Electronic Journal of Qualitative Theory of Differential Equations Proc. 7th Coll. QTDE, 2004, No. 17 1-12; http://www.math.u-szeged.hu/ejqtde/

Exponential Estimates of Solutions of Difference Equations with Continuous Time

Hajnalka Péics

University of Novi Sad

Faculty of Civil Engineering, Subotica

Serbia and Montenegro

E-mail: peics@gf.su.ac.yu

Dedicated to Professor László Hatvani on his 60th birthday

Abstract

In this paper ^{1} we study the scalar difference equation with continuous time of the form

x(t) = a(t)x(t − 1) + b(t)x(p(t)),

where a, b : [t_{0}, ∞) → R are given real functions for t_{0} > 0 and p : [t_{0}, ∞) → R is a given function such that p(t) ≤ t, lim_{t→∞} p(t) = ∞. Using the method of characteristic equation we obtain an exponential estimate of solutions of this equation which can be applied to the difference equations with constant delay and to the case t − p_{2} ≤ p(t) ≤ t − p_{1} for real numbers 1 < p_{1} ≤ p_{2}.

We generalize the main result to the equation with several delays of the form

x(t) = a(t)x(t − 1) + ∑_{i=1}^{m} b_{i}(t)x(p_{i}(t)),

where a, b_{i} : [t_{0}, ∞) → R are given functions for i = 1, 2, ..., m, and p_{i} : [t_{0}, ∞) → R are given such that p_{i}(t) ≤ t, lim_{t→∞} p_{i}(t) = ∞ for i = 1, 2, ..., m. We apply the obtained result to particular cases such as p_{i}(t) = t − p_{i} for i = 1, 2, ..., m, where 1 ≤ p_{1} < p_{2} < ... < p_{m} are real numbers.

AMS (MOS) subject classification: 39A11, 39B22

1. Introduction

The characteristic equation is a useful tool in the qualitative analysis of the theory of differential and difference equations. Knowing the solutions or the behavior of solutions of the characteristic equations we can obtain some properties of solutions of the considered differential or difference equations. This method is usually applied in the investigations of the theory of oscillation, asymptotic behavior, stability, etc. Using the solutions of the characteristic equation we can, from a new point of view, describe the asymptotic behavior of solutions of the difference equation. We apply the obtained results for some particular cases.

1This paper is in final form and no version of it will be submitted for publication elsewhere.

Assume that t_{0} > 0 and a, b : [t_{0}, ∞) → R are given real functions. Let p : [t_{0}, ∞) → R be given function such that, for every T > t_{0} there exists a δ > 0 such that p(t) ≤ t − δ for every t ∈ [t_{0}, T], and lim_{t→∞} p(t) = ∞. Now, we investigate the scalar difference equation with continuous arguments

x(t) = a(t)x(t − 1) + b(t)x(p(t)),  (1)

where x(t) ∈ R.

Let N be the set of nonnegative integers, R the set of real numbers and R_{+} = (0, ∞).

For given m ∈ N, t ∈ R_{+} and a function f : R → R we use the standard notation

∏_{ℓ=t}^{t−1} f(ℓ) = 1, ∏_{ℓ=t−m}^{t} f(ℓ) = f(t − m)f(t − m + 1)...f(t)

and

∑_{τ=t}^{t−1} f(τ) = 0, ∑_{τ=t−m}^{t} f(τ) = f(t − m) + f(t − m + 1) + ... + f(t).

The difference operator Δ is defined by

Δf(t) = f(t + 1) − f(t).

For a function g : R_{+} × R_{+} → R, the difference operator Δ_{t} is given by

Δ_{t}g(t, a) = g(t + 1, a) − g(t, a).

Set

t_{−1} = min {inf{p(s) : s ≥ t_{0}}, t_{0} − 1}

and

t_{m} = inf{s : p(s) > t_{m−1}} for all m = 1, 2, ...

Then {t_{m}}_{m=−1}^{∞} is an increasing sequence such that

lim_{m→∞} t_{m} = ∞, ⋃_{m=1}^{∞}[t_{m−1}, t_{m}) = [t_{0}, ∞)

and p(t) ∈ ⋃_{i=0}^{m}[t_{i−1}, t_{i}) for all t_{m} ≤ t < t_{m+1}, m = 0, 1, 2, ...

