On some applications of convolution to linear differential equations with Levitan almost periodic coefficients
Keywords
Almost periodic function in view of the Lebesgue measure, convolution, linear differential equation, Levitan almost periodic function, N-almost periodic function, Levitan almost periodic coefficientsAbstract
We investigate some properties of Levitan almost periodic functions with particular emphasis on their behavior under convolution. These considerations allow us to establish the main result concerning Levitan almost periodic solutions to linear differential equations of the first order. In particular, we state a condition, which guarantees that a special linear equation possesses a Levitan almost periodic solution. We also compare the class of Levitan almost periodic functions and the class of almost periodic functions with respect to the Lebesgue measure, and simultaneously, give an answer to the open question posed by Basit and G\"unzler in the paper \cite{Basit}.References
J. Andres, A.M. Bersani and R.F. Grande, Hierarchy of almost-periodic function spaces, Rend. Mat. Appl. 26 (7) (2006), 121–188.
B. Basit and H. Günzler, Difference property for perturbations of vector-valued Levitan almost periodic functions and their analogs, Russ. J. Math. Phys. 12 (4) (2005), 424–438.
A.S. Besicovitch, On generalized almost periodic functions, Proc. London Math. Soc. 25 (2) (1926), 495–512.
H. Bohr, Zur Theorie der fastperiodischen Funktionen, I. Teil Eine Verallgemeinerung der Theorie der Fourierreihen, Acta Math. 45 (1925), 29–127.
H. Bohr, Zur Theorie der fastperiodischen Funktionen, II. Teil Zusammenhang der fastperiodischen Functionen mit Funktionen von unendkich vielen Variablen; gleichmassige Approximation durch trigonometrische Summen, Acta Math. 46 (1925), 101–214.
H. Bohr, Zur Theorie der fastperiodischen Funktionen, III. Teil Dirichletentwicklung analytischer Functionen, Acta Math. 47 (1926), 237–281.
G. Bruno and A. Pankov, On convolution operators in the spaces of almost periodic functions, Z. Anal. Anwendungen 19 (2) (2000), 359–367.
D. Bugajewski and T. Diagana, Almost automorphy of the convolution operator and applications to differential and functional differential equations, Nonlinear Studies 13(2) (2006), 129–140.
D. Bugajewski, X. Gan and P. Kasprzak, Mappings of higher order and nonlinear equations in some spaces of almost periodic functions, Nonlinear Anal. 75 (2012), 5294–5310.
D. Bugajewski and A. Nawrocki, Some remarks on almost periodic functions in view of the Lebesgue measure with applications to linear differential equations, Ann. Acad. Sci. Fenn. Math. (in press).
T. Caraballo and D. Cheban, Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard’s separation condition I, J. Differential Equations 246 (2009), 108–128.
C. Corduneau, Almost Periodic Functions, AMS/Chelsea Publication Series, Chelsea Publishing Company, New York, 1989.
T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, New York, 2013.
J. Favard, Sur les equations differentielles a coefficients presque-periodiques, Acta Math. 51 (1927), 31–81.
G. N’Guerekata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plenum Publishers, New York, 2001.
G. N’Guerekata, Topics in Almost Automorphy, Springer, New York, 2005.
B.M. Levitan, Almost Periodic Functions, Gosudarstv. Izdat. Tekhn.-Teor. Lit., Moscow, 1953 (in Russian).
B.M. Levitan and V.V. Zhikov, Almost Periodic Functions and Differenial Equations, Cambridge University Press, Cambridge, 1982.
B.M. Levitan and V.V. Zhikov, Favard theory, Russian Math. Surveys 32 (1977), 129–180 (in Russian).
M.G. Lyubarskiı̆, An extension of Favard’s theory to the case of a system of linear differential equations with unbounded Levitan almost periodic coefficients, Dokl. Akad. Nauk SSSR 206 (1972), 808–810 (in Russian).
W. Marzantowicz and J. Signerska, Firing map of an almost periodic input function, Discret. Contin. Dyn. Syst. (2011), 1032–1041.
Y. Meyer, Quasicrystals, almost periodic patterns, mean-periodic functions and irregular sampling, Afr. Diaspora J. Math. (N.S.) 13 (2012), 1–45.
V.V. Stepanov, Über einige Verallgemeinerungen der fastperiodischen Funktionen, Ann. Math. 95 (1926), 437–498.
S. Stoiński, Almost periodic function in the Lebesgue measure, Comment. Math. Prace Mat. 34 (1994), 189–198.
S. Stoiński, Almost Periodic Functions, Scientific Publisher UAM, Poznań, 2008 (in Polish).
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0