doi:10.1155/2010/182696
Research Article
Asymptotic Bounds for Linear Difference Systems
Jan ˇ
Cerm ´ak
Institute of Mathematics, Brno University of Technology, Technick´a 2, 61669 Brno, Czech Republic
Correspondence should be addressed to Jan ˇCerm´ak,[email protected]
Received 1 February 2010; Accepted 12 April 2010
Academic Editor: Josef Diblik
Copyrightq2010 Jan ˇCerm´ak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 constantor boundedcoefficients 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:
yn1 Anyn
p
k0
Bknyαn k, n∈Nn0, 1.1
whereNn0 {n0, n01, . . .},n0 ∈N,p∈N,AnandBknare givenm×mreal matrices,
andα:Nn0 → Zis nondecreasing, unbounded asn → ∞, and satisfying
αn p < n ∀n∈Nn0. 1.2
special choices of parameters. For example, ifAnis an asymptotically constant matrix and
Bkn, k 0,1, . . . , p are identically zero matrices, then1.1is of Poincar´e type studied,
e.g., in1–3 with respect to its asymptotic properties. Similarly, asymptotic stability of the system1.1is 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 of1.1have 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
1.1is of a variable order which becomes infinite asn → ∞i.e., whenn−αn → ∞as
n → ∞. Such difference equations are studied only rarelyespecially 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 in8 and applied in9 .
This paper is structured as follows. InSection 2, we mention some related mathemat-ical 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 inSection 2and 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 inSection 3. Furthermore, we apply some of our results to the simplest
Eulerdiscretization 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 onRmand let · be an induced matrix norm. Throughout this
paper we assume that the auxiliary linear system
yn1 Anyn 2.1
is uniformly asymptotically stablefor a precise definition and related properties we refer to
10 . We recall here that the system2.1is uniformly asymptotically stable if and only if there exist real scalarsM >0 and 0< μ <1 such that
YnY−1s≤Mμn−s ∀n≥s∈Nn0, 2.2
whereYnis the fundamental matrix of2.1. Then along with the investigated difference system1.1we consider the scalar difference inequality:
p
k0
Bknωαn k≤
This inequality formally arises from 1.1 and later we show that it can provide an upper bound sequence for any solutionyn of1.1. Before doing this we have to specify some properties ofωnwhich turn out to be necessary for the validity of such estimates. We define the functionβ:Nn0 → Nn0by the relation
βn supw∈Nn0, αw p≤n
2.4
and the orbitΩn0 {βjn0, j∈N}, whereβjmeans thejth iterate ofβ. Since the property 1.2implies thatβn> nfor alln∈Nn0, this orbit is unboundedasj → ∞. Further, we
introduce the operator
Δ−ωn Δωn− |Δωn|
2 2.5
and assume that there exists a positive solutionωnof2.3such that
Δ−ωn≤Δ−ωn1, ∀n∈Nn
0, 2.6
n∈Ωn0
Δ−ωn
ωβn >−∞. 2.7
The discussion on the existence of ωn having such properties is performed inSection 4. Now we mention only an obvious fact, namely, that if ωn is nondecreasing, then the assumptions2.6and2.7become trivial.
Finally, we consider the functional relation
ϕβnϕn 1, n∈Ωn0 2.8
known as the Abel equationsee, e.g.,11 . This relation enables us to set up an increasing sequence of the valuesϕβjn
0starting from the valueϕn0. In addition, we consider an
arbitrary extension of this solutionϕnof2.8fromΩn0toNn0requiring
ϕn≤ϕn1, n∈Nn0. 2.9
As we can see later, this sequence may also be involved in an upper bound term for solutions
ynof1.1.
3. Asymptotic Estimate for the Solutions of
1.1
The goal of this section is to derive an asymptotic bound of solutions of the system1.1. The following holds.
Theorem 3.1. Consider the difference equation1.1, whereAnis nonsingular for alln ∈Nn0
monotonous solution of 2.3having the properties2.6and2.7. Finally, letϕnbe a sequence satisfying2.8and2.9. Ifynis a solution of1.1, then
yn OωnMϕn asn−→ ∞, 3.1
where the positive real constantMis given by2.2.
