We establish necessary and sufficient conditions for the solvability of a family of differential equations with periodic operator coefficient and periodic boundary conditions by using the notion of relative spectrum of a linear bounded operator in a Banach space and the ergodic theorem. It is shown that if the existence condition is satisfied, then the required periodic solutions can be constructed by using the deduced formula for the generalized inverse operator of a linear bounded operator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. G. Krein, Lectures on the Theory of Stability of Solutions of Differential Equations in Banach Spaces [in Russian], Institute of Mathematics, Academy of Sciences of Ukr. SSSR, Kiev (1964).
E. Asplund, “A non-closed relative spectrum,” Ark. Mat., 3, 425–427 (1958).
P. R. Halmos, A Hilbert Space Problem Book [Russian translation], Mir, Moscow (1970).
I. Ts. Gokhberg and N. Ya. Krupnik, Introduction to the Theory of One-Dimensional Singular Integral Operators [in Russian], Shtiintsa, Kishinev (1973).
A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Utrecht (2004).
J. von Neumann, “On regular rings,” Proc. Amer. Math. Soc., 22, 707–713 (1936).
K. Yosida, Functional Analysis [Russian translation], Mir, Moscow (1967).
E. Deutch, “Semi-inverses, reflexive semi-inverses, and pseudoinverses of an arbitrary linear transformation,” Linear Algebra Its Appl., 4, 313–322 (1971).
M. I. Kadets and B. S. Mityagin, “Complemented subspaces in Banach spaces,” Usp. Mat. Nauk, 28, No. 6, 77–95 (1973).
A. Pełczyński and T. Figiel, “On the Enflo method for the construction of Banach spaces,” Usp. Mat. Nauk., 28, No. 6, 95–109 (1973).
V. S. Korolyuk and A. F. Turbin, Mathematical Foundations of the Phase Lumping of Complex Systems [in Russian], Naukova Dumka, Kiev (1978).
I. Ts. Gokhberg and A. S. Markus, “On the stability of some properties of normally solvable operators,” Mat. Sb., 40 (82), No. 4, 453–466 (1956).
I. Ts. Gokhberg and M. G. Krein, “Basic concepts of defect numbers, root numbers, and indices of linear operators,” Usp. Mat. Nauk, 12, No. 2, 43–115 (1957).
F. V. Atkinson, “Normal solvability of linear equations in normed spaces,” Mat. Sb. Nov. Ser., 28, No. 1, 3–14 (1951).
A. Pełczyński, “On some problems of Banach,” Usp. Mat. Nauk, 28, No. 6, 67–77 (1973).
A. G. Baskakov, “Semigroups of difference operators in spectral analysis of linear differential operators,” Funkts. Anal. Prilozhen., 30, No. 1, 1–11 (1996).
A. G. Baskakov, “Estimates of bounded solutions to linear differential operators,” Differents. Uravn., 39, No. 3, 413–415 (2003).
Yu. Latushkin and Yu. Tomilov, “Fredholm differential operators with unbounded coefficients,” J. Different. Equat., 208, 388–429 (2005).
B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations [in Russian], Moscow University, Moscow (1978).
A. A. Boichuk and A. A. Pokutnyi, “Bounded solutions of linear differential equations in Banach spaces,” Nelin. Kolyv., 9, No. 1, 3–14 (2006); English translation: Nonlin. Oscillat., 9, No. 1, 1–12 (2006).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 3, pp. 329–338, March, 2013.
Rights and permissions
About this article
Cite this article
Boichuk, A.A., Pokutnyi, A.A. Application of the Ergodic Theory to the Investigation of Boundary-Value Problems with Periodic Operator Coefficients. Ukr Math J 65, 366–376 (2013). https://doi.org/10.1007/s11253-013-0783-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-013-0783-9