pith. sign in

arxiv: 1907.02512 · v1 · pith:N6RYS4SVnew · submitted 2019-07-03 · 🧮 math.DS

Levitan Almost Periodic Solutions of Linear Differential Equations

David Cheban This is my paper

Pith reviewed 2026-05-25 09:59 UTC · model grok-4.3

classification 🧮 math.DS
keywords Levitan almost periodiclinear differential equationsbounded solutionscocyclesnonautonomous dynamical systemsdifference equations
0
0 comments X

The pith

A linear differential equation with Levitan almost periodic coefficients admits a Levitan almost periodic solution if it has at least one bounded solution, without assuming separation of homogeneous solutions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends Levitan's original theorem from Bohr almost periodic coefficients to the weaker Levitan almost periodic case. It establishes that the existence of one bounded solution on the real line guarantees a Levitan almost periodic solution for the inhomogeneous linear equation. The argument proceeds in the setting of cocycles over nonautonomous dynamical systems and removes the separation-from-zero requirement that was previously imposed on bounded solutions of the homogeneous equation. The same conclusion is shown to hold for the corresponding difference equations.

Core claim

If the linear differential equation x' = A(t)x + f(t) has Levitan almost periodic coefficients A(t) and f(t) and possesses at least one bounded solution, then it possesses a Levitan almost periodic solution. The separation from zero of bounded solutions of the homogeneous equation x' = A(t)x is not assumed. The result is obtained by studying the equation inside the framework of general nonautonomous dynamical systems realized by cocycles.

What carries the argument

Cocycles over nonautonomous dynamical systems, used to construct the Levitan almost periodic solution directly from the existence of any bounded solution.

If this is right

  • The same existence statement holds for linear difference equations with Levitan almost periodic coefficients.
  • The conclusion applies when the coefficients are Levitan almost periodic but not necessarily Bohr almost periodic.
  • Bounded solutions can be used to produce Levitan almost periodic solutions without first verifying separation of the homogeneous solutions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The cocycle approach may extend the result to other classes of recurrent coefficients beyond Levitan almost periodicity.
  • Verification of almost periodic behavior becomes simpler in applications where checking separation is difficult.
  • The technique could be tested on concrete examples such as equations with almost periodic but non-uniformly continuous coefficients.

Load-bearing premise

The coefficients A(t) and f(t) are Levitan almost periodic and the equation is analyzed via cocycles over nonautonomous dynamical systems.

What would settle it

A specific linear differential equation with Levitan almost periodic coefficients that has a bounded solution on the real line but no Levitan almost periodic solution.

read the original abstract

The known Levitan's Theorem states that the linear differential equation $$ x'=A(t)x+f(t) \ \ \ (*) $$ 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 $$ x'=A(t)x\ .\ \ \ (**) $$ In this paper we prove that linear differential equation (*) with Levitan almost periodic coefficients has a Levitan almost periodic solution, if it has at least one bounded solution. In this case, the separation from zero of bounded solutions of equation (**) is not assumed. The analogue of this result for difference equations also is given. We study the problem of existence of Bohr/Levitan almost periodic solutions for equation (*) in the framework of general nonautonomous dynamical systems (cocycles).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper extends Levitan's theorem on linear differential equations x'=A(t)x+f(t) with Bohr almost periodic coefficients to the case of Levitan almost periodic coefficients. It claims that existence of at least one bounded solution implies existence of a Levitan almost periodic solution, without assuming separation from zero of bounded solutions to the homogeneous equation x'=A(t)x. The proof is carried out in the framework of nonautonomous dynamical systems via cocycles on the hull; an analogous result is stated for difference equations.