Proof. The vectorzn yn/ωnprovides a solution of the linear difference system:
employing inequalities2.2-2.3we arrive at the estimate
Substituting this back into3.5we have
where we have used the property2.6and relations
n−1
Applying the functionϕnsatisfying2.8and2.9to these inequalities we get
ϕβn0 ≤ϕn≤ϕ
β1n0 , 3.11
that is,
by the use ofϕβn
0 ϕn0 . To summarize this,
|zn| ≤LMϕn 3.13
for a real constantL >0 and alln∈Nn0. The property3.1is 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 sequenceωn which obviously satisfies both assumptions
2.6and2.7and substitute it into2.3, we get the following.
Corollary 3.2. Consider the difference equation1.1, whereAnis nonsingular for alln∈Nn0
and assume that the auxiliary difference system2.1is uniformly asymptotically stable; that is,2.2
holds with0< μ <1andM >0. IfM≤1and
p
k0
Bkn ≤1−μ, 3.14
then any solutionynof 1.1is bounded.
4. Applications
The difference equations of the type1.1play a significant role in numerical analysis of linear differential equations with a delayed argument. Consider the equation
xt Ctxt Dtxτt, t∈I t0,∞ 4.1
with continuous matrix functionsC,Dand continuous delay argumentτ satisfyingτt < t
for all t > t0. Important numerical discretizations of 4.1 originate from the class of Θ
-methods. If we consider the mesh points tn t0 nh n ∈ N, h > 0 is the stepsize
andynmeans the approximation of the exact valuextn, then theΘ-method applied to 4.1 leads just to the difference equation 1.1, where An, Bkn, and αn depend on
Ct,Dt and τt. Since τtn generally does not belong to the given mesh, we have to
employ an interpolationstandard procedures are piecewise constant and piecewise linear interpolation. The choice of a suitable interpolation influences the value of parameterpin
1.1; usual values arep0,p1, andp2. General references concerning theΘ-methods and other discretizations of delay differential equations are provided by books12,13 , where relevant formulae and some of their properties can be found.
We illustrate the previous considerations by the one-legΘ-method0≤Θ≤1with a piecewise linear interpolation applied to the pantograph equation:
whereCandDare constant real matrices. The corresponding discretization becomes
On this account, we are going to discuss the question of a possible usefulness of
Theorem 3.1for asymptotic description of such a system. First we state the following.
Lemma 4.1. Consider the inequality 2.3, where0 < μ < 1, Bkn ≤ bk, andαn λn,
is the monotonous solution of 2.3satisfying the properties2.6and2.7.
and2.6obviously holds with respect tor <0. To verify2.7we utilize the relation
λ−1n−p−1< βn< λ−1n1 4.9
following from the definition ofβn. Then using the mean value theorem we get
−Δ−ωn n 1
From here we can deduce that
−Δ−ωn
We show that Lemma 4.1enables 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 system1.1in a special form
yn1 Ayn
the matrixAand assume thatAhas a complete set of eigenvectors. Then the system2.1is uniformly asymptotically stable if and only ifρA<1 and the inequality2.2becomes
YnY−1s≤MρAn−s ∀n≥s∈Nn0 4.13
see10 . Similarly we can modify the relation4.5withμbeing replaced byρA. It remains to dispose with the form ofϕnsatisfying2.8and2.9. Because of the inequality4.9we consider two auxiliary Abel equations with a continuous argument, namely,
ϕ∗
λ−1t1 ϕ∗t 1,
ϕ∗λ−1t−p−1 ϕ∗t 1.
4.14
We can easily check that the functions
whereC1, C2are arbitrary real constants, satisfy these equations on corresponding domains.
Taking into account the property2.9we can see that asymptotics of ϕnsatisfying2.8
and2.9are described via−logλn. 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 Ais a nonsingular constant matrix with a complete set of eigenvectors such thatρA < 1 and let Bkn ≤ bk,n ∈ Nn0,k
0,1, . . . , p. Ifynis a solution of4.12, then
yn OnrM−logλn as n−→ ∞, 4.16
whererlogλ1−ρA/b,bpk0bkandMis given by4.13.
Remark 4.3. The previous ideas and procedures can be extended into a more general case. Consider the differential equation4.2with the delayed argumentτt λtreplaced by a generalτtsuch thatτtis continuous and nonincreasing onI and 0< τt ≤ λ <1 for allt∈I. Then using the above mentioned discretization we arrive at4.12withλnbeing replaced byαnhaving some specific propertiesin particular,n−αn → ∞asn → ∞. Employing the Schr ¨oder transformationuηt, whereηtis a differentiable solution of the relationητt ληt, we can similarly as in the caseαn λnverify that the property
4.16remains preserved with nbeing replaced byηnon 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 estimate4.16follows from the generalTheorem 3.1. Our next intention is to show that considering a simplified form of4.12we can modify the proof ofTheorem 3.1to obtain a stronger estimate of solutions.
