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volume 7, issue 4, article 122, 2006.

Received 07 September, 2005;

accepted 27 January, 2006.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

SOME NEW DISCRETE NONLINEAR DELAY INEQUALITIES AND APPLICATION TO DISCRETE DELAY EQUATIONS

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

c

2000Victoria University ISSN (electronic): 1443-5756 267-05

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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

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, Dif- ference equations.

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 particu- larly 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]). Ifuandf are non-negative functions on[0,∞) satisfying

u²(x)≤c²+ 2 Z x

0

f(s)u(s)ds, x∈[0,∞), for some constantc≥0, then

u(x)≤c+ Z x

0

f(s)ds, x∈[0,∞).

While Ou-Iang’s inequality is interesting in its own right, it also has nu- merous 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],

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Ma-Yang [10], Pachpatte [14] – [18], Tsamatos-Ntouyas [19], and Yang [20].

Among such extensions, the discretization is of particular interest because anal- ogous 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-Iang-type inequalities giving explicit bounds to unknown discrete functions. These on the one hand generalize some existing results in the litera- ture 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.

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2. Discrete Inequalities with Delay

Throughout this paper,R+= (0,∞)⊂R,Z+=R+∩Z, and for anya, b∈R, Râ = [a,∞),Zâ=Râ∩Z,Z[a,b]=Z∩[a, b]. IfXandY are sets, the collection of functions of X intoY, the collection of continuous functions ofX into Y, and that of continuously differentiable functions of X into Y are denoted by F(X, Y),C(X, Y), andC¹(X, Y), respectively. As usual, ifuis a real-valued function onZ[a,b], the difference operator∆onuis 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)withpnon-decreasing;

(A2)w∈C(R0,R0)is non-decreasing withw(r)>0forr >0;

(A3)σ ∈ F(Z0,Z)withσ(s)≤sfor alls∈Z0and−∞< a:= inf{σ(s) : s∈Z⁰} ≤0;

(A4)ψ ∈ F(Z[a,0],R0); and

(A5)φ ∈C¹(R0,R0)withφ⁰ non-decreasing andφ⁰(r)>0forr >0.

Initial Conditions

(I1)x(s) = ψ(s)for alls∈Z[a,0];

(I2)ψ(σ(s))≤φ⁻¹(p(s))for alls ∈Z⁰withσ(s)≤0.

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Theorem 2.1. Under Assumptions (A1) – (A5), ifx ∈ F(Za,R0)is a function satisfying the nonlinear delay inequality

(2.1) φ(x(n))

≤p(n) +

n−1

X

s=0

φ⁰(x(σ(s))){f(s) +g(s)x(σ(s)) +h(s)w(x(σ(s)))}

for alln ∈Z0with initial conditions (I1) – (I2), then

(2.2) x(n)≤Φ⁻¹ (

Φ

"

exp

n−1

X

s=0

g(s)

!

φ⁻¹(p(n)) +

n−1

X

s=0

f(s)

!#

+ exp

n−1

X

s=0

g(s)

!_n−1 X

t=0

h(t) )

for alln ∈Z[0,α], whereΦ∈C(R0,R)is defined by Φ(r) :=

Z r 1

ds

w(s), r >0,

andα≥0is chosen such that the RHS of (2.2) is well-defined, that is, Φ

"

exp

n−1

X

s=0

g(s)

!

φ⁻¹(p(n)) +

n−1

X

s=0

f(s)

!#

+ exp

n−1

X

s=0

g(s)

!_n−1 X

t=0

h(t)∈ I_mΦ for alln ∈Z^[0,α].

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Proof. Fixε >0andN ∈Z[0,α]. Defineu:Z[0,N]→R0by

(2.3) u(n) :=φ⁻¹ (

ε+p(N)

+

n−1

X

t=0

φ⁰(x(σ(t))) [f(t) +g(t)x(σ(t)) +h(t)w(x(σ(t)))]

) .

By (A5),uis non-decreasing onZ[0,N]. For anyn ∈Z[0,N], by (A5) again, (2.4) u(n)≥φ⁻¹(ε+p(N))>0.

Asφ(u(n))> φ(x(n)), we have

(2.5) u(n)> x(n).

Next, observe that ifσ(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 alln ∈Z^[0,N].

