On some classes of difference equations of infinite order
- Alexander V Vasilyev^{1} and
- Vladimir B Vasilyev^{2}Email author
https://doi.org/10.1186/s13662-015-0542-3
© Vasilyev and Vasilyev 2015
Received: 4 March 2015
Accepted: 17 June 2015
Published: 10 July 2015
Abstract
We consider a certain class of difference equations on an axis and a half-axis, and we establish a correspondence between such equations and simpler kinds of operator equations. The last operator equations can be solved by a special method like the Wiener-Hopf method.
Keywords
difference equation symbol solvabilityMSC
39B32 42A381 Introduction
Difference equations of finite order arise very often in various problems in mathematics and applied sciences, for example in mathematical physics and biology. The theory for solving such equations is very full for equations with constant coefficients [1, 2], but fully incomplete for the case of variable coefficients. Some kinds of such equations were obtained by the second author by studying general boundary value problems for mode elliptic pseudo differential equations in canonical non-smooth domains, but there is no solution algorithm for all situations [3–5]. There is a certain intermediate case between the two mentioned above, namely it is a difference equation with constant coefficients of infinite order. Here we will briefly describe these situations.
Further, such equations can be equations with a continuous variable or a discrete one, and this property separates such an equation on a class of properly difference equations and discrete equations. In this paper we will consider the case of a continuous variable x, and a solution on the right-hand side will be considered in the space \(L_{2}({\mathbb {R}})\) for all equations.
1.1 Difference equation of a finite order with constant coefficients
1.2 Difference equation of infinite order with constant coefficients
Lemma 1
The operator \(\mathcal{D}\) is a linear bounded operator \(L_{2}({\mathbb {R}})\to L_{2}({\mathbb{R}})\) if \(\{a_{k}\}_{-\infty}^{+\infty}\in {l}^{1}\).
Proof
The proof of this assertion can be obtained immediately. □
1.3 Difference and discrete equations
Obviously there are some relations between difference and discrete equations. Particularly, if \(\{\beta_{k}\}_{-\infty}^{+\infty }={\mathbb{Z}}\), then the operator (5) is a discrete convolution operator. For studying discrete operators in a half-space the authors have developed a certain analytic technique [9–11]. Below we will try to enlarge this technique for more general situations.
2 General difference equations
For studying this equation we will use methods of the theory of multi-dimensional singular integral and pseudo differential equations [3, 6, 12] which are non-usual in the theory of difference equations. Our next goal is to study multi-dimensional difference equations, and this one-dimensional variant is a model for considering other complicated situations. This approach is based on the classical Riemann boundary value problem and the theory of one-dimensional singular integral equations [13–15].
2.1 Background
2.2 Topological barrier
2.3 Difference equations on a half-axis
To describe a solving technique for (11) we recall the following (cf. [13, 14]).
Definition
Example 1
The boundary values of the integral \(\Phi(z)\) satisfy the Plemelj-Sokhotskii formulas [13, 14], and thus the projectors P and Q are corresponding projectors on the spaces of analytic functions [15].
Theorem 2
Proof
Remark 1
Remark 2
The condition \(\sigma(\xi)\in C(\dot{\mathbb{R}})\) is not a strong restriction. Such symbols exist for example in the case that \(\sigma (\xi)\) is represented by a finite sum, and \(\beta_{k}\in{\mathbb {Q}}\). Then \(\sigma(\xi)\) is a continuous periodic function.
3 General solution
Since \(\sigma(\xi)\in C(\dot{\mathbb{R}})\), and Indσ is an integer, we consider the case \(\ae\equiv\operatorname{Ind} \sigma\in{\mathbb{N}}\) in this section.
Theorem 3
Proof
Remark 3
This result does not depend on the choice of the continuation l.
Corollary 4
4 Solvability conditions
Theorem 5
Proof
But there is some inaccuracy. Indeed, this solution belongs to the space \(A_{+}({\mathbb{R}})\), but more exactly it belongs to its subspace \(A^{k}_{+}({\mathbb{R}})\). This subspace consists of functions analytic in \({\mathbb{C}}_{+}\) with zeros of the order −æ in the point \(z=i\). To obtain a solution from \(L_{2}({\mathbb{R}}_{+})\) we need some corrections in the last formula. Since the operator P is related to the Cauchy type integral we will use certain decomposition formulas for this integral (see also [12–14]).
5 Conclusion
It seems this approach to difference equations may be useful for studying the case that the variable x is a discrete one. We have some experience in the theory of discrete equations [9–11], and we hope that we can be successful in this situation also. Moreover, in our opinion the developed methods might be applicable for multi-dimensional difference equations.
Declarations
Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions. This work is supported by Russian Fund of Basic Research and government of Lipetsk region of Russia, project No. 14-41-03595-a.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Milne-Thomson, LM: The Calculus of Finite Differences. Chelsea, New York (1981) MATHGoogle Scholar
- Jordan, C: Calculus of Finite Differences. Chelsea, New York (1950) Google Scholar
- Vasil’ev, VB: Wave Factorization of Elliptic Symbols: Theory and Applications. Introduction to the Theory of Boundary Value Problems in Non-Smooth Domains. Kluwer Academic, Dordrecht (2000) View ArticleGoogle Scholar
- Vasilyev, VB: General boundary value problems for pseudo differential equations and related difference equations. Adv. Differ. Equ. 2013, 289 (2013) MathSciNetView ArticleGoogle Scholar
- Vasilyev, VB: On some difference equations of first order. Tatra Mt. Math. Publ. 54, 165-181 (2013) MathSciNetGoogle Scholar
- Mikhlin, SG, Prößdorf, S: Singular Integral Operators. Akademie Verlag, Berlin (1986) View ArticleGoogle Scholar
- Sobolev, SL: Cubature Formulas and Modern Analysis: An Introduction. Gordon & Breach, Montreux (1992) MATHGoogle Scholar
- Dudgeon, DE, Mersereau, RM: Multidimensional Digital Signal Processing. Prentice Hall, Englewood Cliffs (1984) Google Scholar
- Vasilyev, AV, Vasilyev, VB: Discrete singular operators and equations in a half-space. Azerb. J. Math. 3(1), 84-93 (2013) MathSciNetGoogle Scholar
- Vasilyev, AV, Vasilyev, VB: Discrete singular integrals in a half-space. In: Current Trends in Analysis and Its Applications, Proc. 9th Congress, Krakow, Poland, August 2013, pp. 663-670. Birkhäuser, Basel (2015) View ArticleGoogle Scholar
- Vasilyev, AV, Vasilyev, VB: Periodic Riemann problem and discrete convolution equations. Differ. Equ. 51(5), 652-660 (2015) View ArticleGoogle Scholar
- Eskin, G: Boundary Value Problems for Elliptic Pseudodifferential Equations. Am. Math. Soc., Providence (1981) Google Scholar
- Gakhov, FD: Boundary Value Problems. Dover, New York (1981) Google Scholar
- Muskhelishvili, NI: Singular Integral Equations. North-Holland, Amsterdam (1976) Google Scholar
- Gokhberg, I, Krupnik, N: Introduction to the Theory of One-Dimensional Singular Integral Equations. Birkhäuser, Basel (2010) MATHGoogle Scholar