Abstract
We obtain an asymptotic expansion for a solution to a nonhomogeneous retarded- or neutraltype differential-difference equation. The case of unbounded delays is considered. The influence is accounted for the roots of the characteristic equation. We establish the exact asymptotics for the remainder depending on the asymptotic properties of the free matrix term of the equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Sgibnev M. S., “Behavior at infinity of a solution to a differential-difference equation,” Siberian Math. J., 53, No. 6, 1139–1154 (2012).
Bellman R. E. and Cooke K. L., Differential-Difference Equations, Academic Press, New York and London (1963).
Hale J. K., Theory of Functional Differential Equations, Springer-Verlag, New York, Heidelberg, and Berlin (1977).
Vlasov V. V. and Ivanov S. A., “Estimates for solutions of nonhomogeneous differential-difference equations of neutral type,” Russian Math. (Iz. VUZ), 50, No. 3, 22–28 (2006).
Lesnykh A. A., “Estimates of the solutions of difference-differential equations of neutral type,” Math. Notes, 81, No. 4, 503–517 (2007).
Vlasov V. V. and Medvedev D. A., “Functional-differential equations in Sobolev spaces related problems of spectral theory,” in: Contemporary Mathematics. Fundamental Trends [in Russian], 2008, 30, pp. 3–173.
Hille E. and Phillips R. S., Functional Analysis and Semigroups, Amer. Math. Soc., Providence (1957).
Feller W., An Introduction to Probability Theory and Its Applications. Vol. 2, John Wiley and Sons, Inc., New York etc. (1966).
Lancaster P., The Theory of Matrices, Academic Press, New York (1969).
Gelfand I. M., Raĭkov D. A., and Shilov G. E., Commutative Normed Rings, Chelsea Publishing Company, New York (1964).
Rogozin B. A. and Sgibnev M. S., “Banach algebras of measures on the real axis,” Siberian Math. J., 21, No. 2, 160–169 (1980).
Horn R. A. and Johnson Ch. R., Matrix Analysis, Cambridge Univ. Press, Cambridge (1985).
Markushevich A. I., The Theory of Analytic Functions. Vol. 1 [in Russian], Nauka, Moscow (1967).
Azbelev N. V., Maksimov V. P., and Rakhmatullina L. F., Elements of the Modern Theory of Functional-Differential Equations. Methods and Applications [Russian], Inst. Komp-yutern. Issled., Moscow (2002).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2014 Sgibnev M.S.
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 3, pp. 650–665, May–June, 2014.
Rights and permissions
About this article
Cite this article
Sgibnev, M.S. Behavior at infinity of a solution to a matrix differential-difference equation. Sib Math J 55, 530–543 (2014). https://doi.org/10.1134/S0037446614030148
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446614030148