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Therefore, for anys∈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)

for someξ ∈[u(s), u(s+ 1)]. Observe that by (2.4) and (A5),φ⁰(ξ)>0. Thus by the monotonicity ofφ⁰, for anys∈Z^[0,N−1],

∆u(s)≤ φ⁰(u(s))

φ⁰(ξ) {f(s) +g(s)u(s) +h(s)w(u(s))}

≤f(s) +g(s)u(s) +h(s)w(u(s)) . Summing up, we have

u(n)−u(0) =

n−1

X

s=0

∆u(s)

≤

n−1

X

s=0

f(s) +

n−1

X

s=0

h(s)w(u(s)) +

n−1

X

s=0

g(s)u(s),

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or

u(n)≤

"

φ⁻¹(ε+p(N)) +

n−1

X

s=0

f(s) +

n−1

X

s=0

h(s)w(u(s))

# +

n−1

X

s=0

g(s)u(s) for all n ∈ Z[0,N]. Hence by the discrete version of the Gronwall-Bellman inequality (see, e.g., [16, Corollary 1.2.5]),

u(n)≤

"

φ⁻¹(ε+p(N)) +

n−1

X

s=0

f(s) +

n−1

X

s=0

h(s)w(u(s))

# exp

n−1

X

s=0

g(s)

≤

"

φ⁻¹(ε+p(N)) +

N−1

X

s=0

f(s) +

n−1

X

s=0

h(s)w(u(s))

# exp

N−1

X

s=0

(2.7) g(s)

for alln ∈ Z[0,N]. Denote byv(n)the RHS of (2.7). Thenv is non-decreasing and for alln∈Z^[0,N],

(2.8) u(n)≤v(n).

Therefore, for anyt∈Z^[0,N−1],

∆v(t) =v(t+ 1)−v(t)

=h(t)w(u(t)) exp

N−1

X

s=0

g(s)

≤h(t)w(v(t)) exp

N−1

X

s=0

g(s).

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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

w(η)∆v(t)

for someη ∈[v(t), v(t+ 1)]. Observe that by (2.4), (2.8), and (A2),w(η)>0.

Therefore, aswis non-decreasing,

∆(Φ◦v)(t)≤ 1

w(η)h(t)w(v(t)) exp

N−1

X

s=0

g(s)

≤h(t) exp

N−1

X

s=0

g(s)

for allt ∈Z^[0,N−1]. Summing up, we have

n−1

X

t=0

∆(Φ◦v)(t)≤

n−1

X

t=0

h(t) exp

N−1

X

s=0

g(s).

On the other hand,

n−1

X

t=0

∆(Φ◦v)(t) = Φ (v(n))−Φ (v(0))

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= Φ (v(n))−Φ

"

exp

N−1

X

s=0

g(s)

!

φ⁻¹(ε+p(N)) +

N−1

X

s=0

f(s)

!#

,

therefore, Φ (v(n))≤Φ

"

exp

N−1

X

s=0

g(s)

!

φ⁻¹(ε+p(N)) +

N−1

X

s=0

f(s)

!#

+

n−1

X

t=0

h(t) exp

N−1

X

s=0

g(s) for alln ∈Z[0,N]. In particular, takingn =N we have

Φ (v(N))≤Φ

"

exp

N−1

X

s=0

g(s)

!

φ⁻¹(ε+p(N)) +

N−1

X

s=0

f(s)

!#

+ exp

N−1

X

s=0

g(s)

!_N−1 X

t=0

h(t).

SinceN ∈Z^[0,α]is arbitrary, Φ (v(n))≤Φ

"

exp

n−1

X

s=0

g(s)

!

φ⁻¹(ε+p(n)) +

n−1

X

s=0

f(s)

!#

+ exp

n−1

X

s=0

g(s)

!_n−1 X

t=0

h(t)

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for alln ∈Z[0,α]. Hence v(n)≤Φ⁻¹

( Φ

"

exp

n−1

X

s=0

g(s)

!

φ⁻¹(ε+p(n)) +

n−1

X

s=0

f(s)

!#

+ exp

n−1

X

s=0

g(s)

!_n−1 X

t=0

h(t) )

and so by (2.5) and (2.8), x(n)< u(n)≤v(n)

≤Φ⁻¹ (

Φ

"

exp

n−1

X

s=0

g(s)

!

