Linear difference equation of second order in a Banach space and operators splitting

L. Yu. Kabantosva

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 3, pp. 285-293.

Résumé

In differential and difference equations classical textbooks, the $n$-th order differential and difference equations reducing by standard substitution to first-order differential and difference equations system is described. Each of the cohering equations can be written in the operator form. Naturally there is a question of coincidence of a number of properties of differential and difference equations (operators) of the second order and the corresponding functional equations (operators) of first order. In this paper we study the second order linear difference equation in the complex Banach space with bounded operator coefficients. The first theorem establishes the simultaneous invertibility of the second-order difference operator and the corresponding first-order difference operator, and the inverse operator formula is given. The research is conducted under conditions of the corresponding “algebraic” operator equation with separated roots. Theorem 2 establishes the second-order operator matrix and block-diagonal operator matrix similarity. In pair of operator roots separation condition in Theorem 3, the necessary and sufficient condition for the second and the first order difference operators invertibility is obtained. In Theorem 4 we obtain the operators under consideration inverse operators formalism (formula). In Theorems 5 and 6 for bounded solutions on the set of non-negative integers an asymptotic formalism of these solutions is obtained using operator-valued functions, this formalism can be called splitting at infinity.

Bibliographie

[1] Baskakov A. G., Duplishcheva A. Yu., “Difference operators and operator-valued matrices of the second order”, Izv. Math., 79:2 (2015), 217–232 | DOI | DOI | MR | Zbl

[2] Krein M. G., Langer G. K., “Certain mathematical principles of the linear theory of damped vibrations of continua”, Appl. Theory of Functions in Continuum Mechanics (Tbilisi, 1963), v. II, Proc. Intern. Sympos., Fluid and Gas Mechanics, Math. Methods, Nauka, M., 1965, 283–322 (in Russian) | Zbl

[3] Daleckij Ju. L., Krejn M. G., Stability of solutions of differential equations in a Banach space, Nauka, M., 1970, 536 pp. (in Russian)

[4] Krejn S. G., Linear differential equations in Banach space, Nauka, M., 1967, 464 pp. (in Russian)

[5] Henry D., Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin–Heidelberg–New York, 1981, 358 pp. | MR | Zbl

[6] Levitan B. M., Zikov V. V., Almost-periodic functions and differential equations, Moscow Univ. Press, M., 1978, 206 pp. (in Russian)

[7] Baskakov A. G., “Semigroups of difference operators in spectral analysis of linear differential operators”, Functional Analysis and Its Applications, 30:3 (1996), 149–157 | DOI | DOI | MR | Zbl

[8] Baskakov A. G., “Linear differential operators with unbounded operator coefficients and semigroups of bounded operators”, Math. Notes, 59:6 (1996), 586–593 | DOI | DOI | MR | Zbl

[9] Baskakov A. G., “Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations”, Russian Math. Surveys, 68:1 (2013), 69–116 | DOI | DOI | MR | Zbl

[10] Baskakov A. G., Pastukhov A. I., “Spectral Analysis of a Weighted Shift Operator with Unbounded Operator Coefficients”, Siberian Math. J., 42:6 (2001), 1026–1036 | DOI | MR | Zbl

[11] Baskakov A. G., “Harmonic and spectral analysis of power bounded operators and bounded semigroups of operators on Banach spaces”, Math. Notes, 97:2 (2015), 164–178 | DOI | DOI | MR | Zbl

[12] Dunford N., Schwartz J. T., Linear operators, v. I, Pure Appl. Math., 7, General theory, Interscience Publ., Inc., New York; Interscience Publ., Ltd., London, 1958, 858 pp. | MR | Zbl