Asymptotic Bounds for Linear Difference Systems

Abstract

This paper describes asymptotic properties of solutions of some linear difference systems. First we consider system of a general form and estimate its solutions by use of a solution of an auxiliary scalar difference inequality assuming that this solution admits certain properties. Then applying this result to linear difference systems of a variable order with constant (or bounded) coefficients we derive effective asymptotic criteria for such systems. Beside it, we give applications of these results to numerical analysis of vector differential equations with infinite lags.

1. Introduction

Stability and asymptotic investigation of linear difference equations is a conventional topic which covers the study of various types of these equations. In this paper, we consider the linear difference system:

(1.1)

where , , , and are given real matrices, and is nondecreasing, unbounded as , and satisfying

(1.2)

Equation (1.1) represents a general pattern, where some other specifications of entry parameters and their properties are necessary. Before doing this we note that many papers on asymptotics of difference equations involve difference systems fitting this pattern under special choices of parameters. For example, if is an asymptotically constant matrix and , are identically zero matrices, then (1.1) is of Poincaré type (studied, e.g., in [1–3] with respect to its asymptotic properties). Similarly, asymptotic stability of the system (1.1) is another frequent topic discussed in the framework of asymptotic analysis of difference systems. We can mention, for example, the papers [4, 5] or [6], where various particular cases of (1.1) have been investigated.

The primary goal of this paper does not consist in the generalization of these results. Although we are going to formulate the asymptotic bound of solutions for a system of the general form (1.1), we show that conditions of this criterion are satisfied especially when (1.1) is of a variable order which becomes infinite as (i.e., when as ). Such difference equations are studied only rarely (especially in the scalar case—see, e.g., [7]). Our wish is to describe some of their asymptotic properties.

The idea of our approach consists in a modification of suitable techniques developed for asymptotic analysis of differential equations with infinite lag (which are essentially continuous counterparts of difference equations of infinite order). Considering the scalar case, some general principles of this approach were initiated in [8] and applied in [9].

This paper is structured as follows. In Section 2, we mention some related mathematical tools which are necessary in our asymptotic investigation of (1.1). Section 3 presents an asymptotic bound of all solutions of (1.1). This bound is expressed via a solution of some auxiliary scalar difference inequality introduced in Section 2 and its validity requires some properties of this solution. In Section 4, we consider some particular cases of (1.1) and derive effective asymptotic formulae for their solutions following from the asymptotic result formulated in Section 3. Furthermore, we apply some of our results to the simplest (Euler) discretization of the vector pantograph equation and compare obtained result with corresponding asymptotic property of the exact pantograph equation.

2. Preliminaries

Let denote a vector norm on and let be an induced matrix norm. Throughout this paper we assume that the auxiliary linear system

(2.1)

is uniformly asymptotically stable (for a precise definition and related properties we refer to [10]). We recall here that the system (2.1) is uniformly asymptotically stable if and only if there exist real scalars and such that

(2.2)

where is the fundamental matrix of (2.1). Then along with the investigated difference system (1.1) we consider the scalar difference inequality:

(2.3)

This inequality formally arises from (1.1) and later we show that it can provide an upper bound sequence for any solution of (1.1). Before doing this we have to specify some properties of which turn out to be necessary for the validity of such estimates. We define the function by the relation

(2.4)

and the orbit , where means the th iterate of . Since the property (1.2) implies that for all , this orbit is unbounded (as ). Further, we introduce the operator

(2.5)

and assume that there exists a positive solution of (2.3) such that

(2.6)

(2.7)

The discussion on the existence of having such properties is performed in Section 4. Now we mention only an obvious fact, namely, that if is nondecreasing, then the assumptions (2.6) and (2.7) become trivial.

Finally, we consider the functional relation

(2.8)

known as the Abel equation (see, e.g., [11]). This relation enables us to set up an increasing sequence of the values starting from the value . In addition, we consider an arbitrary extension of this solution of (2.8) from to requiring

(2.9)

As we can see later, this sequence may also be involved in an upper bound term for solutions of (1.1).

3. Asymptotic Estimate for the Solutions of (1.1)

The goal of this section is to derive an asymptotic bound of solutions of the system (1.1). The following holds.

Theorem 3.1.

Consider the difference equation (1.1), where is nonsingular for all and assume that the auxiliary difference system (2.1) is uniformly asymptotically stable. Further, consider the inequality (2.3) with the real scalar given by (2.2) and let be a positive monotonous solution of (2.3) having the properties (2.6) and (2.7). Finally, let be a sequence satisfying (2.8) and (2.9). If is a solution of (1.1), then

(3.1)

where the positive real constant is given by (2.2).

Proof.

The vector provides a solution of the linear difference system:

(3.2)

If we multiply (3.2) by ( is the inverse matrix to the fundamental matrix of (2.1)) from the left and rewrite it with replaced by , we obtain

(3.3)

Indeed, applying the difference operator on the left-hand side of (3.3) and using we arrive at (3.2).

We define recursively and put , , (here we use the notation , where , ). Thus .