φ⁻¹(ε+p(n)) +

n−1

X

s=0

f(s)

!#

+ exp

n−1

X

s=0

g(s)

!_n−1 X

t=0

h(t) )

for alln ∈Z[0,α]. Finally, lettingε →0⁺, we conclude that x(n)≤Φ⁻¹

( Φ

"

exp

n−1

X

s=0

g(s)

!

φ⁻¹(p(n)) +

n−1

X

s=0

f(s)

!#

+ exp

n−1

X

s=0

g(s)

!_n−1 X

t=0

h(t) )

for alln ∈Z^[0,α].

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Remark 1. In many cases the non-decreasing function w satisfiesR∞ 1

ds w(s) =

∞. For example, w= constant > 0, w(s) = √

s, etc., are such functions. In such casesΦ(∞) = ∞and so we may takeα → ∞, that is, (2.2) is valid for alln∈Z0.

Theorem 2.2. Under Assumptions (A1) – (A5), ifx ∈ F(Za,R0)is a function satisfying the nonlinear delay inequality

φ(x(n))≤p(n) +

n−1

X

s=0

φ⁰(x(σ(s))) (

f(s) +g(s)x(σ(s))

+h(s)

s−1

X

t=0

k(t)w(x(σ(t))) )

for alln ∈Z⁰with initial conditions (I1) – (I2), then

(2.9) x(n)≤Φ⁻¹ (

Φ

"

exp

n−1

X

s=0

g(s)

!

φ⁻¹(p(n)) +

n−1

X

s=0

f(s)

!#

+ exp

n−1

X

s=0

g(s)

!_n−1 X

s=0 s−1

X

t=0

h(s)k(t) )

for alln∈Z^[0,β], whereΦ∈C(R⁰,R)is as defined in Theorem2.1, andβ ≥0 is chosen such that the RHS of (2.9) is well-defined, that is,

Φ

"

exp

n−1

X

s=0

g(s)

!

φ⁻¹(p(n)) +

n−1

X

s=0

f(s)

!#

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+ exp

n−1

X

s=0

g(s)

!_n−1 X

s=0 s−1

X

t=0

h(s)k(t)∈ I_mΦ

for alln ∈Z[0,β].

Proof. Fixε >0andM ∈Z[0,β]. Defineu:Z[0,M]→R0 by

(2.10) u(n) :=φ⁻¹ (

ε+p(M) +

n−1

X

δ=0

φ⁰(x(σ(δ)))·

"

f(δ) +g(δ)x(σ(δ))

+h(δ)

δ−1

X

t=0

k(t)w(x(σ(t)))

#) .

By (A5),uis non-decreasing onZ^[0,M]. For anyn ∈Z^[0,M], by (A5) again, (2.11) u(n)≥φ⁻¹(ε+p(M))>0.

Asφ(u(n))> φ(x(n)), we have

(2.12) u(n)> x(n).

Using the same arguments as in the derivation of (2.6) in the proof of Theorem 2.1, we have

(2.13) x(σ(n))≤u(n) for alln ∈Z[0,M].

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Hence for anys∈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

t=0

k(t)w(x(σ(t))) )

≤φ⁰(u(s)) (

f(s) +g(s)u(s) +h(s)

s−1

X

t=0

k(t)w(u(t)) )

.

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 ofφ⁰, for anys∈Z^[0,M−1],

∆u(s)≤ φ⁰(u(s)) φ⁰(ξ)

(

f(s) +g(s)u(s) +h(s)

s−1

X

t=0

k(t)w(u(t)) )

≤f(s) +g(s)u(s) +h(s)

s−1

X

t=0

k(t)w(u(t)) . Summing up, we have

u(n)−u(0) =

n−1

X

s=0

∆u(s)

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≤

n−1

X

s=0

f(s) +

n−1

X

s=0

h(s)

s−1

X

t=0

k(t)w(u(t)) +

n−1

X

s=0

g(s)u(s),

or u(n)≤

"

φ⁻¹(ε+p(M)) +

n−1

X

s=0

f(s) +

n−1

X

s=0

h(s)

s−1

X

t=0

k(t)w(u(t))