Letp 0 andB0n Bin4.12 a numerical interpretation of such an equation is
discussed later. If the matrixAhas a complete set of eigenvectors, then it is diagonalizable. Therefore we can choose the basis such thatAis diagonal and consider the difference system
yn1 Ayn Byλn, 0< λ <1, n∈Nn0, 4.17
wherey yi,A diagai, andB bijare constant possibly complexmatrices. To
formulate a stronger estimate of its solutions, instead of the scalar inequality2.3we employ the vector inequality
|B|ψλn≤I− |A|ψn 4.18
and discuss the existence of a positive solutionψ ψiof4.18. We note that the symbols |A|, |B| mean matrices|A| diag|ai|,|B| |bij|and the inequality4.18between two
vectors represents minequalities between their corresponding coordinates. Similarly, by a positivenonnegative vector or matrix we understand such a vector or matrix whose all coordinates or elements are positivenonnegative.
Definition 4.4. We say thatm×mmatrixQ qijis reducible if there exists a partition of the
set{1,2, . . . , m}into two complementary systems{i1, . . . , iu}and{j1, . . . , jv}uvmsuch
thatqij 0 wheneveri∈ {i1, . . . , iu}andj ∈ {j1, . . . , jv}. OtherwiseQis 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 Q has a positive eigenvalue such that the corresponding eigenvector is positive. This eigenvalue is the spectral radius ofQand it is the simple root of the characteristic equation.
IfρA<1, we rewrite the inequality4.18as
Qψλn≤ψn, 4.19
whereQ I−|A|−1|B|. In addition, ifBis irreducible, thenQis irreducible and nonnegative. Hence, by the Frobenius theorem,Qhas the eigenvalueρQwith the corresponding positive eigenvectorξ. Using this notation we have the following.
Proposition 4.6. LetAdiagaibe a diagonal matrix with|ai|<1, i1, . . . , mand letB bij
be irreducible. Then the inequality4.18admits the positive solution:
ψn
⎧ ⎨ ⎩
ξρQ−logλn ifρQ≥1,
ξρQ−logλn1/1−λ ifρQ<1.
4.20
Proof. LetρQ≥1. Then all coordinates ofψnare nondecreasing and we can write
|B|ψλn |B|ξρQ−logλλn ≤ |B|ξρQ−logλλn. 4.21
It suffices to show that
|B|ξρQ−logλλn I− |A|ξρQ−logλn 4.22
which is equivalent to
QξρQξ. 4.23
The required spectral properties ofQnow follow from the Frobenius theorem. The caseρQ<1 can be proved quite analogously.
Theorem 4.7. Consider the difference system4.17, whereAdiagaiandB bijare constant
matrices such that|ai|<1,i1, . . . , mandBis irreducible. Then
yn On−logλρQ , Q I− |A|−1|B| as n−→ ∞ 4.24
holds for any solutionynof 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 system4.17can be rewritten as
by the use ofProposition 4.6. This inequality is just the inequality3.5. Consequently, using the same proof procedures as given in the corresponding part of the proof ofTheorem 3.1and taking into accountLemma 4.1, we can show the boundedness ofSas → ∞.
Ifai0, then
zin
1
ψin m
j1
bijψjλn−1zjλn−1, 4.31
andProposition 4.6implies
|zin| ≤
S
ψin m
j1
bijψjλn−1≤S
ψin−1
ψin .
4.32
The boundedness ofSas → ∞now follows from the corresponding properties ofψin.
Remark 4.8. Our final goal is to give a numerical interpretation toTheorem 4.7. Consider again the pantograph equation4.2with agenerally complexdiagonal matrixC. Applying the forward Euler method with a piecewise constant interpolation to such an equation4.2we obtain the linear difference system4.17, whereAIhCandBhD. Since|ai|<1 if and
only if 2 Reci<−h|ci|2, we can reformulateTheorem 4.7as follows.
