Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 4, Article 122, 2006
SOME NEW DISCRETE NONLINEAR DELAY INEQUALITIES AND
APPLICATION TO DISCRETE DELAY EQUATIONS
WING-SUM CHEUNG∗ AND SHIOJENN TSENG
D EPARTMENT OF M ATHEMATICS
U NIVERSITY OF H ONG KONG
H ONG KONG
wscheung@hku.hk
D EPARTMENT OF M ATHEMATICS
TAMKANG U NIVERSITY
TAMSUI , TAIWAN 25137
tseng@math.tku.edu.tw
Received 07 September, 2005; accepted 27 January, 2006
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Gronwall-Bellman-Ou-Iang-type Inequalities, Discrete inequalities, Difference equations.
2000 Mathematics Subject Classification. 26D10, 26D15, 39A10, 39A70.
1. I NTRODUCTION
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.
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
∗ Author of correspondence. Research is supported in part by the Research Grants Council of the Hong Kong SAR (Project No.
HKU7017/05P).
267-05
2
W ING -S UM C HEUNG AND S HIOJENN T SENG
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], 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 Gronwall-Bellman-Ou-Iangtype 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.
2. D ISCRETE I NEQUALITIES WITH D ELAY
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.
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
J. Inequal. Pure and Appl. Math., 7(4) Art. 122, 2006
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D ISCRETE N ONLINEAR D ELAY I NEQUALITY
3
for all n ∈ Z0 with initial conditions (I1) – (I2), then
( "
!
!#
n−1
n−1
X
X
(2.2)
x(n) ≤ Φ−1 Φ
exp
g(s)
φ−1 (p(n)) +
f (s)
s=0
s=0
+ exp
n−1
X
! n−1
)
X
g(s)
h(t)
s=0
t=0
for all n ∈ Z[0,α] , where Φ ∈ C(R0 , R) is defined by
Z r
ds
, r > 0,
Φ(r) :=
1 w(s)
and α ≥ 0 is chosen such that the RHS of (2.2) is well-defined, that is,
"
!
!#
! n−1
n−1
n−1
n−1
X
X
X
X
Φ
exp
g(s)
φ−1 (p(n)) +
f (s)
+ exp
g(s)
h(t) ∈ Im Φ
s=0
s=0
s=0
t=0
for all n ∈ Z[0,α] .
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
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
(2.6)
x (σ(n)) ≤ u(n) for all n ∈ Z[0,N ] .
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)
J. Inequal. Pure and Appl. Math., 7(4) Art. 122, 2006
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4
W ING -S UM C HEUNG AND S HIOJENN T SENG
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) ≤
Summing up, we have
u(n) − u(0) =
n−1
X
∆u(s)
s=0
≤
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
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
u(n) ≤ φ−1 (ε + p(N )) +
f (s) +
h(s)w (u(s)) exp
g(s)
"
(2.7)
≤ φ−1 (ε + p(N )) +
s=0
s=0
N
−1
X
n−1
X
s=0
f (s) +
s=0
#
h(s)w (u(s)) exp
N
−1
X
g(s)
s=0
s=0
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 ] ,
u(n) ≤ v(n) .
(2.8)
Therefore, for any t ∈ Z[0,N −1] ,
∆v(t) = v(t + 1) − v(t)
= h(t)w (u(t)) exp
N
−1
X
g(s)
s=0
≤ h(t)w (v(t)) exp
N
−1
X
g(s) .
s=0
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)
J. Inequal. Pure and Appl. Math., 7(4) Art. 122, 2006
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D ISCRETE N ONLINEAR D ELAY I NEQUALITY
5
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
∆(Φ ◦ v)(t) ≤
X
1
h(t)w (v(t)) exp
g(s)
w(η)
s=0
≤ h(t) exp
N
−1
X
g(s)
s=0
for all t ∈ Z[0,N −1] . Summing up, we have
n−1
X
∆(Φ ◦ v)(t) ≤
t=0
n−1
X
h(t) exp
N
−1
X
t=0
g(s) .
s=0
On the other hand,
n−1
X
∆(Φ ◦ v)(t) = Φ (v(n)) − Φ (v(0))
t=0
"
= Φ (v(n)) − Φ
exp
N
−1
X
!
g(s)
φ−1 (ε + p(N )) +
s=0
N
−1
X
!#
f (s)
,
s=0
therefore,
"
Φ (v(n)) ≤ Φ
exp
N
−1
X
!