For a given nonnegative integer m, fix a point t ≥ t_{0}, and define the natural numbers k_{m}(t) such that

k_{m}(t) := [t − t_{m}], m = 0, 1, 2...

Then, for t ∈ [t_{m}, t_{m+1}), we have

t − k_{m}(t) − 1 < t_{m} and t − k_{m}(t) ≥ t_{m}, m = 0, 1, 2, ...

and for the arbitrary t ≥ t_{0}, m = 0 we have

t − k_{0}(t) − 1 < t_{0} and t − k_{0}(t) ≥ t_{0}.

Set

T_{m}(t) := {t − k_{m}(t), t − k_{m}(t) + 1, ..., t − 1, t}, m = 0, 1, 2, ....

For a given function φ : [t_{−1}, t_{0}) → R, Equation (1) has the unique solution x^{φ} satisfying the initial condition

x^{φ}(t) = φ(t) for t_{−1} ≤ t < t_{0}.  (2)

We present the characteristic equation associated with the initial value problem (1) and (2), and using them obtain an asymptotic estimate of solutions of Equation (1) which can be applied to the difference equations with constant delay and to the case t − p_{2} ≤ p(t) ≤ t − p_{1} for real numbers 1 < p_{1} ≤ p_{2}.

After that we generalize the main result to the equation with several delays

x(t) = a(t)x(t − 1) + ∑_{i=1}^{m} b_{i}(t)x(p_{i}(t)),

where a, b_{i} : [t_{0}, ∞) → R are given functions for i = 1, 2, ..., m, and p_{i} : [t_{0}, ∞) → R are given such that, for every T > t_{0} there exists a δ > 0 such that p_{i}(t) ≤ t − δ for every t ∈ [t_{0}, T], and lim_{t→∞} p_{i}(t) = ∞ for i = 1, 2, ..., m. We apply the obtained results to particular cases such as p_{i}(t) = t − p_{i} for i = 1, 2, ..., m, where 1 ≤ p_{1} < p_{2} < ... < p_{m} are real numbers.

Let x = x^{φ} be a solution of the initial value problem (1) and (2) such that x(t) ≠ 0 for t ≥ t_{0}. Then

1 = a(t)\frac{x(t − 1)}{x(t)} + b(t)\frac{x(p(t))}{x(t)} for t ≥ t_{0}.

Define the new function

λ(t) := \frac{x(t − 1)}{x(t)} for t ≥ t_{0}.  (3)

Since, now

x(t) = φ(t − k_{0}(t) − 1) ∏_{ℓ=t−k_{0}(t)}^{t} \frac{1}{λ(ℓ)} for t ≥ t_{0},

the function λ defined by (3) is a solution of the characteristic equation of the form

1 − a(t)λ(t) =
= b(t)\frac{φ(p(t) − k_{0}(p(t)) − 1)}{φ(t − k_{0}(t) − 1)} ∏_{ℓ=t−k_{0}(t)}^{t} λ(ℓ)  ∏_{ℓ=p(t)−k_{0}(p(t))}^{p(t)} \frac{1}{λ(ℓ)},  (4)

for t ≥ t_{0}. Characteristic equation (4) associated with the initial value problem (1) and (2) is a generalization of the characteristic equation given in [3] and [4] for discrete difference equations.

2. Preliminaries

For given scalar functions a, ρ : [t_{0}, ∞) → R, ρ(t) ≠ 0, for given initial function φ and nonnegative numbers n we define the numbers

R_{n} := sup_{t_{n}≤t<t_{n+1}} \{ρ(t) ∑_{τ=t−k_{n}(t)}^{t} \frac{Δρ(τ − 1)}{ρ(τ)ρ(τ − 1)} ∏_{ℓ=τ+1}^{t} a(ℓ)\}  (5)

and

M_{0} := sup_{t_{−1}≤t<t_{0}} ρ(t)|φ(t)|.

We shall need the following hypotheses.

• (H_{1}) Let a : [t_{0}, ∞) → R be a given real function satisfying 0 < a(t) < 1, for all t ≥ t_{0} and b : [t_{0}, ∞) → R an arbitrary given real function for all t ≥ t_{0}.

