Representation of Solutions of Difference Equations with Continuous Time

Hajnalka Péics †

University of Novi Sad Faculty of Civil Engineering Subotica Department of Mathematics 24000 Subotica, Kozara 2/A, Yugoslavia

Abstract. This paper describes a representation of solutions of the system of nonautonomous functional equations x(t) = A(t)x(t − 1) + B(t)x(p(t)), in form of series, using the Cauchy matrix of the linear system y(t) = A(t)y(t − 1). A representation of analitical solutions of the equation x(t) = ax(t − 1) + bx(pt) with constant coefficients is also investigated.

AMS No. 39A11, 39B22 KEYWORDS: difference equations with continuous time, asymptotic behaviour

1. Introduction

Consider the system of difference equations with continuous time

x(t) = A(t)x(t − 1) + B(t)x(p(t)),  (1)

where x(t) is an n-dimensional column vector, A(t) = (a_{ij}(t)) and B(t) = (b_{ij}(t)) are n×n real matrix functions and p(t) is a nonnegative real function, such that lim_{t→∞} p(t) = ∞ and for every T > t_{0} there exists a δ > 0 such that p(t) ≤ t − δ for every t ∈ [t_{0}, T]. The purpose of this paper is to obtain a series representation for the solutions of system (1), which can be applied to study the asymptotic behaviour of solutions. The equation with constant coefficients is also investigated because it is important in its own right. Similar problems were studied for the pantograph differential equation of the form

ẋ(t) = A(t)x(t) + B(t)x(p(t)).

For the differential equation with constant coefficients the well-known Dirichlet series solution is given. The reader interested in this topic can consult Carr and Dyson [2], Fox, Mayers, Ockendon and Taylor [3], Kato and McLeod [4] and Terjéki [7].

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

Let t_{0} be a positive real number and set

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

By a solution of (1) we mean an n-dimensional column vector function x where the components x_{i}(t), i = 1, ..., n, are defined for t ≥ t_{−1} and satisfy the system (1) for t ≥ t_{0}. For a given vector valued function ϕ(t), where the real components ϕ_{i}, i = 1, ..., n are given on t_{−1} ≤ t < t_{0}, the system (1) has a unique solution x^{ϕ} satisfying the initial condition

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

Fix a point t such that t ≥ t_{0}, and define natural number k_{0}(t) such that

k_{0}(t) := [t − t_{0}].

Then,

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

and set

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

For a given real number t and for a given positive integer n use the notation

t^{[n]} = t(t − 1)(t − 2)...(t − n + 1).

The Cauchy matrix of the initial value problem

y(t) = A(t)y(t − 1), t ≥ t_{0},  (3)
y(t) = ϕ(t), t_{0} − 1 ≤ t < t_{0}  (4)

is W (τ; t), where

W (τ; t) = A(t)A(t − 1)...A(τ + 1)

for t ≥ t_{0}, τ ∈ T_{0}(t), with W (t; t) = E and the n-dimensional unite matrix E.

2. Main Results

First of all we prove a simple but fundamental result.

Theorem 1. Let y_{0}(t) denote the solution of the initial value problem (3) and (4) with ϕ(t) ≢ 0 for t_{−1} ≤ t < t_{0}, and the sequence {y_{n}(t), n = 1, 2, ...} is defined by

y_{n}(t) = A(t)y_{n}(t − 1) + B(t)y_{n−1}(p(t)), t ≥ t_{0},
y_{n}(t) ≡ 0, t_{−1} ≤ t < t_{0}, n = 1, 2, ...

Then

x(t) = ∑_{n=0}^{∞} y_{n}(t)  (5)

is a solution of the initial value problem (1) and (2). Moreover, this series is finite on every finite subinterval of [t_{0}, ∞).