# +

n−1

X

s=0

g(s)u(s)

for all n ∈ Z[0,M]. Hence by the discrete version of the Gronwall-Bellman inequality (see, e.g., [16, Corollary 1.2.5]),

u(n)≤

"

φ⁻¹(ε+p(M)) +

n−1

X

s=0

f(s)

+

n−1

X

s=0

h(s)

s−1

X

t=0

k(t)w(u(t))

# exp

n−1

X

s=0

g(s)

≤

"

φ⁻¹(ε+p(M)) +

M−1

X

s=0

f(s)

+

n−1

X

s=0

h(s)

s−1

X

t=0

k(t)w(u(t))

# exp

M−1

X

s=0

(2.14) g(s)

for alln∈Z^[0,M]. Denote byv(n)the RHS of (2.14). Thenv is non-decreasing and for alln∈Z[0,M],

(2.15) u(n)≤v(n).

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Therefore, for anyδ∈Z^[0,M−1],

∆v(δ) =v(δ+ 1)−v(δ)

=h(δ)

δ−1

X

t=0

k(t)w(u(t))

! exp

M−1

X

s=0

g(s)

≤h(δ)

δ−1

X

t=0

k(t)w(v(t))

! exp

M−1

X

s=0

g(s)

≤h(δ)w(v(δ))

δ−1

X

t=0

k(t)

! exp

M−1

X

s=0

g(s).

On the other hand, by the Mean Value Theorem,

∆(Φ◦v)(δ) = Φ (v(δ+ 1))−Φ (v(δ))

= Φ⁰(η)∆v(δ) = 1

w(η)∆v(δ)

for someη∈[v(δ), v(δ+ 1)]. Observe that by (2.11), (2.14), and (A2),w(η)>

0. Therefore, aswis non-decreasing,

∆(Φ◦v)(δ)≤ 1

w(η)h(δ)w(v(δ))

δ−1

X

t=0

k(t)

! exp

M−1

X

s=0

g(s)

≤h(δ)

δ−1

X

t=0

k(t)

! exp

M−1

X

s=0

g(s)

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for allδ ∈Z^[0,M−1]. Summing up, we have

n−1

X

δ=0

∆(Φ◦v)(δ)≤

n−1

X

δ=0

h(δ)

δ−1

X

t=0

k(t)

! exp

M−1

X

s=0

g(s),

or

Φ (v(n))≤Φ (v(0)) +

n−1

X

δ=0

h(δ)

δ−1

X

t=0

k(t)

! exp

M−1

X

s=0

g(s)

= Φ

"

φ⁻¹(ε+p(M)) +

M−1

X

s=0

f(s)

! exp

M−1

X

s=0

g(s)

#

+

n−1

X

δ=0

h(δ)

δ−1

X

t=0

k(t)

! exp

M−1

X

s=0

g(s)

for alln ∈Z[0,M]. In particular, takingn =M this yields

Φ (v(M))≤Φ

"

φ⁻¹(ε+p(M)) +

M−1

X

s=0

f(s)

! exp

M−1

X

s=0

g(s)

#

+

M−1

X

δ=0

h(δ)

δ−1

X

t=0

k(t)

! exp

M−1

X

s=0

g(s).

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SinceM ∈Z[0,β]is arbitrary,

Φ (v(n))≤Φ

"

φ⁻¹(ε+p(n)) +

n−1

X

s=0

f(s)

! exp

n−1

X

s=0

g(s)

#

+

n−1

X

δ=0

h(δ)

δ−1

X

t=0

k(t)

! exp

n−1

X

s=0

g(s)

for alln ∈Z[0,β]. Hence

v(n)≤Φ⁻¹ (

Φ

"

φ⁻¹(ε+p(n)) +

n−1

X

s=0

f(s)

! exp

n−1

X

s=0

g(s)

#

+

n−1

X

δ=0

h(δ)

δ−1

X

t=0

k(t)

! exp

n−1

X

s=0

g(s) )

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

s=0

f(s)

! exp

n−1

X

s=0

g(s)

#

+

n−1

X

δ=0

h(δ)

δ−1

X

t=0

k(t)

! exp

n−1

X

s=0

g(s) )

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for alln ∈Z[0,β]. Finally, lettingε→0⁺, we conclude that x(n)≤Φ⁻¹

( Φ

"

exp

n−1

X

s=0

g(s)

!