φ−1 (ε + p(N )) +
g(s)
N
−1
X
!#
f (s)
s=0
s=0
+
n−1
X
h(t) exp
N
−1
X
g(s)
s=0
t=0
for all n ∈ Z[0,N ] . In particular, taking n = N we have
"
Φ (v(N )) ≤ Φ
exp
N
−1
X
!
g(s)
φ−1 (ε + p(N )) +
s=0
N
−1
X
!#
f (s)
s=0
+
exp
N
−1
X
! 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)
s=0
φ−1 (ε + p(n)) +
n−1
X
!#
f (s)
s=0
+
exp
n−1
X
s=0
J. Inequal. Pure and Appl. Math., 7(4) Art. 122, 2006
! n−1
X
g(s)
h(t)
t=0
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6
W ING -S UM C HEUNG AND S HIOJENN T SENG
for all n ∈ Z[0,α] . Hence
( "
v(n) ≤ Φ−1
Φ
exp
n−1
X
!
g(s)
φ−1 (ε + p(n)) +
s=0
n−1
X
!#
f (s)
s=0
+ exp
n−1
X
! n−1
)
X
g(s)
h(t)
s=0
t=0
and so by (2.5) and (2.8),
( "
x(n) < u(n) ≤ v(n) ≤ Φ−1
Φ
exp
n−1
X
!
φ−1 (ε + p(n)) +
g(s)
s=0
n−1
X
!#
f (s)
s=0
+ exp
n−1
X
! n−1
)
X
g(s)
h(t)
s=0
t=0
for all n ∈ Z[0,α] . Finally, letting ε → 0+ , we conclude that
!#
!
( "
n−1
n−1
X
X
f (s)
g(s)
φ−1 (p(n)) +
x(n) ≤ Φ−1 Φ
exp
s=0
s=0
+ exp
n−1
X
! n−1
)
X
g(s)
h(t)
s=0
t=0
for all n ∈ Z[0,α] .
R∞
ds
Remark 2.2. In many cases the non-decreasing function w satisfies 1 w(s)
= ∞. For exam√
ple, 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 2.3. Under Assumptions (A1) – (A5), if x ∈ F(Za , R0 ) is a function satisfying the
nonlinear delay inequality
(
)
n−1
s−1
X
X
φ (x(n)) ≤ p(n) +
φ0 (x (σ(s))) f (s) + g(s)x (σ(s)) + h(s)
k(t)w (x (σ(t)))
s=0
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
h(s)k(t)
g(s)
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
Φ
exp
g(s)
φ−1 (p(n)) +
f (s)
s=0
s=0
+
exp
n−1
X
s=0
J. Inequal. Pure and Appl. Math., 7(4) Art. 122, 2006
! n−1 s−1
XX
g(s)
h(s)k(t) ∈ Im Φ
s=0 t=0
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D ISCRETE N ONLINEAR D ELAY I NEQUALITY
7
for all n ∈ Z[0,β] .
Proof. Fix ε > 0 and M ∈ Z[0,β] . Define u : Z[0,M ] → R0 by
(
"
n−1
X
(2.10) u(n) := φ−1 ε + p(M ) +
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
By (A5), u is non-decreasing on Z[0,M ] . For any n ∈ Z[0,M ] , by (A5) again,
u(n) ≥ φ−1 (ε + p(M )) > 0 .
(2.11)
As φ (u(n)) > φ (x(n)), we have
u(n) > x(n) .
(2.12)
Using the same arguments as in the derivation of (2.6) in the proof of Theorem 2.1, we have
x (σ(n)) ≤ u(n) for all n ∈ Z[0,M ] .
(2.13)
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
(
≤ φ0 (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))
∆u(s) ≤
f (s) + g(s)u(s) + h(s)
k(t)w (u(t))
φ0 (ξ)
t=0
≤ f (s) + g(s)u(s) + h(s)
s−1
X
k(t)w (u(t)) .