Corollary 4.9. Consider4.2, whereC diagciandD dijare constant matrices such that
2 Reci < −h|ci|2, i 1, . . . , m and D is irreducible. Letyn be an approximate value of xnh
calculated via the recurrence relation4.17, whereAIhC,BhD, andh >0is the stepsize. Then
yn On−logλρQ , Q I− |IhC|−1h|D|as n−→ ∞. 4.33
In particular, ifCis a real diagonal matrix with negative diagonal elements,I hC
is nonnegative note that this is actually a stepsize restriction, and D is a nonnegative irreducible matrix, then4.33becomes
yn On−logλρC−1D asn−→ ∞. 4.34
It should be emphasized that exactly the samenonimprovableasymptotic estimate holds for the solutionxtof the exact pantograph equation4.2provided that all eigenvalues of
Chave negative real parts and the maximal geometric multiplicity of eigenvalues ofC−1D
with the maximal real part is equal to 1see15,16 . Consequently, the estimate4.34of the corresponding numerical solution is optimal in this sense.
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 least9,20–22 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.
Acknowledgments
The author is grateful to an anonymous referee for numerous suggestions and comments which considerably helped to improve the content of this paper. The research was supported by the research plan MSM 0021630518 “Simulation modelling of mechatronic systems” of the Ministry of Education, Youth and Sports of the Czech Republic and by the Grant no. 201/08/0469 of the Czech Grant Agency.
References
1 S. N. Elaydi, “Asymptotics for linear difference equations. I. Basic theory,” Journal of Difference Equations and Applications, vol. 5, no. 6, pp. 563–589, 1999.
2 G. Farkas, “On asymptotics of solutions of Poincar´e difference systems,”Journal of Difference Equations and Applications, vol. 7, no. 2, pp. 183–191, 2001.
3 M. Pituk, “Asymptotic behavior of a nonhomogeneous linear recurrence system,” Journal of Mathematical Analysis and Applications, vol. 267, no. 2, pp. 626–642, 2002.
4 M. Kipnis and D. Komissarova, “Stability of a delay difference system,” Advances in Difference Equations, Article ID 31409, p. 9, 2006.
5 H. Matsunaga, “A note on asymptotic stability of delay difference systems,”Journal of Inequalities and Applications, no. 2, pp. 119–125, 2005.
6 Y. Nagabuchi, “Asymptotic behaviors of a linear difference system with multiple delays,”Funkcialaj Ekvacioj, vol. 46, no. 1, pp. 89–119, 2003.
7 S. Heller, “An asymptotic solution of the difference equationan1−anan/2,”Journal of Mathematical Analysis and Applications, vol. 34, pp. 464–469, 1971.
8 J. ˇCerm´ak, P. Kundr´at, and M. Urb´anek, “Delay equations on time scales: essentials and asymptotics of the solutions,”Journal of Difference Equations and Applications, vol. 14, no. 6, pp. 567–580, 2008.
9 J. ˇCerm´ak and J. J´ansk ´y, “On the asymptotics of the trapezoidal rule for the pantograph equation,”
Mathematics of Computation, vol. 78, no. 268, pp. 2107–2126, 2009.
10 S. N. Elaydi,An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 2005.
11 M. Kuczma, B. Choczewski, and R. Ger,Iterative Functional Equations, vol. 32 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 1990.
12 A. Bellen and M. Zennaro,Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation, The Clarendon Press Oxford University Press, New York, NY, USA, 2003.
13 J. Kuang and Y. Cong,Stability of Numerical Methods For Delay Differential Equations, Science Press, Beijing, China, 2005.
14 F. R. Gantmacher,Matrix Theory, Nauka, Moscow, Russia, 4th edition, 1988.
15 A. Iserles, “On the generalized pantograph functional-differential equation,” European Journal of Applied Mathematics, vol. 4, no. 1, pp. 1–38, 1993.
16 Y. Liu, “Asymptotic behavior of functional-differential equations with proportional time delays,”
European Journal of Applied Mathemtics, vol. 7, pp. 11–30, 1996.
17 D. B. Benditkis, J. Dibl´ık, and D. Ya. Khusainov, “Weak delay in systems with an aftereffect,”Functional Differential Equations, vol. 9, no. 3-4, pp. 385–404, 2002.
19 J. ˇCerm´ak and M. Urb´anek, “On the asymptotics of solutions of delay dynamic equations on time scales,”Mathematical and Computer Modelling, vol. 46, no. 3-4, pp. 445–458, 2007.
20 M. Buhmann and A. Iserles, “Stability of the discretized pantograph differential equation,”
Mathematics of Computation, vol. 60, no. 202, pp. 575–589, 1993.
21 M. Z. Liu and D. Li, “Properties of analytic solution and numerical solution of multi-pantograph equation,”Applied Mathematics and Computation, vol. 155, no. 3, pp. 853–871, 2004.