Significance. If the central claim holds, the result relaxes a standard hypothesis in the theory of almost periodic solutions for linear nonautonomous systems and shows that the weaker Levitan topology is compatible with direct construction of invariant sections from bounded orbits. The use of the cocycle formalism on the hull is a methodological strength that avoids reliance on separation.

minor comments (2)
  1. The abstract states the main theorem but does not outline the key steps of the cocycle argument or the precise way the Levitan hull is used to produce the solution; a one-sentence sketch would improve readability.
  2. Notation for the Levitan hull and the cocycle map is introduced without an explicit reference to the standard definition used in the nonautonomous dynamical systems literature; adding a short sentence with the relevant reference would clarify the setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of its contribution, and the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a mathematical existence theorem for Levitan almost periodic solutions of linear nonautonomous ODEs under Levitan almost periodic coefficients, relaxing the separation hypothesis from Levitan's classical result. The argument is conducted via the standard cocycle formalism on the hull of the coefficients; this is an external methodological framework, not derived from the target statement. No equations reduce by construction to fitted parameters, no self-citations are invoked as load-bearing uniqueness theorems, and the central claim is not a renaming or re-derivation of its own inputs. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the claim rests on standard definitions and properties of Levitan almost periodic functions and the cocycle formalism for nonautonomous systems; no free parameters, invented entities, or ad-hoc axioms are indicated.

axioms (1)
  • standard math Standard properties of Levitan almost periodic functions and cocycles in nonautonomous dynamical systems.
    Invoked as the setting in which the existence result is proved without separation.

pith-pipeline@v0.9.0 · 5672 in / 1251 out tokens · 36707 ms · 2026-05-25T09:59:12.963313+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    I. U. Bronsteyn, Extensions of Minimal Transformation Group. Kishinev, Stiintsa, 1974, 311 p. (in Russian) [English translation in Group, Sijthoff & Noordhoff, Alphen aan den Rijn, 1979, viii+319.]

    work page 1974

  2. [2]

    U., Nonautonomous Dynamical Systems

    Bronstein I. U., Nonautonomous Dynamical Systems . Stiintsa, Chi¸ sin˘ au, 1984. (in Russian)

    work page 1984

  3. [3]

    International Journal of Pure and Applied Mathematics, Vol

    Inese Bula and Juris Viksina, Example of Strictly Convex Metric Spaces with Non Convex Balls. International Journal of Pure and Applied Mathematics, Vol. 25, No.1, 2005, pp.85- 91

    work page 2005

  4. [4]

    Matematica Moravica, Vol

    Inese Bula, Strictly Convex Metric Spaces and Fixed Poin ts. Matematica Moravica, Vol. 3, 1999, pp.5-16

    work page 1999

  5. [5]

    Caraballo and D

    T. Caraballo and D. Cheban, Almost periodic and almost au tomorphic solutions of lin- ear differential/difference equations without Favard’s sep aration condition. I, J. Differential Equations, 246 (2009), 108–128

    work page 2009

  6. [6]

    D. N. Cheban, On the Comparability of Points of Dynamical Systems with Regard to the Character of Recurrence Property in the Limit. Mathematical Sciences, issue No.1, 1977. Kishinev, ”Shtiintsa”, pp. 66-71

    work page 1977

  7. [7]

    Cheban, Levitan almost periodic and almost automorph ic solutions of V -monotone dif- ferential equations, J

    D. Cheban, Levitan almost periodic and almost automorph ic solutions of V -monotone dif- ferential equations, J. Dynam. Differential Equations 20 (2008), 669–697

    work page 2008

  8. [8]

    D. N. Cheban, Asymptotically Almost Periodic Solutions of Differential E quations. Hindawi Publishing Corporation, New York, 2009, ix+186 pp

    work page 2009

  9. [9]

    D. N. Cheban, Bohr/Levitan Almost Periodic and Almost Au tomorphic Solutions of Linear Stochastic Differential Equations without Favard’s Separa tion Condition. Chapter in book ”Advance in Evolution Equations: Evolution Equations: Alm ost Periodicity and Beyond”. Editors: G. N’Guerekata, Jin Lian and Alexander Pankov. Dedicated to the memory of V. V. Zhiko...