Proof. First we show that the series (5) is absolutely convergent on [t_{0}, ∞). Define

M(F, T) := sup_{t_{0}≤t≤T} ||F (t)||

for any matrix or vector function F for T > t_{0} and

M(W, T) := sup_{t_{0}≤τ≤t≤T} ||W (τ; t)||

for the Cauchy matrix W (τ; t). Since for t ≥ t_{0}

y_{n}(t) = ∑_{τ=t−k_{0}(t)}^{t} W (τ; t)B(τ)y_{n−1}(p(τ)), n = 1, 2, ...

hence, for T > t_{0} and t_{0} ≤ t ≤ T, the following inequality holds:

||y_{n}(t)|| ≤ ∑_{τ=t−k_{0}(t)}^{t} M(W, t)M(B, t)||y_{n−1}(p(τ))||.

By using mathematical induction we will show that

y_{n}(t) = 0 for t_{0} ≤ t < t_{0} + (n − 1)δ.  (6)

For n = 2 we have

||y_{2}(t)|| ≤ M(W, T)M(B, T) ∑_{τ=t−k_{0}(t)}^{t} ||y_{1}(p(τ))||.

For t_{0} ≤ t < t_{0} + δ we have p(t) < t_{0} and y_{1}(p(t)) = 0. Therefore,

y_{2}(t) = 0 for t_{0} ≤ t < t_{0} + δ.

Suppose that statement (6) is valid for n = k and prove it for n = k + 1. Then

||y_{k+1}(t)|| ≤ M(W, T)M(B, T) ∑_{τ=t−k_{0}(t)}^{t} ||y_{k}(p(τ))||.

For t_{0} ≤ t < t_{0} + kδ we have p(t) ≤ t − δ < t_{0} + (k − 1)δ and by the inductional hypothesis y_{k}(p(t)) = 0, and so y_{k+1}(t) = 0.

Then, exists a natural number N such that

y_{m}(t) = 0 for all m ≥ N and t_{0} ≤ t ≤ T.

Therefore,

x(t) = ∑_{n=0}^{N−1} y_{n}(t) for t_{0} ≤ t ≤ T

and the convergence is clear. Moreover,

x(t) = ∑_{n=0}^{∞} y_{n}(t) = y_{0}(t) + ∑_{n=1}^{∞} y_{n}(t) =

= A(t)y_{0}(t − 1) + ∑_{n=1}^{∞} A(t)y_{n}(t − 1) + ∑_{n=1}^{∞} B(t)y_{n−1}(p(t)) =

= A(t) ∑_{n=0}^{∞} y_{n}(t − 1) + B(t) ∑_{n=0}^{∞} y_{n}(p(t)) =

= A(t)x(t − 1) + B(t)x(p(t)),

and the proof is complete. In the space of vector or matrix functions f(t) let the operators S_{p} and W^{∗} be defined by

S_{p}f(t) = f(p(t)), W^{∗}f(t) = ∑_{τ=t−k_{0}(t)}^{t} W (τ; t)f(τ).

Then

y_{n} = W^{∗}(BS_{p}y_{n−1}) = (W^{∗}BS_{p})^{n}y_{0}, n = 1, 2, ...

Therefore Theorem 1 implies the next corollary. Corollary 1. The unique solution of the initial value problem (1) and (2) is given by

x(t) = ∑_{n=0}^{∞}(W^{∗}BS_{p})^{n}W (t − k_{0}(t) − 1; t)ϕ(t − k_{0}(t) − 1).  (7)

In the next result we give conditions garanteeing that series (7) is absolutely and uniformly convergent on the interval [t_{0}, ∞).

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 21, p. 4 Theorem 2. Suppose that there exist positive constants M, b and a such that 0 < a < 1 and there exists a positive scalar function f(t) such that

sup_{t_{−1}≤θ≤t_{0}} ∑_{τ=θ}^{∞} f(τ) = f_{0} < ∞,

||W (τ; t)|| ≤ Ma^{t}, t_{0} − 1 ≤ τ ≤ t_{0} ≤ t,  (8)

||W (τ; t)|| ≤ Ma^{t−τ} , t_{0} ≤ τ ≤ t,

||B(t)|| ≤ b + f(t), t ≥ t_{0},

M (\frac{b}{1 − a} + f_{0}) < 1.  (9)

Then series (7) is absolutely and uniformly convergent for t ≥ t_{0}. If, in addition, there exists a positive constant p such that

0 < pt ≤ p(t) for t ≥ t_{0},  (10)

then the solution of the initial value problem (1) and (2) tends to zero, as t → ∞.