t=0
Summing up, we have
u(n) − u(0) =
n−1
X
∆u(s)
s=0
≤
n−1
X
f (s) +
s=0
n−1
X
h(s)
s=0
s−1
X
k(t)w (u(t)) +
t=0
n−1
X
g(s)u(s) ,
s=0
or
"
−1
u(n) ≤ φ
(ε + p(M )) +
n−1
X
s=0
J. Inequal. Pure and Appl. Math., 7(4) Art. 122, 2006
f (s) +
n−1
X
s=0
h(s)
s−1
X
t=0
#
k(t)w (u(t)) +
n−1
X
g(s)u(s)
s=0
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W ING -S UM C HEUNG AND S HIOJENN T SENG
for all n ∈ Z[0,M ] . Hence by the discrete version of the Gronwall-Bellman inequality (see, e.g.,
[16, Corollary 1.2.5]),
"
−1
u(n) ≤ φ
(ε + p(M )) +
"
(2.14)
−1
≤ φ
(ε + p(M )) +
n−1
X
n−1
X
f (s) +
h(s)
s−1
X
#
k(t)w (u(t)) exp
s=0
s=0
t=0
M
−1
X
n−1
X
s−1
X
f (s) +
s=0
h(s)
s=0
n−1
X
g(s)
s=0
#
k(t)w (u(t)) exp
t=0
M
−1
X
g(s)
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 ] ,
u(n) ≤ v(n) .
(2.15)
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
M
−1
X
g(s)
s=0
!
k(t)w (v(t)) exp
M
−1
X
g(s)
s=0
t=0
≤ h(δ)w (v(δ))
δ−1
X
!
k(t) exp
t=0
M
−1
X
g(s) .
s=0
On the other hand, by the Mean Value Theorem,
∆(Φ ◦ v)(δ) = Φ (v(δ + 1)) − Φ (v(δ))
1
= Φ0 (η)∆v(δ) =
∆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
s=0
for all δ ∈ Z[0,M −1] . Summing up, we have
n−1
X
∆(Φ ◦ v)(δ) ≤
δ=0
J. Inequal. Pure and Appl. Math., 7(4) Art. 122, 2006
n−1
X
δ=0
h(δ)
δ−1
X
t=0
!
k(t) exp
M
−1
X
g(s) ,
s=0
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D ISCRETE N ONLINEAR D ELAY I NEQUALITY
9
or
Φ (v(n)) ≤ Φ (v(0)) +
n−1
X
δ−1
X
h(δ)
M
−1
X
k(t) exp
t=0
δ=0
"
=Φ
!
φ−1 (ε + p(M )) +
s=0
M
−1
X
M
−1
X
!
f (s) exp
s=0
+
n−1
X
δ−1
X
h(δ)
#
g(s)
s=0
!
k(t) exp
M
−1
X
t=0
δ=0
g(s)
g(s)
s=0
for all n ∈ Z[0,M ] . In particular, taking n = M this yields
"
!
#
M
−1
M
−1
X
X
Φ (v(M )) ≤ Φ
φ−1 (ε + p(M )) +
f (s) exp
g(s)
s=0
s=0
M
−1
X
+
h(δ)
φ−1 (ε + p(n)) +
Φ (v(n)) ≤ Φ
n−1
X
!
f (s) exp
n−1
X
+
v(n) ≤ Φ−1
Φ
φ−1 (ε + p(n)) +
s=0
#
g(s)
n−1
X
!
δ−1
X
h(δ)
k(t) exp
t=0
δ=0
n−1
X
g(s) .
s=0
s=0
for all n ∈ Z[0,β] . Hence
( "
k(t) exp
M
−1
X
t=0
δ=0
Since M ∈ Z[0,β] is arbitrary,
"
!
δ−1
X
!
f (s) exp
n−1
X
n−1
X
g(s)
s=0
#
g(s)
s=0
s=0
+
n−1
X
δ−1
X
h(δ)
!
k(t) exp
t=0
δ=0
n−1
X
)
g(s)
s=0
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
n−1
X
#
g(s)
s=0
h(δ)
δ=0
δ−1
X
!
k(t) exp
t=0
n−1
X
)
g(s)
s=0
for all n ∈ Z[0,β] . Finally, letting ε → 0+ , we conclude that
( "
!
!#
n−1
n−1
X
X
exp
g(s)
φ−1 (p(n)) +
f (s)
x(n) ≤ Φ−1 Φ
s=0
s=0
+
exp
n−1
X
s=0
for all n ∈ Z[0,β] .
J. Inequal. Pure and Appl. Math., 7(4) Art. 122, 2006
! n−1 δ−1
)
XX
g(s)
h(δ)k(t)
δ=0 t=0
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10
W ING -S UM C HEUNG AND S HIOJENN T SENG
Remark 2.4. Similar to the previous remark, in case Φ(∞) = ∞, (2.9) holds for all n ∈ Z0 .