φ⁻¹(p(n)) +

n−1

X

s=0

f(s)

!#

+ exp

n−1

X

s=0

g(s)

!_n−1 X

δ=0 δ−1

X

t=0

h(δ)k(t) )

for alln ∈Z[0,β].

Remark 2. Similar to the previous remark, in caseΦ(∞) = ∞, (2.9) holds for alln∈Z0.

Theorem 2.3. Under Assumptions (A1), (A3) and (A4), ifx ∈ F(Z^a,R⁰)is a function satisfying the nonlinear delay inequality

x^r(n)≤c^r+

n−1

X

s=0

x^r(σ(s)){f(s) +g(s)x^r(σ(s))} , n ∈Z⁰, with initial conditions (I1) and

(I3) ψ(σ(s))≤c for alls ∈Z0 withσ(s)≤0, wherer, c > 0are constants, then

(2.16) x(n)≤

"

c^−r

n−1

Y

s=0

(1−f(s))−

n

X

s=1

g(s)

n−1

Y

t=s

(1−f(t))

#−¹_r

for all n ∈ Z[0,γ], whereγ ≥ 0 is chosen such that the RHS of (2.16) is well- defined.

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Proof. Defineu∈ F(Z0,R0)by (2.17) u^r(n) := c^r+

n−1

X

s=0

x^r(σ(s)){f(s) +g(s)x^r(σ(s))} , n∈Z0 . Clearly,u≥0is non-decreasing and

(2.18) x(n)≤u(n) for alln ∈Z0.

Similar to the derivation of (2.6) in the proof of Theorem2.1, we easily establish x(σ(n))≤u(n) for alln ∈Z0 .

By (2.17), for anyn∈Z0,

∆u^r(n) =u^r(n+ 1)−u^r(n)

=x^r(σ(n)){f(n) +g(n)x^r(σ(n))}

≤u^r(n){f(n) +g(n)u^r(n)}

≤u^r(n+ 1){f(n) +g(n)u^r(n)} . Asu(0) =c, by elementary analysis, we infer from (2.17) that (2.19) u(n)≤y(n) for alln ∈Z^[0,ρ]

where Z[0,ρ] is the maximal lattice on which the unique solution y(n) to the discrete Bernoulli equation

(2.20)







∆y^r(n) =y^r(n+ 1){f(n) +g(n)y^r(n)}, n ∈Z0

y(0) =c

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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

t=s

(1−f(t))

#⁻¹_r

for alln∈Z[0,γ]. The assertion now follows from (2.18), (2.19) and (2.21).

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3. Immediate Consequences

Direct application of the results in Section2yields 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) x^α(n)

≤p(n) +

n−1

X

s=0

x^α−1(σ(s)){f(s) +g(s)x(σ(s)) +h(s)w(x(σ(s)))}

for alln ∈Z⁰with initial conditions (I1) and

(I4) ψ(σ(s))≤p^α¹(s) for alls∈Z0withσ(s)≤0, whereα≥1is a constant, then

(3.2) x(n)≤Φ⁻¹ (

Φ

"

exp 1 α

n−1

X

s=0

g(α)

!

p^α¹(n) + 1 α

n−1

X

s=0

f(s)

!#

+ exp 1 α

n−1

X

s=0

g(α)

! 1 α

n−1

X

t=0

h(t) )

for all n ∈ Z[0,µ], where µ ≥ 0is chosen such that the RHS of (3.2) is well- defined for alln∈Z[0,µ], andΦis defined as in Theorem2.1.

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Proof. Let φ : R0 → R0 be defined by φ(r) = r^α, r ∈ R0. Thenφ satisfies Assumption (A5). By (3.1) we have

φ(x(n))≤p(n)+

n−1

X

s=0

φ⁰(x(σ(s)))

f(s)

α +g(s)

α x(σ(s)) + h(s)

α w(x(σ(s)))

.

Furthermore, it is easy to see that

φ(x(s))≤p^α¹(s) =φ⁻¹(p(s)) for alls∈Z0withσ(s)≤0. Thus Theorem2.1applies and the assertion follows.

Remark 3.