Proof. Let p_{0} be a real number such that

0 ≤ p_{0} < 1 and p_{0}t ≤ p(t).

Introduce the sequence {γ_{n}} as follows.

γ_{0} := 1, γ_{n} := Mγ_{n−1} (\frac{b}{1 − a^{1−p_{0}^{n}}} + f_{0}) , n = 1, 2, ...

In virtue of (9) it is easy to see that the series

∑_{n=0}^{∞} γ_{n}

is finite. Let

y_{0}(t) = W (t − k_{0}(t) − 1; t)ϕ(t − k_{0}(t) − 1)

and let {y_{n}(t)} be defined as in Theorem 1. We assert that

||y_{n}(t)|| ≤ Mγ_{n}a^{p_{0}^{n}t}||ϕ||, n = 0, 1, 2, ...  (11)

Inequality (8) implies assertion (11) for n = 0. Suppose that (11) is true for n − 1. Then

||y_{n}(t)|| ≤ ∑_{τ=t−k_{0}(t)}^{t} ||W (τ; t)||||B(τ)||||y_{n−1}(p(τ))||

≤ ∑_{τ=t−k_{0}(t)}^{t} M^{2}a^{t−τ} (b + f(τ))γ_{n−1}a^{p_{0}^{n−1}p(τ)}||ϕ||

≤ M^{2}a^{t}γ_{n−1}||ϕ|| ∑_{τ=t−k_{0}(t)}^{t} a^{(p_{0}^{n}−1)τ} (b + f(τ))

= M^{2}a^{t}γ_{n−1}||ϕ|| ( ∑_{τ=t−k_{0}(t)}^{t} ba^{(p_{0}^{n}−1)τ} + ∑_{τ=t−k_{0}(t)}^{t} a^{(p_{0}^{n}−1)τ} f(τ))

≤ M^{2}a^{t}γ_{n−1}||ϕ|| (\frac{b}{a^{p_{0}^{n}−1} − 1}a^{(p_{0}^{n}−1)τ} |_{t−k_{0}(t)}^{t+1} + a^{(p_{0}^{n}−1)t} ∑_{τ=t−k_{0}(t)}^{t} f(τ))

≤ M^{2}a^{t}γ_{n−1}||ϕ|| (\frac{b}{a^{p_{0}^{n}−1} − 1}a^{(p_{0}^{n}−1)(t+1)} + a^{(p_{0}^{n}−1)t}f_{0})

= M^{2}γ_{n−1}||ϕ||a^{p_{0}^{n}t} (\frac{ba^{p_{0}^{n}−1}}{a^{p_{0}^{n}−1} − 1} + f_{0})

= M^{2}γ_{n−1}||ϕ||a^{p_{0}^{n}t} (\frac{b}{1 − a^{1−p_{0}^{n}}} + f_{0})

= Mγ_{n}a^{p_{0}^{n}t}||ϕ||,

and (11) is true for all positive integers n. It means that (7) is absolutely and uniformly convergent on [t_{0}, ∞) and the first part of the theorem is proved. If (10) is satisfied then we can choose p_{0} = p and for all ϵ > 0 we can find an integer N such that

2M ∑_{n=N}^{∞} γ_{n} < ϵ.

Then

||x(t)|| ≤ ∑_{n=0}^{∞} ||y_{n}(t)|| ≤ ∑_{n=0}^{∞} Mγ_{n}a^{p^{n}t}||ϕ|| ≤

≤ (M ∑_{n=N}^{∞} γ_{n} + M ∑_{n=0}^{N−1} a^{p^{N−1}t}γ_{n}) ||ϕ|| < ϵ||ϕ||,

if t is so large that

2Ma^{p^{N−1}t} ∑_{n=0}^{N−1} γ_{n} < ϵ.

This proves the second part of the theorem.

If we apply the above results to the scalar equation with constant coefficients

x(t) = ax(t − 1) + bx(pt),  (12)

where a, b, p are real constants such that 0 < a < 1 and 0 < p < 1, the form of the functions y_{n}(t) will be too complicate, not suitable for further investigation. Therefore, to solve Equation (12) by this method, we need a computer. But we can obtain a nice series representation form for the analitical solutions of Equation (12). Of course, it is neccessary for the initial function to be analitical.