Theorem 2.5. 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
with initial conditions (I1) and
ψ (σ(s)) ≤ c
(I3)
for all s ∈ Z0 with σ(s) ≤ 0 ,
where r, c > 0 are constants, then
#− r1
"
n−1
n
n−1
Y
X
Y
(1 − f (s)) −
g(s)
(1 − f (t))
(2.16)
x(n) ≤ c−r
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 well-defined.
Proof. Define u ∈ F(Z0 , R0 ) by
(2.17)
r
r
u (n) := c +
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 ,
∆ur (n) = ur (n + 1) − ur (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
is defined. Now the unique solution for (2.20) is (see, e.g., [1])
"
#− r1
n−1
n
n−1
Y
X
Y
(2.21)
y(n) = c−r
(1 − f (s)) −
g(s)
(1 − f (t))
s=0
s=1
t=s
for all n ∈ Z[0,γ] . The assertion now follows from (2.18), (2.19) and (2.21).
J. Inequal. Pure and Appl. Math., 7(4) Art. 122, 2006
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D ISCRETE N ONLINEAR D ELAY I NEQUALITY
11
3. I MMEDIATE C ONSEQUENCES
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)
α
x (n) ≤ p(n) +
n−1
X
xα−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
1
ψ (σ(s)) ≤ p α (s) for all s ∈ Z0 with σ(s) ≤ 0 ,
(I4)
where α ≥ 1 is a constant, then
( "
(3.2)
x(n) ≤ Φ−1
Φ
!
!#
n−1
n−1
1
1X
1X
exp
g(α)
p α (n) +
f (s)
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 well-defined for all
n ∈ Z[0,µ] , 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 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 3.2.
(i) In Corollary 3.1, if we set α = 2, p(n) ≡ c2 , g(n) ≡ 0, we have
x2 (n) ≤ c2 +
n−1
X
x (σ(s)) {f (s) + h(s)w (x (σ(s)))} ,
n ∈ Z0
s=0
implies
( "
x(n) ≤ Φ−1
#
)
n−1
n−1
1X
1X
Φ c+
f (s) +
h(s) ,
2 s=0
2 s=0
n ∈ Z[0,µ] .
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 .
Corollary 3.3. Under Assumptions (A1) – (A4) with p ∈ F(Z0 , R+ ), if x ∈ F(Za , R1 ) satisfies
the nonlinear delay inequality
(3.3)
α
x (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
J. Inequal. Pure and Appl. Math., 7(4) Art. 122, 2006
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12
W ING -S UM C HEUNG AND S HIOJENN T SENG
for all n ∈ Z0 with initial conditions (I1) and
1
(I5)
ψ (σ(s)) ≤ ln (p(s)) for all s ∈ Z0 with σ(s) ≤ 0 ,
α
where α > 0 is a constant, then
(
"
!
!!
n−1
n−1
X
X
1
1
1
(3.4) x(n) ≤ exp Φ−1 Φ
exp
g(s)
ln p(n) +
f (s)
α s=0
α
α s=0
! n−1
#)
n−1
1X
1X
g(s)
h(t)
+ exp
α s=0
α t=0
for all n ∈ Z[0,ν] , where ν ≥ 0 is chosen such that the RHS of (3.4) is well-defined for all
n ∈ Z[0,ν] , and Φ is defined as in Theorem 2.1.
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
1
ψ (σ(s)) ≤ ln (p(s)) = φ−1 (p(s)) for all s ∈ Z0 with σ(s) ≤ 0 .
α
Thus Theorem 2.1 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
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.
Remark 3.4. In case Φ(∞) = ∞, (3.4) holds for all n ∈ Z0 .
Corollary 3.5. Under Assumptions (A1) – (A4), if x ∈ F(Za , R0 ) satisfies the nonlinear delay
inequality
(
)
n−1
s−1
X
X
(3.6) xα (n) ≤ p(n) +
xα−1 (σ(s)) f (s) + g(s)x (σ(s)) + h(s)
k(t)w (x (σ(t)))
s=0
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
(3.7)
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
X
1X
1X
+ exp
g(s)
h(s)
k(t)
α s=0
α s=0
t=0
J. Inequal. Pure and Appl. Math., 7(4) Art. 122, 2006
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D ISCRETE N ONLINEAR D ELAY I NEQUALITY
13
for all n ∈ Z[0,η] , where η ≥ 0 is chosen such that the RHS of (3.7) is well-defined 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 Assumption (A5).