(i) In Corollary3.1, if we setα = 2,p(n)≡c²,g(n)≡0, we have x²(n)≤c² +

n−1

X

s=0

x(σ(s)){f(s) +h(s)w(x(σ(s)))}, n∈Z0

implies

x(n)≤Φ⁻¹ (

Φ

"

c+ 1 2

n−1

X

s=0

f(s)

# + 1

2

n−1

X

s=0

h(s) )

, n ∈Z^[0,µ].

This is the discrete analogue of a result of Pachpatte in [14]. Furthermore, ifσ =id, this reduces to a result of Pachpatte in [18].

(ii) In caseΦ(∞) =∞, (3.2) holds for alln ∈Z0.

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Corollary 3.2. Under Assumptions (A1) – (A4) with p ∈ F(Z0,R+), if x ∈ F(Z^a,R¹)satisfies the nonlinear delay inequality

(3.3) x^α(n)

≤p(n) +

n−1

X

s=0

x^α(σ(s)){f(s) +g(s) lnx(σ(s)) +h(s)w(lnx(σ(s)))}

for alln ∈Z0with initial conditions (I1) and

(I5) ψ(σ(s))≤ 1

αln (p(s)) for alls∈Z0 withσ(s)≤0, whereα >0is a constant, then

(3.4) x(n)≤exp (

Φ⁻¹

"

Φ

exp 1 α

n−1

X

s=0

g(s)

!

× 1

αlnp(n) + 1 α

n−1

X

s=0

f(s)

!!

+ exp 1 α

n−1

X

s=0

g(s)

! 1 α

n−1

X

t=0

h(t)

#)

for all n ∈ Z[0,ν], where ν ≥ 0 is chosen such that the RHS of (3.4) is well- defined for alln∈Z^[0,ν], andΦis defined as in Theorem2.1.

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Proof. Lettingy(n) = lnx(n), (3.3) becomes

(3.5) exp (αy(n))

≤p(n) +

n−1

X

s=0

exp (αy(σ(s))){f(s) +g(s)y(σ(s)) +h(s)w(y(σ(s)))}.

Let φ : R⁰ → R⁰ be defined by φ(r) = exp(αr), r ∈ R⁰. Then φ satisfies Assumption (A5). Hence from (3.5), we have

φ(y(n))≤p(n)+

n−1

X

s=0

φ⁰(y(σ(s)))

f(s)

α + g(s)

α y(σ(s)) + h(s)

α w(y(σ(s)))

.

Furthermore, it is easy to see that ψ(σ(s))≤ 1

α ln (p(s)) =φ⁻¹(p(s)) for alls∈Z⁰withσ(s)≤0. Thus Theorem2.1applies and we have

y(n)≤Φ⁻¹ (

Φ

"

exp 1 α

n−1

X

s=0

g(s)

! 1

αlnp(n) + 1 α

n−1

X

s=0

f(s)

!#

+ exp 1 α

n−1

X

s=0

g(s)

! 1 α

n−1

X

t=0

h(t) )

for alln ∈Z[0,ν], and from this the assertion follows.

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Remark 4. In caseΦ(∞) =∞, (3.4) holds for alln ∈Z0.

Corollary 3.3. Under Assumptions (A1) – (A4), if x ∈ F(Z^a,R⁰)satisfies the nonlinear delay inequality

(3.6) x^α(n)≤p(n) +

n−1

X

s=0

x^α−1(σ(s)) (

f(s) +g(s)x(σ(s))

+h(s)

s−1

X

t=0

k(t)w(x(σ(t))) )

for alln ∈ Z0 with initial conditions (I1) and (I4), whereα ≥1is a constant, then

(3.7) x(n)≤Φ⁻¹

Φ

"

exp 1 α

n−1

X

s=0

g(s)

!

p^α¹(n) + 1 α

n−1

X

s=0

f(s)

!#

+ exp 1 α

n−1

X

s=0

g(s)

! 1 α

n−1

X

s=0

h(s)

s−1

X

t=0

k(t)

!)

for all n ∈ Z^[0,η], where η ≥ 0is chosen such that the RHS of (3.7) is well- defined for alln∈Z[0,η], andΦis defined as in Theorem2.1.