• (H_{2}) Let p : [t_{0}, ∞) → R be a given function such that, for every T > t_{0} there exists a δ > 0 such that p(t) ≤ t − δ for every t ∈ [t_{0}, T], and lim_{t→∞} p(t) = ∞.

• (H_{∗}) There exists a real function ρ : [t_{−1}, ∞) → (0, ∞), which is bounded on the initial interval [t_{−1}, t_{0}), and such that

|b(t)|ρ(t − 1) ≤ (1 − a(t))ρ(p(t)) for all t ≥ t_{0},

where functions a and b are given in (H_{1}).

• (H_{∗∗}) There exists a real number R such that

∏_{n=0}^{j}(1 + R_{n}) ≤ R,  (6)

for all positive integers j, where the numbers R_{n} are defined by (5).

Theorem A. (Péics [5]) Suppose that conditions (H_{1}), (H_{2}), (H_{∗}) and (H_{∗∗}) hold.

Let x = x^{φ} be the solution of the initial value problem (1) and (2) with bounded function φ. Then

|x(t)| ≤ \frac{M_{0}R}{ρ(t)} for all t ≥ t_{0}.

Zhou and Yu in [6] obtained for estimating function the exponential function for the case when the lag function is between two constant delays. Applying Theorem A, by finding an appropriate estimating function ρ, we can determine the rate of convergence of the solutions of generalized difference equations for different types of the lag function. We will provide two examples.

If the lag function p is such that p_{1}t ≤ p(t) ≤ p_{2}t, for real numbers 0 < p_{1} ≤ p_{2} < 1, then the estimating function is a power function, namely ρ(t) = t^{k}.

Corollary A. (Péics [5]) Suppose that condition (H_{1}) holds. For given real numbers

p_{1}, p_{2} and t_{0} such that 0 < p_{1} ≤ p_{2} < 1 and t_{0} > p_{1}/(1 − p_{1}), let p be given real function such that p_{1}t ≤ p(t) ≤ p_{2}t for all t ≥ t_{0}. Suppose that there exist real numbers Q and α such that 0 < Q ≤ 1, 0 < α < 1,

|b(t)| ≤ Q(1 − a(t)), α ≤ 1 − a(t) for all t ≥ t_{0}

and

log \frac{1}{Q} (log \frac{1}{p_{1}} − log \frac{1}{p_{2}} ) < log \frac{1}{p_{1}} log \frac{1}{p_{2}} .

Let x = x^{φ} be a solution of the initial value problem (1) and (2) with bounded function φ, and let

k = \frac{log Q}{log p_{1}} , M_{0} = sup_{t_{−1}≤t<t_{0}}\{t^{k}|φ(t)|\}.

Then

|x(t)| ≤ \frac{C}{(t − \frac{p_{1}}{1−p_{1}} )^{k}} for all t ≥ t_{0},

where

C = M_{0} ∏_{n=0}^{∞} (1 + \frac{kt_{0}^{k}(1 − p_{1})^{k+1}}{αp_{1}^{k}(t_{0}(1 − p_{1}) − p_{2}^{n})^{k+1}} (\frac{p_{2}^{k+1}}{p_{1}^{k}} )^{n}) .

Remark 1. We can prove similarly the above result by choosing ρ(t) = t^{k}. Then

the statement of Theorem A holds for t ≥ t_{0}/p_{1} that does not disturb the asymptotic behavior of solutions but the comparableness with the function ρ is clearer and the rate of the convergence is better.

If the lag function p is such that ^{p_{2}} \sqrt{t} ≤ p(t) ≤ ^{p_{1}} \sqrt{t}, for natural numbers 1 < p_{1} ≤ p_{2}, then the estimating function is a logarithm function. Namely, ρ(t) = log^{k} t.

Corollary B. (Péics [5]) Suppose that condition (H_{1}) holds. Let t_{0} ≥ 1 be given

real number, p_{1}, p_{2} be given natural numbers such that 1 < p_{1} ≤ p_{2}. Let p be given real function such that ^{p_{2}} \sqrt{t} ≤ p(t) ≤ ^{p_{1}} \sqrt{t} for all t ≥ t_{0}. Suppose that there exist real numbers Q and α such that 0 < Q ≤ 1, 0 < α < 1 and

|b(t)| ≤ Q(1 − a(t)), α ≤ 1 − a(t) for all t ≥ t_{0}.

