Opuscula Math. 36, no. 4 (2016), 525-539
http://dx.doi.org/10.7494/OpMath.2016.36.4.525
Opuscula Mathematica
On solvability of some difference-discrete equations
Alexander V. Vasilyev
Vladimir B. Vasilyev
Abstract. Multidimensional difference equations in a discrete half-space are considered. Using the theory of periodic Riemann problems a general solution and solvability conditions in discrete Lebesgue spaces are obtained. Some statements of boundary value problems of discrete type are given.
Keywords: multidimensional difference-discrete equation, symbol, factorization, periodic Riemann problem.
Mathematics Subject Classification: 39A14, 42B05.
- D.E. Dudgeon, R.M. Mersereau, Multidimensional Digital Signal Processing, Prentice-Hall, Inc., Englewood Cliffs, 1984.
- G. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Equations, AMS, Providence, 1981.
- F.D. Gakhov, Boundary Value Problems, Dover Publications, NY, 1981.
- I. Gokhberg, N. Krupnik, Introduction to the Theory of One Dimensional Singular Integral Equations, Birkhäuser, Basel, 2010.
- C. Jordan, Calculus of Finite Differences, Chelsea Publishing Company, New York, NY, 1950.
- V.G. Kurbatov, Functional Differential Operators and Equations, Kluwer Academic Publishers, Dordrecht, 1999.
- S.G. Mikhlin, S. Prössdorf, Singular Integral Operators, Berlin, Akademie-Verlag, 1986.
- L.M. Milne-Thomson, The Calculus of Finite Differences, Chelsea Publishing Company, New York, NY, 1981.
- N.I. Muskhelishvili, Singular Integral Equations, North Holland, 1976.
- B. Noble, Methods Based on Wiener-Hopf Technique for the Solution of Partial Differential Equations, Pergamon Press, London-New York-Paris-Los Angeles, 1958.
- S.L. Sobolev, Cubature Formulas and Modern Analysis: An Introduction, Gordon and Breach Sci. Publ., Montreux, 1992.
- E. Titchmarsh, Introduction to the Theory of Fourier Integrals, Chelsea Publishing Company, New York, NY, 1986.
- V.B. Vasil'ev, Wave Factorization of Elliptic Symbols: Theory and Applications. Introduction to the Theory of Boundary Value Problems in Non-Smooth Domains, Dordrecht-Boston-London, Kluwer Academic Publishers, 2000.
- V.B. Vasilyev, General boundary value problems for pseudo differential equations and related difference equations, Adv. Difference Equ. (2013) 2013:289.
- V.B. Vasilyev, On some difference equations of first order, Tatra Mt. Math. Publ. 54 (2013), 165-181.
- A.V. Vasilyev, V.B. Vasilyev, Discrete singular operators and equations in a half-space, Azerb. J. Math. 3 (2013) 1, 84-93.
- A.V. Vasilyev, V.B. Vasilyev, Discrete singular integrals in a half-space, [in:] Current Trends in Analysis and Its Applications, Proc. 9th ISAAC Congress, Krakow, Poland, 2013; Birkhäuser, Basel, 2015, 663-670.
- A.V. Vasilyev, V.B. Vasilyev, Periodic Riemann problem and discrete convolution equations, Differ. Equ. 51 (2015) 5, 652-660.
- A.V. Vasilyev, V.B. Vasilyev, On some classes of difference equations of infinite order, Adv. Difference Equ. (2015) 2015:211.
- V.S. Vladimirov, Methods of the Theory of Functions of Many Complex Variables, Dover Publications, NY, 2007.
- Alexander V. Vasilyev
- National Research Belgorod State University, Department of Mathematical Analysis, Studencheskaya 14/1, Belgorod 307008, Russia
- Vladimir B. Vasilyev
- Chair of Pure Mathematics, Lipetsk State Technical University, Lipetsk 398600, Russia
- Communicated by Josef Diblík.
- Received: 2015-09-17.
- Accepted: 2016-01-26.
- Published online: 2016-04-01.
