Asymptotic Representation of the Solutions of Linear Volterra Difference Equations
© Győri and L. Horváth. 2008
Received: 26 February 2008
Accepted: 4 April 2008
Published: 15 April 2008
This article analyses the asymptotic behaviour of solutions of linear Volterra difference equations. Some sufficient conditions are presented under which the solutions to a general linear equation converge to limits, which are given by a limit formula. This result is then used to obtain the exact asymptotic representation of the solutions of a class of convolution scalar difference equations, which have real characteristic roots. We give examples showing the accuracy of our results.
The literature on the asymptotic theory of the solutions of Volterra difference equations is extensive, and application of this theory is rapidly increasing to various fields. For the basic theory of difference equations, we choose to refer to the books by Agarwal , Elaydi , and Kelley and Peterson . Recent contribution to the asymptotic theory of difference equations is given in the papers by Kolmanovskii et al. , Medina , Medina and Gil , and Song and Baker ; see [8–19] for related results.
In our case, we split the sum in (1.1) only into two terms, and the condition (1.5) is not assumed. In fact, we show an example in Section 4, where (1.5) does not hold and hence in [20, Theorem 3.1] is not applicable. At the same time our main theorem gives a limit formula. It is also interesting to note that our proof is simpler than it was applied in .
In the paper  (see also ), it is shown that if satisfies (1.8) and (1.9), and the initial sequence is suitable, then for the solution of (1.6) and (1.7) the sequence , is bounded. Furthermore, some extra conditions guarantee that the limit is finite and satisfies a limit formula.
In our paper, we improve considerably the result in . First, we give explicit necessary and sufficient conditions for the existence of a for which (1.8) and (1.9) are satisfied. Second, we prove the existence of the limit and give a limit formula for under the condition only . These two statements are formulated in our second main theorem stated in Section 3. The proof of the existence of is based on our first main result.
The article is organized as follows. In Section 2, we briefly explain some notation and definitions which are used to state and to prove our results. In Section 3, we state our two main results, whose proofs are relegated to Section 5.
Our theory is illustrated by examples in Section 4, including an interesting nonconvolution equation. This example shows the significance of the middle sum in (1.3), since only this term contributes to the limit of the solution of (1.1) in this case.
2. Mathematical Preliminaries
In this section, we briefly explain some notation and well-known mathematical facts which are used in this paper.
Let be the set of integers, and . stands for the set of all -dimensional column vectors with real components and is the space of all by real matrices. The zero matrix in is denoted by , and the identity matrix by . Let be the matrix in whose elements are all . The absolute value of the vector and the matrix is defined by and , respectively. The vector and the matrix is nonnegative if and , , respectively. In this case, we write and . can be endowed with any norms, but they are equivalent. A vector norm is denoted by and the norm of a matrix in induced by this vector norm is also denoted by . The spectral radius of the matrix is given by , which is independent of the norm employed to calculate it.
A partial ordering is defined on by letting if and only if . The partial ordering enables us to define the , and so forth for the sequences of vectors and matrices, which can also be determined componentwise and elementwise, respectively. It is known that for , and if and .
3. The Main Results
Here, we assume
Now, we are in a position to state our first main result.
By a solution of the Volterra difference equation (3.7) we mean a sequence satisfies (3.7) for any .
It can be easily seen that for any initial sequence , (3.7) has exactly one solution satisfying (3.8). This unique solution is denoted by and it is called the solution of the initial value problem (3.7), (3.8).
Now, we are ready to state our second result which will be proved in Section 5. This result shows that the implicit condition (A) and the explicit condition (B) are equivalent and the solutions of (3.7) can be asymptotically characterized by as .
4. Examples and the Discussion of the Results
In this section, we illustrate our results by examples and the interested reader could also find some discussions.
Our Theorem 3.1 is given for system of equations, however the next example shows that this result is also new even in scalar case.
Then, it can be easily seen that problem (4.1), (4.2) is equivalent to problem (3.1), (3.2).
and hence in [20, Theorem 3.1] is not applicable.
5. Proofs of the Main Theorems
5.1. Proof of Theorem 3.1
To prove Theorem 3.1 we need the next result from .
Now, we prove some lemmas.
thus (5.2) is satisfied.
Since the matrix is invertible, which shows the uniqueness part of the lemma. On the other hand, by Lemma 5.1 we have that is a bounded sequence, and hence is finite. Thus, is well defined and satisfies (5.28). The proof is complete.
Now, we prove Theorem 3.1.
and hence the proof of Theorem 3.1 is complete.
5.2. Proof of Theorem 3.3
Theorem 3.3 will be proved after some preparatory lemmas.
whenever , and this shows . Finally, we consider the case . Then, follows from the condition . (d) The series of functions (5.58) can be differentiated term-by-term within , and therefore , . Together with (c) this gives the claim. (e) We have only to apply (d), (c), and (b). The proof is complete.
We are now in a position to prove Theorem 3.3.
is finite and satisfies the required relation (3.22). The proof is now complete.
and this is a contradiction.
This work was supported by Hungarian National Foundation for Scientific Research Grant no. K73274.