Let x = x^{φ} be a solution of the initial value problem (1) and (2) with bounded function φ, and let

k = −\frac{log Q}{log p_{2}} , M_{0} = sup_{t_{−1}≤t<t_{0}} \{log^{k} t|φ(t)|\} .

Then

|x(t)| ≤ \frac{C}{log^{k} t} for all t ≥ t_{0},

where

C = M_{0} ∏_{n=0}^{∞} (1 + \frac{kp_{2}^{k}p_{2}^{nk} log^{k} t_{0}}{α(t_{0}^{p_{1}^{n}} − 1) log^{k+1}(t_{0}^{p_{1}^{n}} − 1)}) .

3. Main Result

Assume that

• (H_{3}) For the function a and b given in (H_{1}), there is a real function λ : [t_{0}, ∞) → (1, ∞) and there is an initial function φ in (2) such that

|b(t)|\frac{φ(p(t) − k_{0}(p(t)) − 1)}{φ(t − k_{0}(t) − 1)} ∏_{ℓ=t−k_{0}(t)}^{t} λ(ℓ) ∏_{ℓ=p(t)−k_{0}(p(t))}^{p(t)} \frac{1}{λ(ℓ)} ≤

≤ 1 − a(t)λ(t), t ≥ t_{0}.  (7)

In the next result we use the concept of the characteristic equation but it is not necessary to have the solution of characteristic equation (4). It is sufficient only to have a solution of Inequality (7) that is a much weaker condition.

Theorem 1. Suppose that conditions (H_{1}), (H_{2}) and (H_{3}) hold. Let x = x^{φ} be the

solution of the initial value problem (1) and (2). Then

|x(t)| ≤ (φ(t − k_{0}(t) − 1) sup_{t_{−1}≤t≤t_{0}} λ(t)) ∏_{ℓ=t−k_{0}(t)}^{t} \frac{1}{λ(ℓ)}, t ≥ t_{0}.

Proof. Introduce the transformation

y(t) := \frac{x(t)}{φ(t − k_{0}(t) − 1)} ∏_{ℓ=t−k_{0}(t)}^{t} λ(ℓ).

Then the function y(t) satisfies the equation

y(t) = a(t)λ(t)y(t − 1) +

+b(t)\frac{φ(p(t) − k_{0}(p(t)) − 1)}{φ(t − k_{0}(t) − 1)} ∏_{ℓ=t−k_{0}(t)}^{t} λ(ℓ) ∏_{ℓ=p(t)−k_{0}(p(t))}^{p(t)} \frac{1}{λ(ℓ)}y(p(t)).

Using hypothesis (7) we obtain that

|y(t)| ≤ a(t)λ(t)|y(t − 1)| + (1 − a(t)λ(t))|y(p(t))|.

Let t ∈ [t_{n}, t_{n+1}), τ ∈ T_{n}(t) and |y(t)| = u(t). Then, the above inequality is equivalent to

Δ_{τ} (u(τ − 1) ∏_{ℓ=t−k_{n}(t)}^{τ−1} \frac{1}{a(ℓ)λ(ℓ)}) ≤

≤ (1 − a(τ)λ(τ))u(p(τ)) ∏_{ℓ=t−k_{n}(t)}^{τ} \frac{1}{a(ℓ)λ(ℓ)}.

Summing up both sides of this inequality from t − k_{n}(t) to t gives that

u(t) ≤ u(t − k_{n}(t) − 1) ∏_{ℓ=t−k_{n}(t)}^{t} a(ℓ)λ(ℓ) +

+ ∑_{τ=t−k_{n}(t)}^{t}(1 − a(τ)λ(τ))u(p(τ)) ∏_{ℓ=τ+1}^{t} a(ℓ)λ(ℓ).

