PIECEWISE LINEAR APPROXIMATIONS OF SOLUTIONS OF GENERALIZED LINEAR DIFFERENTIAL EQUATIONS

(Dedicated to the memory of Mikhail Drakhlin)

Zdeněk Halas and Milan Tvrdý

Institute of Mathematics, Academy of Sciences of the Czech Republic, Praha, Czech Republic

Abstract. This contribution deals with systems of linear generalized linear differential equations of the form

where is an -complex matrix valued function, is an -comp-lex vector valued function, and have bounded variation on The integrals are understood in the Kurzweil-Stieltjes sense.

Our aim is to present some new results on the continuous dependence of solutions to linear generalized differential equations on parameters and initial data. In particular, we generalize in several aspects the known result by Ashordia. Our main goal consists in the more general notion of a solution to the given system. In particular, both and need not be of bounded variation on and, in general, they can be regulated functions.

1. INTRODUCTION

Starting with Kurzweil [8], generalized differential equations has been extensively studied by many authors, like e.g. Hildebrandt [6], Schwabik, Tvrdý and Vejvoda [14]-[16], [18]-[20], Hönig [7], Ashordia [2], [3]. In particular, see the monographs [16], [14], [19] and [7] and the references therein. Moreover, during several recent decades, the inte-rest in their special cases like equations with impulses or discrete systems increased considerably, cf. e.g. monographs [10], [22], [4], [13] or [1].

Importance of generalized linear differential equations with regulated solutions consists in that they enable us to treat in a unified way both continuous and discrete systems and, in addition, also systems with fast oscillating data.

Throughout the paper we keep the following notation: As usual, is the set of natural numbers and stands for the set of complex

numbers. is the space of complex matrices of the type and For a matrix , the elements of are denoted by

and the norm of is defined by

The symbols and stand respectively for the identity and the zero matrix of the proper type. For an -matrix , denotes its determinant.

If then by and we denote the corresponding closed and open intervals, respectively. Furthermore, and are the corresponding half-open intervals. When the intervals and occur, they are understood to be empty.

For a function we set The set is called a subdivision of if If,

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for each and each the function possesses in the limits

we say that the function is regulated on the interval The set of all -matrix valued functions regulated on the interval is denoted by Furthermore, we denote for and

for It is known that, for each the set of all points of its discontinuity on the interval is at

most countable. Moreover, for each there are at most finitely many points such that and at most finitely many points such that

Clearly, each function regulated on is bounded on i.e. for

For a function we denote by its variation over We say that has a bounded variation on if The set of -complex matrix valued functions of bounded variation on is denoted by By we denote the set of functions such that each component of is absolutely continuous on the interval

Similarly, stands for the set of functions that are continuous on Analogously to the spaces of functions of bounded variation,

and Obviously,

and Finally, a function is called a finite step function on if there is a division of such that is constant on every interval The set of all finite step functions on

is denoted by is the set of all -matrix valued functions whose arguments are finite step functions and It is known that the set is dense in with respect to the supremum norm, i.e.

(1.1)

2. KURZWEIL-STIELTJES INTEGRAL

The integrals which occur in this paper are the Perron-Stieltjes ones. For the original definition, see A.J. Ward [21] or S. Saks [12] . We use the equivalent summation definition due to J. Kurzweil [8] (cf. also e.g. [9] or [16]). We call this integral the Kurzweil-Stieltjes integral, in short KS-integral. For the reader's convenience, let us recall the properties of the KS-integral needed later.

It is well-known that the KS-integral exists if and Taking into account [17, Theorem 2.8] or [19, Theorem 2.3.8], we can

state the following crucial existence assertion.

Theorem 2.1. If and at least one of the functions has a bounded variation on then the integral exists. Furthermore,

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(2.1)and

In particular, if and then

Further basic properties of the KS-integral with respect to scalar regulated functions were described in [17] (see also [19]).

Given a - matrix valued function and a - matrix valued function defined on and such that all the integrals

exist, the symbol (or simply ) stands for the -matrix with the entries

The extension of the results obtained in [17] or [18] for scalar real valued functions to complex vector valued or matrix valued functions is obvious and hence for the basic facts concerning integrals with respect to regulated functions we refer to the corresponding assertions from [17] or [18]. The next natural assertion will be very useful for our purposes and, to our opinion, it is not available in the literature.

Lemma 2.2. Let for and let

(2.2)

(2.3)

(2.4)Then

Proof. Let be given. By (1.1), (2.2) and (2.4), we can find and such that

Furthermore, since using (2.1) we can see that, for and the relations

wherefrom our assertion immediately follows. □

3. GENERALIZED LINEAR DIFFERENTIAL EQUATIONS

Let and Consider an integral equation (3.1)

We say that a function is a solution of (3.1) on the interval if the integral has a sense and equality (3.1) is satisfied for all

The equation (3.1) is usually called a generalized linear differential equation. Such equations with solutions having values in the space of real -vectors have been thoroughly investigated e.g. in the monographs [14] or [16]. In this

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section we will describe the basics needed later. A special attention is paid to the features whose extension to the complex case seems not to be so straightforward.