- Agarwal RP: Difference Equations and Inequalities: Theory, Methods, and Application, Monographs and Textbooks in Pure and Applied Mathematics. Volume 228. 2nd edition. Marcel Dekker, New York, NY, USA; 2000:xvi+971.Google Scholar
- Elaydi S: An Introduction to Difference Equations, Undergraduate Texts in Mathematics. 3rd edition. Springer, New York, NY, USA; 2005:xxii+539.Google Scholar
- Kelley WG, Peterson AC: Difference Equations: An Introduction with Applications. Academic Press, Boston, Mass, USA; 1991:xii+455.MATHGoogle Scholar
- Kolmanovskii VB, Castellanos-Velasco E, Torres-Muñoz JA: A survey: stability and boundedness of Volterra difference equations. Nonlinear Analysis: Theory, Methods & Applications 2003, 53(7-8):861-928. 10.1016/S0362-546X(03)00021-XMATHView ArticleGoogle Scholar
- Medina R: Asymptotic behavior of Volterra difference equations. Computers & Mathematics with Applications 2001, 41(5-6):679-687. 10.1016/S0898-1221(00)00312-6MATHMathSciNetView ArticleGoogle Scholar
- Medina R, Gil MI: The freezing method for abstract nonlinear difference equations. Journal of Mathematical Analysis and Applications 2007, 330(1):195-206. 10.1016/j.jmaa.2006.07.074MATHMathSciNetView ArticleGoogle Scholar
- Song Y, Baker CTH: Admissibility for discrete Volterra equations. Journal of Difference Equations and Applications 2006, 12(5):433-457. 10.1080/10236190600563260MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP, Pituk M: Asymptotic expansions for higher-order scalar difference equations. Advances in Difference Equations 2007, 2007:-12.Google Scholar
- Berezansky L, Braverman E: On exponential dichotomy, Bohl-Perron type theorems and stability of difference equations. Journal of Mathematical Analysis and Applications 2005, 304(2):511-530. 10.1016/j.jmaa.2004.09.042MATHMathSciNetView ArticleGoogle Scholar
- Bodine S, Lutz DA: Asymptotic solutions and error estimates for linear systems of difference and differential equations. Journal of Mathematical Analysis and Applications 2004, 290(1):343-362. 10.1016/j.jmaa.2003.09.068MATHMathSciNetView ArticleGoogle Scholar
- Bodine S, Lutz DA: On asymptotic equivalence of perturbed linear systems of differential and difference equations. Journal of Mathematical Analysis and Applications 2007, 326(2):1174-1189. 10.1016/j.jmaa.2006.03.070MATHMathSciNetView ArticleGoogle Scholar
- Driver RD, Ladas G, Vlahos PN: Asymptotic behavior of a linear delay difference equation. Proceedings of the American Mathematical Society 1992, 115(1):105-112. 10.1090/S0002-9939-1992-1111217-0MATHMathSciNetView ArticleGoogle Scholar
- Elaydi S, Murakami S, Kamiyama E: Asymptotic equivalence for difference equations with infinite delay. Journal of Difference Equations and Applications 1999, 5(1):1-23. 10.1080/10236199908808167MATHMathSciNetView ArticleGoogle Scholar
- Elaydi S: Asymptotics for linear difference equations. II. Applications. In New Trends in Difference Equations. Taylor & Francis, London, UK; 2002:111-133.Google Scholar
- Graef JR, Qian C: Asymptotic behavior of a forced difference equation. Journal of Mathematical Analysis and Applications 1996, 203(2):388-400. 10.1006/jmaa.1996.0387MATHMathSciNetView ArticleGoogle Scholar
- Győri I: Sharp conditions for existence of nontrivial invariant cones of nonnegative initial values of difference equations. Applied Mathematics and Computation 1990, 36(2):89-111. 10.1016/0096-3003(90)90014-TMathSciNetView ArticleGoogle Scholar
- Kolmanovskii V, Shaikhet L: Some conditions for boundedness of solutions of difference Volterra equations. Applied Mathematics Letters 2003, 16(6):857-862. 10.1016/S0893-9659(03)90008-5MATHMathSciNetView ArticleGoogle Scholar
- Li ZH: The asymptotic estimates of solutions of difference equations. Journal of Mathematical Analysis and Applications 1983, 94(1):181-192. 10.1016/0022-247X(83)90012-4MATHMathSciNetView ArticleGoogle Scholar
- Trench WF: Asymptotic behavior of solutions of Poincaré recurrence systems. Computers & Mathematics with Applications 1994, 28(1–3):317-324.MATHMathSciNetView ArticleGoogle Scholar
- Applelby JAD, Győri I, Reynolds DW: On exact convergence rates for solutions of linear systems of Volterra difference equations. Journal of Difference Equations and Applications 2006, 12(12):1257-1275. 10.1080/10236190600986594MATHMathSciNetView ArticleGoogle Scholar
- Philos ChG, Purnaras IK: The behavior of solutions of linear Volterra difference equations with infinite delay. Computers & Mathematics with Applications 2004, 47(10-11):1555-1563. 10.1016/j.camwa.2004.06.007MATHMathSciNetView ArticleGoogle Scholar
- Philos ChG, Purnaras IK: On linear Volterra difference equations with infinite delay. Advances in Difference Equations 2006, 2006:-28.Google Scholar
- Győri I, Horváth L: Limit theorems for discrete sums and convolutions. submittedGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.