Let , . Summing the relation (3.3) from to we get

(3.4)

Notice that for any . Then considering appropriate norms and employing inequalities (2.2)-(2.3) we arrive at the estimate

(3.5)

where . Then summation by parts formula yields

(3.6)

Substituting this back into (3.5) we have

(3.7)

where we have used the property (2.6) and relations

(3.8)

Since this estimate holds for any , we can write

(3.9)

Then (2.7) implies that ; hence

(3.10)

for a suitable real and any , that is, any satisfying . Applying the function satisfying (2.8) and (2.9) to these inequalities we get

(3.11)

that is,

(3.12)

by the use of . To summarize this,

(3.13)

for a real constant and all . The property (3.1) is proved.

As we have remarked earlier, a discussion of conditions of Theorem 3.1 will be performed in the following section. We mention here only an immediate consequence of this assertion. If we choose a constant sequence (which obviously satisfies both assumptions (2.6) and (2.7)) and substitute it into (2.3), we get the following.

Corollary 3.2.

Consider the difference equation (1.1), where is nonsingular for all and assume that the auxiliary difference system (2.1) is uniformly asymptotically stable; that is, (2.2) holds with and . If and

(3.14)

then any solution of (1.1) is bounded.

4. Applications

The difference equations of the type (1.1) play a significant role in numerical analysis of linear differential equations with a delayed argument. Consider the equation

(4.1)

with continuous matrix functions , and continuous delay argument satisfying for all . Important numerical discretizations of (4.1) originate from the class of -methods. If we consider the mesh points (, is the stepsize) and means the approximation of the exact value , then the -method applied to (4.1) leads just to the difference equation (1.1), where , , and depend on , and . Since generally does not belong to the given mesh, we have to employ an interpolation (standard procedures are piecewise constant and piecewise linear interpolation). The choice of a suitable interpolation influences the value of parameter in (1.1); usual values are , , and . General references concerning the -methods and other discretizations of delay differential equations are provided by books [12, 13], where relevant formulae and some of their properties can be found.

We illustrate the previous considerations by the one-leg -method () with a piecewise linear interpolation applied to the pantograph equation:

(4.2)

where and are constant real matrices. The corresponding discretization becomes

(4.3)

where

(4.4)

, and for the well-posedness of the method we assume that is nonsingular. Then (4.3) is obviously the difference system of the type (1.1), where is constant matrix, are variable, but bounded matrix functions, ( means an integer part), and .

On this account, we are going to discuss the question of a possible usefulness of Theorem 3.1 for asymptotic description of such a system. First we state the following.

Lemma 4.1.

Consider the inequality (2.3), where , , and , , , . Let

(4.5)

Then

(4.6)

is the monotonous solution of (2.3) satisfying the properties (2.6) and (2.7).

Proof.

First we prove that is the solution of (2.3). Let . Then and substituting into (2.3) we have

(4.7)

The case can be verified analogously. Further, we show that satisfies (2.6) and (2.7). It is enough to consider the case only, because in the opposite case is identically zero. If , then

(4.8)

and (2.6) obviously holds with respect to . To verify (2.7) we utilize the relation

(4.9)

following from the definition of . Then using the mean value theorem we get

(4.10)

From here we can deduce that

(4.11)

Further, it follows from (4.9) that converges, and hence (2.7) holds.

We show that Lemma 4.1 enables us to formulate an effective asymptotic result for solutions of the discretization (4.3). As remarked earlier, this recurrence relation can be rewritten as the difference system (1.1) in a special form

(4.12)

with a constant matrix and bounded matrices . Let be the spectral radius of the matrix and assume that has a complete set of eigenvectors. Then the system (2.1) is uniformly asymptotically stable if and only if and the inequality (2.2) becomes

(4.13)

(see [10]). Similarly we can modify the relation (4.5) with being replaced by . It remains to dispose with the form of satisfying (2.8) and (2.9). Because of the inequality (4.9) we consider two auxiliary Abel equations with a continuous argument, namely,

(4.14)

We can easily check that the functions

(4.15)

where are arbitrary real constants, satisfy these equations on corresponding domains. Taking into account the property (2.9) we can see that asymptotics of satisfying (2.8) and (2.9) are described via . This observation is sufficient for our purpose.

Summarizing previous considerations, we have derived the following.

Corollary 4.2.

Consider the difference equation (4.12), where is a nonsingular constant matrix with a complete set of eigenvectors such that and let , , . If is a solution of (4.12), then

(4.16)

where , and is given by (4.13).

Remark 4.3.

The previous ideas and procedures can be extended into a more general case. Consider the differential equation (4.2) with the delayed argument replaced by a general such that is continuous and nonincreasing on and for all . Then using the above mentioned discretization we arrive at (4.12) with being replaced by having some specific properties (in particular, as ). Employing the Schröder transformation , where is a differentiable solution of the relation , we can similarly as in the case verify that the property (4.16) remains preserved with being replaced by on its right-hand side. A detailed modification of the above utilized technique to this general case is only a technical matter and we omit it.