For our purposes the following property is crucial (3.2)

(Remind that we put Its importance is well illustrated by the next assertion which is a fundamental existence result for equation (3.1).

Theorem 3.1. [18, Proposition 2.5]. Let satisfy (3.2) Then, for each and each equation (3,1) has a unique solution on and Moreover,

Furthermore, analogously to [16, Theorem III.1.7] where we have:

Theorem 3.2. Let satisfy (3.2). Then the relations (3.3)

and

(3.4)hold for each and each solution of on

Proof. First, notice that for such that , we have

Therefore, (3.3) follows easily from the fact that the set

has at most finitely many elements. Now, let be a solution of (3.1). Put and otherwise. Then, as in the proof of [16, Theorem

III.1.7], we have

Consequently,

and

where and for The function is nondecreasing and, since is left-continuous on is also left-continuous

on Therefore we can use the generalized Gronwall inequality (see e.g. [16, Lemma I.4.30] or [14, Corollary 1.43]) to get the estimate (3.4).

Corollary 3.3. Let satisfy (3.2). Then for and each solu-tion of (3,1) on the estimate

(3.5)is true, where is defined by (3.3).Proof. By (3.4), we have which inserted

into the inequality yields estimate (3.5). □

Lemma 3.4. Let satisfy (3.2) and let be defined by (3.3) Then

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(3.6)Proof. We have

Thus, it remains to prove that the inequality (3.7)

is true, as well. To this aim, first, let us notice that for each there is a such that and

(3.8)Indeed, let and let Let be such that

and let be such that for Then Furthermore,

On the other hand, .

Therefore, we can conclude that (3.8) is true.Now, due to (3.2), there is such that Inserting this

instead of into (3.8) and having in mind that we get

It follows that

i.e. (3.7) is true. This completes the proof □

4. CONTINUOUS DEPENDENCE OF A SOLUTION ON PARAMETER

The main result of this paper is provided by the next theorem. It concerns continuous dependence of solutions of generalized linear differential equations on a parameter and generalizes that due to M. Ashordia [2, Theorem 1]. Unlike [2] and [3], we do not utilize

the variation-of-constants formula and therefore we need not assume that, in addition to (3.2), also the condition for all is satisfied. Furthermore, both the nonhomogeneous part of the equation and solutions may be regulated functions (not necessarily of bounded variation).

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Theorem 4.1. Let for Assume (2.3), (2.4), (3.2),

(4.1)

(4.2) Then, equation has a unique solution on Furthermore, for each

sufficiently large, there exists a unique solution on to the equation (4.3)

and (2.2) is true. Proof. Step 1 . As in the first part of the proof of [2, Theorem~1], we can show that

there is a such that for all nd all In particular, (4.3) has a unique solution for

Step 2 . For put

Then, by Lemma 3.4, we have

Since, due to assumption (2.4), we conclude that there is a  such that

To summarize (4.4)

Step 3 . Set Then, for

where

By (4.1) and (4.2), we can see that(4.5)

Furthermore, since and we have

and, by Lemma 2.2, To summarize,

(4.6)On the other hand, using Theorem 3.2 and taking into account the relation (4.4),

we get

wherefrom, with respect to (4.5) and (4.6), the relation follows. Finally, having in mind assumptions (4.1) and (4.2), we conclude that the relation

is true, as well. This completes the proof. □

Notation 4.2. For a denote by a division of given by

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(4.7)

For given and we define

(4.8)

(4.9)

Then, obviously, and and (4.10)

The equations (4.3) with and given by (4.8) and (4.9) are just initial value problems for linear ordinary differential systems

(4.11)In this view, next theorem says that the solutions of (3.1) can be uniformly

approximated by solutions of linear ordinary differential equations provided the functions and are continuous.

Theorem 4.3. Assume that and Let and be such that (4.2) holds. Furthermore, let the sequence of divisions of the interval be given by (4.7) and let

be defined by (4.8) and (4.9), respectively.Then, equation (3.1) has a unique solution on Furthermore, for each equation (4.11) has a solution on and (2.2) holds.