By (3.6),
(
)
s−1
n−1
X
X
g(s)
h(s)
f
(s)
φ (x(n)) ≤ p(n) +
φ0 (x (σ(s)))
+
x (σ(s)) +
k(t)w (x (σ(t)))
α
α
α t=0
s=0
for all n ∈ Z0 . Furthermore, it is easy to see that
1
ψ (σ(s)) ≤ p α (s) = φ−1 (p(s))
for all s ∈ Z0 with σ(s) ≤ 0 .
Thus Theorem 2.3 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] .
Remark 3.6.
(i) In Corollary 3.5, 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
n ∈ Z0
t=0
implies
( "
x(n) ≤ Φ−1
Φ c+
n−1
1X
2
s=0
#
f (s) +
n−1
1X
2
s=0
h(s)
s−1
X
)
n ∈ Z[0,η] .
k(t) ,
t=0
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 3.7. Under Assumptions (A1) – (A4) with p ∈ F(Z0 , R+ ), if x ∈ F(Za , R1 ) satisfies
the nonlinear delay inequality
(
)
n−1
s−1
X
X
(3.8) xα (n) ≤ p(n)+
xα (σ(s)) f (s) + g(s) ln x (σ(s)) + h(s)
k(t)w (ln x (σ(t)))
s=0
t=0
for all n ∈ Z0 with initial conditions (I1) and
1
(I6)
ψ (σ(s)) ≤ ln (p(s)) for all s ∈ Z0 with σ(s) ≤ 0 ,
α
where α > 0 is any constant, then
(
"
!
!!
n−1
n−1
X
X
1
1
1
(3.9) x(n) ≤ exp Φ−1 Φ
exp
g(s)
ln p(n) +
f (s)
α s=0
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
J. Inequal. Pure and Appl. Math., 7(4) Art. 122, 2006
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14
W ING -S UM C HEUNG AND S HIOJENN T SENG
for all n ∈ Z[0,λ] , where λ ≥ 0 is chosen such that the RHS of (3.9) is well-defined for all
n ∈ Z[0,λ] , and Φ is defined as in Theorem 2.1.
Proof. Letting y(n) = ln x(n), (3.8) becomes
exp (αy(n)) ≤ p(n) +
(3.10)
n−1
X
(
exp (αy (σ(s))) f (s) + g(s)y (σ(s))
s=0
+ h(s)
s−1
X
)
k(t)w (y (σ(t)))
t=0
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
(
)
n−1
s−1
X
X
f (s) g(s)
h(s)
+
y (σ(s)) +
k(t)w (y (σ(t)))
φ (y(n)) ≤ p(n) +
φ0 (y (σ(s)))
α
α
α t=0
s=0
for all n ∈ Z0 . Furthermore, it is easy to check that
1
ln (p(s)) = φ−1 (p(s)) for all s ∈ Z0 with σ(s) ≤ 0 .
α
Thus Theorem 2.3 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
ψ (σ(s)) ≤
for all n ∈ Z[0,λ] , and from this the assertion follows.
Remark 3.8.
(i) In Corollary 3.7, if we set α = 2, p(n) ≡ c2 , g(n) ≡ 0, then
(
)
n−1
s−1
X
X
2
2
2
x (n) ≤ c +
x (σ(s)) f (s) + h(s)
k(t)w (ln x (σ(t))) ,
s=0
implies
(
"
x(n) ≤ exp Φ−1 Φ
n ∈ Z0
t=0
!
#)
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 .
4. A PPLICATION
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
J. Inequal. Pure and Appl. Math., 7(4) Art. 122, 2006
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D ISCRETE N ONLINEAR D ELAY I NEQUALITY
15
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
α
|x(n)| = 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)))
s=0
≤ p(n) + K
n−1
X
|G (n, s, x (σ(s)))|
s=0
≤ p(n) + K
n−1
X
[f (s) + g(s) |x (σ(s))| + h(s)w (|x (σ(s))|)] |x (σ(s))|α−1
s=0
for all n ∈ J(x) := the maximal existence lattice on which x is defined. Applying Corollary
3.1, this yields
!
!#
( "
n−1
n−1
X
X
1
K
K
g(α)
p α (n) +
f (s)
|x(n)| ≤ Φ−1 Φ
exp
α s=0
α s=0
! n−1
)
n−1
KX
KX
g(α)
h(t)
+ exp
α s=0
α t=0
for all n ∈ J(x) ∩ Z[0,µ] . This gives the boundedness of solutions of (4.1).
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