Proof. Let φ : R0 → R0 be defined by φ(r) = r^α, r ∈ R0. Thenφ satisfies

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Assumption (A5). By (3.6), φ(x(n))≤p(n) +

n−1

X

s=0

φ⁰(x(σ(s)))

(f(s)

α +g(s)

α x(σ(s)) + h(s)

α

s−1

X

t=0

k(t)w(x(σ(t))) )

for alln ∈Z⁰. Furthermore, it is easy to see that

ψ(σ(s))≤p^α¹(s) =φ⁻¹(p(s)) for alls∈Z0 withσ(s)≤0. Thus Theorem2.2applies and we have

x(n)≤Φ⁻¹ (

Φ

"

exp 1 α

n−1

X

s=0

g(s)

!

p^α¹(n) + 1 α

n−1

X

s=0

f(s)

!#

+ exp 1 α

n−1

X

s=0

g(s)

!

· 1 α

n−1

X

s=0 s−1

X

t=0

h(s)k(t) )

for alln ∈Z[0,η]. Remark 5.

(i) In Corollary3.3, if we putα = 2,p(n)≡c²,g(n)≡0, we have x²(n)≤c²+

n−1

X

s=0

x(σ(s)) (

f(s) +h(s)

s−1

X

t=0

k(t)w(x(σ(t))) )

, n∈Z⁰

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implies

x(n)≤Φ⁻¹ (

Φ

"

c+ 1 2

n−1

X

s=0

f(s)

# +1

2

n−1

X

s=0

h(s)

s−1

X

t=0

k(t) )

, n∈Z[0,η].

This is the discrete analogue of a result of Pachpatte in [14]. Furthermore, ifσ =idandw=id, this reduces to a result of Pachpatte in [18].

(ii) In caseΦ(∞) =∞, (3.7) holds for alln ∈Z⁰.

Corollary 3.4. Under Assumptions (A1) – (A4) with p ∈ F(Z0,R+), if x ∈ F(Za,R1)satisfies the nonlinear delay inequality

(3.8) x^α(n)≤p(n) +

n−1

X

s=0

x^α(σ(s)) (

f(s) +g(s) lnx(σ(s))

+h(s)

s−1

X

t=0

k(t)w(lnx(σ(t))) )

for alln ∈Z0with initial conditions (I1) and (I6) ψ(σ(s))≤ 1

αln (p(s)) for alls∈Z⁰ withσ(s)≤0, whereα >0is any constant, then

(3.9) x(n)≤exp (

Φ⁻¹

"

Φ

exp 1 α

n−1

X

s=0

g(s)

!

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× 1

αlnp(n) + 1 α

n−1

X

s=0

f(s)

!!

+ exp 1 α

n−1

X

s=0

g(s)

!

· 1 α

n−1

X

s=0

h(s)

s−1

X

t=0

k(t)

#)

for all n ∈ Z^[0,λ], where λ ≥ 0 is chosen such that the RHS of (3.9) is well- defined for alln∈Z[0,λ], andΦis defined as in Theorem2.1.

Proof. Lettingy(n) = lnx(n), (3.8) becomes

(3.10) exp (αy(n))≤p(n) +

n−1

X

s=0

exp (αy(σ(s))) (

f(s) +g(s)y(σ(s))

+h(s)

s−1

X

t=0

k(t)w(y(σ(t))) )

for alln ∈Z0. Letφ : R0 →R0 be defined byφ(r) = exp(αr),r ∈ R0. Then φsatisfies Assumption (A5). Hence from (3.10), we have

φ(y(n))≤p(n) +

n−1

X

s=0

φ⁰(y(σ(s)))

×

(f(s)

α + g(s)

α y(σ(s)) + h(s) α

s−1

X

t=0

k(t)w(y(σ(t))) )

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for alln ∈Z0. Furthermore, it is easy to check that ψ(σ(s))≤ 1

α ln (p(s)) =φ⁻¹(p(s)) for alls∈Z0withσ(s)≤0. Thus Theorem2.2applies and we have

y(n)≤Φ⁻¹ (

Φ

"

exp 1 α

n−1

X

s=0

g(s)

! 1

αlnp(n) + 1 α

n−1

X

s=0

f(s)

!#

+ exp 1 α

n−1

X

s=0

g(s)

!

· 1 α

n−1

X

s=0 s−1

X

t=0

h(s)k(t) )

for alln ∈Z[0,λ], and from this the assertion follows.