Proof. Step 1 . Since is uniformly continuous on we have

(4.12)

Let and let be given. Furthermore, let

Then and, according to (4.7), (4.8) and (4.12), we get for

As was chosen independently of we can conclude that (2.4) is true.Step   2 . Analogously we can show that (4.1) holds for and

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Step   3 . By (4.10), condition (2.3) is satisfied. Moreover, as and are continuous, by Theorem 3.1, equations (3.1) and (4.3), have unique solutions and we can complete the proof using Theorem 4.1. □

Remark 4.4. Piecewise linear approximations (4.8) and (4.9) of can be constructed also in a general case, when need not be continuous. Some results for homogeneous equations were in this direction obtained by M. Pelant in his unpublished thesis [11]. If and/or are not continuous on the situation is rather more complicated and, in general, the solutions of (4.3) do not converge to the solution of (3.1), but to solutions of a somewhat modified equation. Nevertheless, using matrix logarithms, one can always define a sequence (4.3) of approximating initial value problems for ODE's whose solutions tend to the solution of (3.1). Of course, even if and are real valued, such approximating piecewise linear functions will be, in general, complex valued. We suppose to expose this theme in more details later.

ACKNOWLEDGEMENTS

This work is supported by the grant No. A100190703 of the Grant Agency of the Academy of Sciences of the Czech Republic and by the Institutional Research Plan No. AV0Z10190503

REFERENCES

1. Agarwal R.P., Bohner M.,Grace S.R. and O'Regan D.: Discrete oscillation theory. New York, NY:Hindawi Publishing Corporation, 2005.

2. Ashordia M.: 'On the correctness of linear boundary value problems for systems of generalized ordinary differential equations'. Proceedings of the Georgian Academy of Sciences. Math. 1 (1993), No. 4, 385-394.

3. Ashordia M.: 'Criteria of correctness of linear boundary value problems for systems of generalized ordinary differential equations'. Czechoslovak Math. J. 46 (1996), 385-404.

4. Bainov D. and Simeonov P.: Impulsive differential equations:periodic solutions and applications. Pitman Monographs and Surveys in Pure and Applied Mathematics 66, Longman Scientific, Harlow, 1993.

5. Boichuk A.A., Perestyuk N.A. and Samoilenko A.M.: 'Periodic solutions of impul-sive differential equations in critical cases' (in Russian). Differ. Uravn. 27 (1991), 1516-1521.

6. Hildebrandt T.H.: 'On systems of linear differentio-Stieltjes integral equations'. Illinois J. Math. 3 (1959) 352-373.

7. Hönig Ch.S.: Volterra Stieltjes-Integral Equations. North Holland and American Elsevier, Mathematics Studies 16, Amsterdam and New York, 1975.

8. Kurzweil J.: 'Generalized ordinary differential equation and continuous dependence on a parameter'. Czechoslovak Math. J. 7 (1957), 418-449.

9. Kurzweil J.: Nichtabsolute konvergente Integrale. BSB B.G. Teubner Verlagsgessel-schaft, Leipzig, 1980.

10. Pandit S.G. and Deo S.G.: Differential Equations Involving Impulses. Lecture Notes in Mathematics 954, Springer-Verlag, Berlin, 1982.

11. Pelant M.: 'On aproximations of solutions of generalized differential equations' (in Czech). PhD. thesis, Charles University, Faculty of Mathematics and Physics

7-8

(Prague), 1997. 12. Saks S: Theory of the Integral. Monografie Matematyczne, Warszawa, 1937. 13. Samoilenko A.M. and Perestyuk A.N.: Impulsive differential equations. World

Scie-ntific Series on Nonlinear Science. Series A. 14. Singapore:World Scientific. 1995.

14. Schwabik Š.: Generalized Ordinary Differential Equations. World Scientific, Singa-pore, 1992.

15. Schwabik Š. and Tvrdý M.: 'On the generalized linear ordinary differential equation'. Časopis pěst. mat. 98 (1973), 206-211.

16. Schwabik Š., Tvrdý M. and Vejvoda O.: Differential and Integral Equations: Boun-dary Value Problems and Adjoint. Academia & Reidel, Praha and Dordrecht, 1979.

17. Tvrdý M.: 'Regulated functions and the Perron-Stieltjes integral. Časopis pěst. mat. 114 (1989), 187-209.

18. Tvrdý M.: 'Generalized differential equations in the space of regulated functions (Boundary value problems and controllability)'. Math. Bohem. 116 (1991), 225-244.

19. Tvrdý M.: Differential and Integral Equations in the Space of Regulated Functions. Mem. Differential Equations Math. Phys. 25 (2002), 1-104.

20. Vejvoda O. and Tvrdý M.: 'Existence of solutions to a linear integro-boundary-diffe-rential equation with additional conditions'. Ann.Mat.Pura Appl. 89 (1971) 169-216.

21. Ward A.J.: The Perron-Stieltjes integral. Math. Z. 41 (1936), 578-604. 22. Zavalishchin S.G. and Sesekin A.N.: Dynamic impulse systems:theory and

applica-tions. Mathematics and its Applications (Dordrecht). 394. Dordrecht: Kluwer, 1997.

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