Define

μ_{n} := sup_{t_{n−1}≤t<t_{n}} u(t) and M_{n} := max{μ_{0}, μ_{1}, ..., μ_{n}}  (8)

for n = 0, 1, 2, .... Then

u(t) ≤ M_{n} ( ∏_{ℓ=t−k_{n}(t)}^{t} a(ℓ)λ(ℓ) + ∑_{τ=t−k_{n}(t)}^{t}(1 − a(τ)λ(τ)) ∏_{ℓ=τ+1}^{t} a(ℓ)λ(ℓ))

= M_{n} ( ∏_{ℓ=t−k_{n}(t)}^{t} a(ℓ)λ(ℓ) + ∑_{τ=t−k_{n}(t)}^{t} Δ_{τ} (∏_{ℓ=τ}^{t} a(ℓ)λ(ℓ)))
= M_{n}.

The above inequality implies that

M_{n+1} ≤ M_{n} and u(t) = |y(t)| ≤ M_{0},

and the assertion of the theorem is valid.

The following corollary represents an asymptotic estimate for the solutions of Equation (1) and gives information about the rate of the convergence of solutions to particular cases such as t − p_{2} ≤ p(t) ≤ t − p_{1}, for real numbers 1 < p_{1} ≤ p_{2}. We obtain that the solutions are exponentially decaying as in the results given by Zhou and Yu in [6].

Corollary 1. Suppose that condition (H_{1}) holds. Let p_{1}, p_{2} and t_{0} be given real

numbers such that 1 ≤ p_{1} < p_{2} and t_{0} ≥ p_{1}. Let p(t) = t − δ(t), with given real function δ such that p_{1} ≤ δ(t) ≤ p_{2} for all t ≥ t_{0}. Suppose that there exists a real number λ > 1 such that

|b(t)| ≤ \frac{1 − λa(t)}{λ^{p_{2}}} for all t ≥ t_{0}.  (9)

Let x = x^{φ} be the solution of the initial value problem (1) and (2), and

M_{0} = sup_{t_{−1}≤t<t_{0}}{λ^{t}|φ(t)|}.

Then

|x(t)| ≤ \frac{M_{0}}{λ^{t}} for all t ≥ t_{0}.

Proof. The relations

t_{n+1} − δ(t_{n+1}) = t_{n}, t − p_{2} ≤ t − δ(t) ≤ t − p_{1}

imply that

t_{0} + np_{1} ≤ t_{n} ≤ t_{0} + np_{2} for n = 1, 2, ...

Introduce the transformation y(t) := x(t)λ^{t}. Let t ∈ [t_{n}, t_{n+1}) and τ ∈ T_{n}(t). Then Equation (1) is equivalent to

Δ_{τ} (y(τ − 1) ∏_{ℓ=t−k_{n}(t)}^{τ−1} \frac{1}{λa(ℓ)}) = b(τ)λ^{δ(τ)}y(τ − δ(τ)) ∏_{ℓ=t−k_{n}(t)}^{τ} \frac{1}{λa(ℓ)}.

Summing up both sides of this equality from t − k_{n}(t) to t gives that

y(t) = y(t − k_{n}(t) − 1) ∏_{ℓ=t−k_{n}(t)}^{t} λa(ℓ) +

+ ∑_{τ=t−k_{n}(t)}^{t} b(τ)λ^{δ(τ)}y(τ − δ(τ)) ∏_{ℓ=τ+1}^{t} λa(ℓ).

Define

μ_{n} := sup_{t_{n−1}≤t<t_{n}} |y(t)| and M_{n} := max{μ_{0}, μ_{1}, ..., μ_{n}}  (10)

for n = 0, 1, 2, .... Since |y(p(τ))| ≤ M_{n} for τ ∈ T_{n}(t) and t_{n} ≤ t < t_{n+1}, by using the summation by parts formula, it follows that

|y(t)| ≤ M_{n} ( ∏_{ℓ=t−k_{n}(t)}^{t} λa(ℓ) + ∑_{τ=t−k_{n}(t)}^{t}(1 − λa(τ)) ∏_{ℓ=τ+1}^{t} λa(ℓ))

= M_{n} ( ∏_{ℓ=t−k_{n}(t)}^{t} λa(ℓ) + ∑_{τ=t−k_{n}(t)}^{t} Δ_{τ} (∏_{ℓ=τ}^{t} λa(ℓ)))
= M_{n}.

The above inequality implies that M_{n+1} ≤ M_{n} and |y(t)| ≤ M_{0}. Therefore

|x(t)| ≤ \frac{M_{0}}{λ^{t}} for all t ≥ t_{0}

and the proof is complete.