Remark 6.

(i) In Corollary3.4, if we setα = 2,p(n)≡c²,g(n)≡0, then x²(n)≤c²+

n−1

X

s=0

x²(σ(s)) (

f(s) +h(s)

s−1

X

t=0

k(t)w(lnx(σ(t))) )

, n ∈Z0

implies

x(n)≤exp (

Φ⁻¹

"

Φ 1

2lnp(n) + 1 2

n−1

X

s=0

f(s)

! +1

2

n−1

X

s=0

h(s)

s−1

X

t=0

k(t)

#)

n ∈Z[0,λ].

This is the discrete version of a result of Pachpatte in [14].

(ii) In caseΦ(∞) =∞, (3.9) holds for alln ∈Z0.

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4. Application

Consider the discrete delay equation (4.1) x^α(n) =F n, x(σ(n)) ,

n−1

X

s=0

G(n, s, x(σ(s)))

!

, n∈Z0

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), andG∈C(Z²0× R,R). IfF, Gsatisfy

|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 somep, f, g, h, wsatisfying (A1) and (A2), and some constantK > 0, then every solution of (4.1) satisfies

|x(n)|^α =

F n, x(σ(n)),

n−1

X

s=0

G(n, s, x(σ(s)))

!

≤p(n) +K

n−1

X

s=0

G(n, s, x(σ(s)))

≤p(n) +K

n−1

X

s=0

|G(n, s, x(σ(s)))|

≤p(n) +K

n−1

X

s=0

[f(s) +g(s)|x(σ(s))|+h(s)w(|x(σ(s))|)]|x(σ(s))|^α−1

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for all n ∈ J(x) := the maximal existence lattice on whichxis defined. Ap- plying Corollary3.1, this yields

|x(n)| ≤Φ⁻¹ (

Φ

"

expK α

n−1

X

s=0

g(α)

!

p^α¹(n) + K α

n−1

X

s=0

f(s)

!#

+ expK α

n−1

X

s=0

g(α)

!K α

n−1

X

t=0

h(t) )

for alln ∈J(x)∩Z[0,µ]. This gives the boundedness of solutions of (4.1).

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References

[1] R.P. AGARWAL, Difference Equations and Inequalities, Marcel Dekker, New York, 2000.

[2] D. BAINOVAND P. SIMEONOV, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, 1992.

[3] E.F. BECKENBACH AND R. BELLMAN, Inequalities, Springer-Verlag, New York, 1961.

[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.

Hungar., 7 (1956), 71–94.

[6] W.S. CHEUNG, On some new integrodifferential inequalities of the Gron- wall 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 So- lutions, Lecture Notes in Mathematics, v.841, Springer-Verlag, Berlin, 1981.

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[10] Q.M. MA AND E.H. YANG, On some new nonlinear delay integral in- equalities, J. Math. Anal. Appl., 252 (2000), 864–878.

[11] D.S. MITRINOVI ´C, Analytic Inequalities, Springer-Verlag, New York, 1970.

[12] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Inequalities Involv- ing Functions and Their Integrals and Derivatives, Kluwer Academic Pub- lishers, Dordrecht, 1991.

[13] L. OU-IANG, The boundedness of solutions of linear differential equa- tionsy⁰⁰+A(t)y= 0, Shuxue Jinzhan, 3 (1957), 409–415.

[14] B.G. PACHPATTE, A note on certain integral inequalities with delay, Pe- riod. Math. Hungar., 31 (1995), 229–234.

[15] B.G. PACHPATTE, Explicit bounds on certain integral inequalities, J.

Math. Anal. Appl., 267 (2002), 48–61.

[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 inequal- ity arising in the theory of differential equations, J. Math. Anal. Appl., 251 (2000), 736–751.

[18] B.G. PACHPATTE, On some new inequalities related to certain inequal- ities in the theory of differential equations, J. Math. Anal. Appl., 189 (1995), 128–144.

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[19] P. Ch. TSAMATOS ANDS.K. NTOUYAS, On a Bellman-Bihari type in- equality with delay, Period. Math. Hungar., 23 (1991), 91–94.

[20] E.H. YANG, Generalizations of Pachpatte’s integral and discrete inequal- ities, Ann. Differential Equations, 13 (1997), 180–188.