We recall that the asymptotic estimate (4.16) follows from the general Theorem 3.1. Our next intention is to show that considering a simplified form of (4.12) we can modify the proof of Theorem 3.1 to obtain a stronger estimate of solutions.

Let and in (4.12) (a numerical interpretation of such an equation is discussed later). If the matrix has a complete set of eigenvectors, then it is diagonalizable. Therefore we can choose the basis such that is diagonal and consider the difference system

(4.17)

where , , and are constant (possibly complex) matrices. To formulate a stronger estimate of its solutions, instead of the scalar inequality (2.3) we employ the vector inequality

(4.18)

and discuss the existence of a positive solution of (4.18). We note that the symbols , mean matrices , and the inequality (4.18) between two vectors represents inequalities between their corresponding coordinates. Similarly, by a positive (nonnegative) vector or matrix we understand such a vector or matrix whose all coordinates or elements are positive (nonnegative).

First we recall the following notion.

Definition 4.4.

We say that matrix is reducible if there exists a partition of the set into two complementary systems and () such that whenever and . Otherwise is called irreducible.

Irreducibility plays an important role in the spectral theory of nonnegative matrices. We utilize here the following assertion which is often referred to as the Frobenius theorem (see, e.g., [14]).

Theorem 4.5.

Every irreducible nonnegative matrix has a positive eigenvalue such that the corresponding eigenvector is positive. This eigenvalue is the spectral radius of and it is the simple root of the characteristic equation.

If , we rewrite the inequality (4.18) as

(4.19)

where . In addition, if is irreducible, then is irreducible and nonnegative. Hence, by the Frobenius theorem, has the eigenvalue with the corresponding positive eigenvector . Using this notation we have the following.

Proposition 4.6.

Let be a diagonal matrix with , and let be irreducible. Then the inequality (4.18) admits the positive solution:

(4.20)

Proof.

Let . Then all coordinates of are nondecreasing and we can write

(4.21)

It suffices to show that

(4.22)

which is equivalent to

(4.23)

The required spectral properties of now follow from the Frobenius theorem.

The case can be proved quite analogously.

Utilizing Proposition 4.6 we can prove the following.

Theorem 4.7.

Consider the difference system (4.17), where and are constant matrices such that , and is irreducible. Then

(4.24)

holds for any solution of (4.17).

Proof.

The proof technique is a modification of the procedures employed in the proof of Theorem 3.1. Therefore we outline a sketch of it. The system (4.17) can be rewritten as

(4.25)

If , we can divide the th equation from (4.25) by to obtain

(4.26)

that is,

(4.27)

Let the symbols , , and have the same meaning as in the proof of Theorem 3.1 (with given by (2.4), where and ). If we choose , , and sum (4.27) from to , we arrive at

(4.28)

Put , where is given by (4.20). Then

(4.29)

hence

(4.30)

by the use of Proposition 4.6. This inequality is just the inequality (3.5). Consequently, using the same proof procedures as given in the corresponding part of the proof of Theorem 3.1 and taking into account Lemma 4.1, we can show the boundedness of as .

If , then

(4.31)

and Proposition 4.6 implies

(4.32)

The boundedness of as now follows from the corresponding properties of .

Remark 4.8.

Our final goal is to give a numerical interpretation to Theorem 4.7. Consider again the pantograph equation (4.2) with a (generally complex) diagonal matrix . Applying the forward Euler method with a piecewise constant interpolation to such an equation (4.2) we obtain the linear difference system (4.17), where and . Since if and only if , we can reformulate Theorem 4.7 as follows.

Corollary 4.9.

Consider (4.2), where and are constant matrices such that , and is irreducible. Let be an approximate value of calculated via the recurrence relation (4.17), where , , and is the stepsize. Then

(4.33)

In particular, if is a real diagonal matrix with negative diagonal elements, is nonnegative (note that this is actually a stepsize restriction), and is a nonnegative irreducible matrix, then (4.33) becomes

(4.34)

It should be emphasized that exactly the same (nonimprovable) asymptotic estimate holds for the solution of the exact pantograph equation (4.2) provided that all eigenvalues of have negative real parts and the maximal geometric multiplicity of eigenvalues of with the maximal real part is equal to 1 (see [15, 16]). Consequently, the estimate (4.34) of the corresponding numerical solution is optimal in this sense.

In general, the problem of a possible mutual relationship between qualitative properties of delay differential and difference equations is not restricted only to numerical investigations and it is studied in a more general context (see, e.g., the papers [17, 18]). However, it is just the numerical analysis of delay differential equations which gives an important motivation for the joint study of both types of delay equations. Related questions concerning joint investigations of delay differential equations and its discretizations in the framework of the time-scale theory are discussed in [8, 19]. Among many papers on numerical investigations of differential equations with a proportional delay we can mention at least [9, 20–22] and the references cited therein. We emphasize that these papers deal with scalar equations of the pantograph type. The investigations of the vector case, mentioned in this paper, are just at the beginning.