Theorem 3. Let C_{0} ≠ 0 be a given real number. Let a, b, p be real numbers such that 0 < a < 1, 0 < p < 1 and |b| < 1 − a. Then

x(t) = ∑_{n=0}^{∞} C_{0}b^{n} ∏_{ℓ=1}^{n}(1 − a^{1−p^{ℓ}})^{−1}a^{p^{n}t}

is a series solution of Equation (12) on [t_{0}, ∞).

Proof. Suppose that a solution of Equation (12) is the series of the form

x(t) = ∑_{n=0}^{∞} C_{n}λ^{p^{n}t}.

Replacing this form in Equation (12) we obtain that

∑_{n=0}^{∞} C_{n}λ^{p^{n}t} = a ∑_{n=0}^{∞} C_{n}λ^{p^{n}(t−1)} + b ∑_{n=0}^{∞} C_{n}λ^{p^{n+1}t},

and therefore,

C_{0}λ^{t} + ∑_{n=1}^{∞} C_{n}λ^{p^{n}t} = \frac{a}{λ}C_{0}λ^{t} + ∑_{n=1}^{∞} \frac{a}{λ^{p^{n}}} C_{n}λ^{p^{n}t} + ∑_{n=1}^{∞} bC_{n−1}λ^{p^{n}t}.

From the above equality follows that

C_{0} (1 − \frac{a}{λ}) = 0, so a = λ, C_{0} ≠ 0.

C_{1} = C_{1}a^{1−p} + bC_{0},
C_{2} = C_{2}a^{1−p^{2}} + bC_{1},
...
C_{n} = C_{n}a^{1−p^{n}} + bC_{n−1},
...

Using mathematical induction we obtain that

C_{n} = \frac{b^{n}C_{0}}{(1 − a^{1−p})(1 − a^{1−p^{2}})...(1 − a^{1−p^{n}})}, n = 1, 2, ...

Then the series solution of Equation (12) is of the form

x(t) = ∑_{n=0}^{∞} C_{0}b^{n} ∏_{ℓ=1}^{n}(1 − a^{1−p^{ℓ}})^{−1}a^{p^{n}t}.

From the above argumentation follows that the necessary and sufficient condition for the convergence is

|b| < 1 − a.

3. Acknowledgements

The author is very grateful to professor József Terjéki (Attila József University, Szeged, Hungary) for valuable comments and help. This research was completed during the visit of the author to the Bolyai Institute at the József Attila University under a fellowship supported by the Hungarian Ministry of Education.

4. References

[1] M. G. Blizorukov, On the Construction of Solutions of Linear Difference Systems with Continuous Time, Differ. Uravn. 32 (1996), No.1, 127-128, translation in Differential Equations 32 (1996), No.1, 133-134.

[2] J. Carr, J. Dyson, The Functional Differential Equation y′(x) = ay(λx) + by(x), Proc. Roy. Soc. Edinburgh, 74A, 13 (1974/75), 165-174.

[3] L. Fox, D. F. Mayers, J. R. Ockendon, A. B. Tayler, On a Functional Differential Equation, J. Inst. Maths Applics, 8 (1971), 271-307.

[4] T. Kato, J. B. McLeod, The Functional Differential Equation y′(x) = ay(λx)+by(x), Bull. Amer. Math. Soc. 77 (1971), 891-937.

[5] H. Péics, On the Asymptotic Behaviour of Difference Equations with Continuous Arguments, Dynamics of Continuous, Discrete and Impulsive Systems (to appear).

[6] H. Péics, On the Asymptotic Behaviour of Solutions of a System of Functional Equations, Mathematica Periodica Hungarica (to appear).

[7] G. P. Pelyukh, A Certain Representation of Solutions to Finite Difference Equations with Continuous Argument, Differ. Uravn. 32 (1996), No.2, 256-264, translation in Differential Equations 32 (1996), No.2, 260-268.

[8] J. Terjéki, Representation of the Solutions to Linear Pantograph Equations, Acta Sci. Math. (Szeged) 60 (1995), 705-713.