferential inequalities of the Gronwall and Wendroff type, J. Math. Anal. Appl., 178 (1993), 438–449.
[7] W.S. CHEUNG AND Q.H. MA, Nonlinear retarded integral inequalities
for functions in two variables, to appear in J. Concrete Appl. Math.
[8] T.H. GRONWALL, Note on the derivatives with respect to a parameter of
the solutions of a system of differential equations, Ann. Math., 20 (1919),
292–296.
[9] H. HARAUX, Nonlinear Evolution Equation: Global Behavior of Solutions, Lecture Notes in Mathematics, v.841, Springer-Verlag, Berlin,
1981.
Some New Discrete Nonlinear
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Shiojenn Tseng
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[10] Q.M. MA AND E.H. YANG, On some new nonlinear delay integral inequalities, J. Math. Anal. Appl., 252 (2000), 864–878.
[11] D.S. MITRINOVIĆ, Analytic Inequalities, Springer-Verlag, New York,
1970.
[12] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.
[13] L. OU-IANG, The boundedness of solutions of linear differential equations y 00 + A(t)y = 0, Shuxue Jinzhan, 3 (1957), 409–415.
Some New Discrete Nonlinear
Delay Inequalities and
Application to Discrete Delay
Equations
[14] B.G. PACHPATTE, A note on certain integral inequalities with delay, Period. Math. Hungar., 31 (1995), 229–234.
Wing-Sum Cheung and
Shiojenn Tseng
[15] B.G. PACHPATTE, Explicit bounds on certain integral inequalities, J.
Math. Anal. Appl., 267 (2002), 48–61.
Title Page
[16] B.G. PACHPATTE, Inequalities for Finite Difference Equations, Marcel
Dekker, New York, 2002.
[17] B.G. PACHPATTE, On some new inequalities related to a certain inequality arising in the theory of differential equations, J. Math. Anal. Appl., 251
(2000), 736–751.
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[18] B.G. PACHPATTE, On some new inequalities related to certain inequalities in the theory of differential equations, J. Math. Anal. Appl., 189
(1995), 128–144.
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[19] P. Ch. TSAMATOS AND S.K. NTOUYAS, On a Bellman-Bihari type inequality with delay, Period. Math. Hungar., 23 (1991), 91–94.
[20] E.H. YANG, Generalizations of Pachpatte’s integral and discrete inequalities, Ann. Differential Equations, 13 (1997), 180–188.
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