Levitan almost periodic solutions of infinite-dimensional linear differential equations

David Cheban

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2019), pp. 56-78.

Résumé

The known Levitan's Theorem states that the finite-dimensional linear differential equation \begin{equation} x'=A(t)x+f(t) \end{equation} with Bohr almost periodic coefficients $A(t)$ and $f(t)$ admits at least one Levitan almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of homogeneous equations \begin{equation} x'=A(t)x. \end{equation} In this paper we prove that infinite-dimensional linear differential equation (1) with Levitan almost periodic coefficients has a Levitan almost periodic solution if it has at least one relatively compact solution and the trivial solution of equation (2) is Lyapunov stable. We study the problem of existence of Bohr/Levitan almost periodic solutions for infinite-dimensional equation (1) in the framework of general nonautonomous dynamical systems (cocycles).

Bibliographie

[1] B. Basit, H. Günzler, Spectral criteria for solutions of evolution equations and comments on reduced spectra, 2010, arXiv: 1006.2169 [math.FA] | MR

[2] S. Bochner, “Curvature and Betti numbers in real and complex vector bundles”, Univ. e Politec. Torino. Rend. Sem. Mat., 15 (1955–56), 225–253 | MR

[3] Group, Sijthoff Noordhoff, Alphen aan den Rijn, 1979, viii+319 pp. | MR

[4] I. U. Bronstein, Nonautonomous Dynamical Systems, Stiintsa, Chişinău, 1984 (in Russian) | MR

[5] Inese Bula, Juris Viksina, “Example of Strictly Convex Metric Spaces with Not Convex Balls”, International Journal of Pure and Applied Mathematics, 25:1 (2005), 85–91 | MR

[6] Inese Bula, “Strictly Convex Metric Spaces and Fixed Points”, Matematica Moravica, 3 (1999), 5–16 | Zbl

[7] T. Caraballo, 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 | DOI | MR | Zbl

[8] D. N. Cheban, “On the Comparability of Points of Dynamical Systems with Regard to the Character of Recurrence Property in the Limit”, Mathematical Sciences, 1, "Shtiintsa", Kishinev, 1977, 66–71

[9] D. N. Cheban, “Levitan almost periodic and almost automorphic solutions of $V$-monotone differential equations”, J. Dynam. Differential Equations, 20 (2008), 669–697 | DOI | MR | Zbl

[10] D. N. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, Hindawi Publishing Corporation, New York, 2009, ix+186 pp. | MR | Zbl

[11] David Cheban, Levitan Almost Periodic Solutions of Linear Differential Equations, 2019, arXiv: 1907.02512 [math.DS] | MR

[12] David N. Cheban, Nonautonomous Dynamics, Nonlinear oscillations and Global attractors, Springer, 2019 (to appear) , 496 pp.

[13] David Cheban, Zhenxin Liu, Periodic, Quasi-Periodic, Almost Periodic, Almost Automorphic, Birkhoff Recurrent and Poisson Stable Solutions for Stochastic Differential Equations, 2019 (to appear)

[14] I. Galina, “On strict convexity”, LU Zinatniskie Raksti, Matematika, 576 (1992), 193–198 | MR

[15] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, New York, 1981 | DOI | MR | Zbl

[16] I. Hitoshi, “On the Existence of Almost Periodic Complete Trajectories for Contractive Almost Periodic Processes”, Journal of Differential Equations, 43:1 (1982), 66–72 | DOI | MR | Zbl

[17] A. F. Izé, “On a Topological Method for the Analysis of the Asymptotic Behavior of Dynamical Systems and Processes.”, Complex Analysis, Functional Analysis and Approximation Theory, North-Holland, 1986, 109–128 | MR | Zbl

[18] B. M. Levitan, “Über Eine Verallgemeinerung der Stetigen Fastperiodischen Funktionen von H. Bohr”, Ann. of Math. (2), 40 (1939), 805–815 | DOI | MR | Zbl

[19] B. M. Levitan, “Some questions of the theory of almost periodic functions. II”, Uspekhi Mat. Nauk, 2:6 (22) (1947), 174–214 | MR

[20] B. M. Levitan, Almost Periodic Functions, Gos. Izdat. Tekhn-Teor. Lit., M., 1953, 396 pp. (in Russian) | MR | Zbl

[21] Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge, 1982, xi+211 pp. | MR | MR | Zbl | Zbl

[22] L. Schwartz, Analyse Mathématique, v. 1, Hermann, 1967 | MR

[23] G. R. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand-Reinhold, London, 1971 | MR | Zbl

[24] B. A. Shcherbakov, Topologic Dynamics and Poisson Stability of Solutions of Differential Equations, Ştiinţa, Chişinău, 1972, 231 pp. (in Russian) | MR

[25] Differential Equations, 11:7, 937–943 | MR | Zbl

[26] B. A. Shcherbakov, Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations, Ştiinţa, Chişinău, 1985, 147 pp. (in Russian) | MR

[27] Funct. Anal. Appl., 13:1 (1979), 74–75 | DOI | MR | Zbl | Zbl

[28] Introduction to Topological Dynamics, Noordhoff, Leyden, 1975 | MR | MR | Zbl

[29] Wataru Takahashi, “A Convexity in Metric Space and Nonexpansive Mappings, I”, Kodai Math. Sem. Rep., 22 (1970), 142–149 | DOI | MR | Zbl

[30] Math. Notes, 9 (1971), 235–238 | DOI | MR | Zbl | Zbl