4. Generalization

In this section we generalize previous results to the difference equations with several delays and show the usefulness of the new results to the classical particular cases.

Assume that t_{0} > 0 is a given real number, a, b_{i} : [t_{0}, ∞) → R are given real functions for i = 1, 2, ..., m. Let p_{i} : [t_{0}, ∞) → R be given functions such that p_{i}(t) ≤ t for every t ∈ [t_{0}, ∞], i = 1, 2, ..., m, and lim_{t→∞} p_{i}(t) = ∞ for i = 1, 2, ..., m. Consider the difference equation with several delays

x(t) = a(t)x(t − 1) + ∑_{i=1}^{m} b_{i}(t)x(p_{i}(t)).  (11)

Set

t_{−1} = min {t_{0} − 1, min_{1≤i≤m} {inf{p_{i}(s), s ≥ t_{0}}}} ,

t_{n} = min_{1≤i≤m} inf{s : p_{i}(s) > t_{n−1}} for all n = 1, 2, ...

Then

p_{i}(t) ∈ ⋃_{j=0}^{n}[t_{j−1}, t_{j}) for t_{n} ≤ t < t_{n+1},

i = 1, 2, ..., m and n = 0, 1, 2, ....

We shall need the following hypotheses.

• (H_{4}) a, b_{i} : [t_{0}, ∞) → R are real functions i = 1, 2, ..., m, such that 0 < a(t) < 1, for all t ≥ t_{0}.

• (H_{5}) p_{i} : [t_{0}, ∞) → R are real functions such that for every T > t_{0} there exists a δ > 0 such that p_{i}(t) ≤ t − δ for every t ∈ [t_{0}, T], i = 1, 2, ..., m, and lim_{t→∞} p_{i}(t) = ∞ for i = 1, 2, ..., m.

The next result gives an asymptotic estimate of solutions of Equation (11) and generalizes the result given in Theorem 1.

Theorem 2. Suppose that conditions (H_{4}) and (H_{5}) hold. Suppose that there is a

real function λ : [t_{0}, ∞) → (1, ∞) and there is an initial function φ in (2) such that

∑_{i=1}^{m} |b_{i}(t)|\frac{φ(p_{i}(t) − k_{0}(p_{i}(t)) − 1)}{φ(t − k_{0}(t) − 1)} ∏_{ℓ=t−k_{0}(t)}^{t} λ(ℓ) ∏_{ℓ=p_{i}(t)−k_{0}(p_{i}(t))}^{p_{i}(t)} \frac{1}{λ(ℓ)} ≤

≤ 1 − a(t)λ(t) for all t ≥ t_{0}.  (12)

Let x = x^{φ} be the solution of the initial value problem (11) and (2). Then

|x(t)| ≤ (φ(t − k_{0}(t) − 1) sup_{t_{−1}≤t≤t_{0}} λ(t)) ∏_{ℓ=t−k_{0}(t)}^{t} \frac{1}{λ(ℓ)} for all t ≥ t_{0}.

Proof. Introduce the transformation

y(t) := \frac{x(t)}{φ(t − k_{0}(t) − 1)} ∏_{ℓ=t−k_{0}(t)}^{t} λ(ℓ).

Then the function y(t) satisfies the equation

y(t) = a(t)λ(t)y(t − 1) +

+ ∑_{i=1}^{m} b_{i}(t)y(p_{i}(t))\frac{φ(p_{i}(t) − k_{0}(p_{i}(t)) − 1)}{φ(t − k_{0}(t) − 1)} ×

× ∏_{ℓ=t−k_{0}(t)}^{t} λ(ℓ) ∏_{ℓ=p_{i}(t)−k_{0}(p_{i}(t))}^{p_{i}(t)} \frac{1}{λ(ℓ)}.

Let t ∈ [t_{n}, t_{n+1}), τ ∈ T_{n}(t) and |y(t)| = u(t). Therefore, it follows that

u(t) ≤ u(t − k_{n}(t) − 1) ∏_{ℓ=t−k_{n}(t)}^{t} a(ℓ)λ(ℓ) +

+ ∑_{τ=t−k_{n}(t)}^{t} ∑_{i=1}^{m} b_{i}(t)u(p_{i}(τ))\frac{φ(p_{i}(t) − k_{0}(p_{i}(t)) − 1)}{φ(t − k_{0}(t) − 1)} ×

× ∏_{ℓ=t−k_{0}(t)}^{t} λ(ℓ) ∏_{ℓ=p_{i}(t)−k_{0}(p_{i}(t))}^{p_{i}(t)} \frac{1}{λ(ℓ)} ∏_{ℓ=τ+1}^{t} a(ℓ)λ(ℓ).

Using notation (8) and hypothesis (12), the same argumentation as in Theorem 1 completes the proof.

The next result is a special case of the previous theorem and generalizes the result given in Corollary 1.

Corollary 2. Suppose that condition (H_{4}) holds. For given real number t_{0} and for

given natural numbers p_{1}, p_{2},...,p_{m} such that 1 ≤ p_{1} < p_{2} < ... < p_{m} and t_{0} ≥ p_{1}, let p_{i}(t) = t − p_{i} for all t ≥ t_{0}, i = 1, 2, ..., m. Suppose that there exists a real number λ > 1 such that

∑_{i=1}^{m} |b_{i}(t)|λ^{p_{i}} ≤ 1 − λa(t) for all t ≥ t_{0}.  (13)

Let x = x^{φ} be the solution of the initial value problem (11) and (2). Then

|x(t)| ≤ \frac{M_{0}}{λ^{t}} for all t ≥ t_{0}.

Proof. Introduce the transformation y(t) := x(t)λ^{t}. Let t ∈ [t_{n}, t_{n+1}) and τ ∈ T_{n}(t). Then, Equation (11) is equivalent to

Δ_{τ} (y(τ − 1) ∏_{ℓ=t−k_{n}(t)}^{τ−1} \frac{1}{λa(ℓ)}) = ∑_{i=1}^{m} b_{i}(τ)λ^{p_{i}}y(τ − p_{i}) ∏_{ℓ=t−k_{n}(t)}^{τ} \frac{1}{λa(ℓ)}.

Summing up both sides of this equality from t − k_{n}(t) to t gives that

y(t) = y(t − k_{n}(t) − 1) ∏_{ℓ=t−k_{n}(t)}^{t} λa(ℓ) +

+ ∑_{τ=t−k_{n}(t)}^{t} ∑_{i=1}^{m} b_{i}(τ)λ^{p_{i}}y(τ − p_{i}) ∏_{ℓ=τ+1}^{t} λa(ℓ).

Using the notation (10) and the summation by parts formula, the same argumentation as in Corollary 1 completes the theorem.

Remark 2. Consider the difference equations of the form

x(t) = a(t)x(t − h) + b(t)x(p(t))

with h ∈ R_{+}. Then, using the transformation y(\frac{t}{h}) = \frac{1}{h}x(t) and s = \frac{t}{h}, we obtain the equation of the form (1) with the unknown function y(s), and the above results can be applied.

5. Acknowledgment

The author expresses special thanks to Dr. József Terjéki, professor at the Bolyai Institute, University of Szeged and Dr. István Győri, professor at the Department of Mathematics and Informatics, University of Veszprém, for valuable comments and help. The research is supported by Serbian Ministry of Science, Technology and Development for Scientific Research Grant no. 101835.

References

[1] I. Győri, G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press - Oxford, 1991.

[2] G. Makay, J. Terjéki, On the Asymptotic Behaviour of the Pantograph Equation, E. J. Qualitative Theory of Diff. Equ. No. 2 (1998), 1-12.

[3] H. Péics, Generalized Characteristic Equation of Linear Delay Difference Equations, Proceedings of the Second International Conference of Difference Equations and Applications, Veszprém, 1995, 499-505.

[4] H. Péics, Applications of Generalized Characteristic Equation of Linear Delay Difference Equations, Matematički Vesnik, Beograd, 50 (1998), 31-36.

[5] H. Péics, On the Asymptotic Behaviour of Difference Equations with Continuous Arguments, Dynamics of Continuous, Discrete and Impulsive Systems, Ser. A Math. Anal., 9, No. 2 (2002), 271-285.

[6] Z. Zhou, J. S. Yu, Decaying Solutions of Difference Equations with Continuous Arguments, Annals of Differential Equations, 14 No.3. (1998), 576-582.

(Received October 6, 2003)