    work page 2019

  10. [10]

    Cheban, Nonautonomous Dynamics: Nonlinear oscillations and Global attractors

    David N. Cheban, Nonautonomous Dynamics: Nonlinear oscillations and Global attractors. Submitted, 2019

    work page 2019

  11. [11]

    LU Zinatniskie Raksti, Matematika, 576(1992), pp

    I.Galina, On strict convexity. LU Zinatniskie Raksti, Matematika, 576(1992), pp. 193-198

    work page 1992

  12. [12]

    Journal of Differential Equations , 43(1):66–72, 1982

    Hitoshi I., On the Existence of Almost Periodic Complet e Trajectories for Contractive Al- most Periodic Processes. Journal of Differential Equations , 43(1):66–72, 1982

    work page 1982

  13. [13]

    Johnson, A linear almost periodic equation with al most automorphic solution, Proc

    R.A. Johnson, A linear almost periodic equation with al most automorphic solution, Proc. Amer. Math. Soc. 82(1981), 199–205

    work page 1981

  14. [14]

    Levitan B., ber eine Verallgemeinerung der stetigen fa stperiodischen Funktionen von H. Bohr. Ann. of Math. (2) 40, (1939). pp.805–815. (German)

    work page 1939

  15. [15]

    B. M. Levitan, Some questions of the theory of almost periodic functions. I I, Uspekhi Mat. Nauk, 2:6(22) (1947), pp.174–214

    work page 1947

  16. [16]

    B. M. Levitan, Almost Periodic Functions. Gos. Izdat. Tekhn-Teor. Lit.. Moscow, 1953, 396 pp. (in Russian)

    work page 1953

  17. [17]

    B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations. Moscow State University Press, Moscow, 1978, 2004 p. (in Rus sian). [English translation: Almost Periodic Functions and Differential Equations. Camb ridge Univ. Press, Cambridge, 1982, xi+211 p.]

    work page 1978

  18. [18]

    L. A. Lusternik and V. J. Sobolev, Elements of Functional Analysis. Hindustan Publishing Corporation (India), 1975, xvi+360 pp

    work page 1975

  19. [19]

    Proceedings of the Second World Congress of Nonlinear Analy sts, Part 2 (Athens, 1996)

    Satoru Murakami, Representation of solutions of linea r functional difference equations in phase space. Proceedings of the Second World Congress of Nonlinear Analy sts, Part 2 (Athens, 1996). Nonlinear Anal. 30 (1997), no. 2, 1153–1164

    work page 1996

  20. [20]

    Ortega, M

    R. Ortega, M. Tarallo, Almost periodic linear different ial equations with non-separated solutions, J. Funct. Anal., 237(2006), no.2, 402–426

    work page 2006

  21. [21]

    Schwartz L., Analyse Math´ ematique, v. 1. Hermann, 1967

    work page 1967

  22. [22]

    R., Topological Dynamics and Ordinary Differential Equations

    Sell G. R., Topological Dynamics and Ordinary Differential Equations . Van Nostrand- Reinhold, London, 1971

    work page 1971

  23. [23]

    B. A. Shcherbakov, Topologic Dynamics and Poisson Stability of Solutions of Di fferential Equations. S ¸tiint ¸a, Chi¸ sin˘ au, 1972, 231 p.(in Russian)

    work page 1972

  24. [24]

    B. A. Shcherbakov, The Comparability of Motions of Dyna mical Systems with Regard to the Nature of their Recurrence. Differential Equations 11(1975), no. 7, 1246–1255.(in Russian) [English translation: Differential Equations, Vol.11. no. 7, pp.937-943]

    work page 1975

  25. [25]

    B. A. Shcherbakov, Poisson Stability of Motions of Dynamical Systems and Solut ions of Differential Equations . S ¸tiint ¸a, Chi¸ sin˘ au, 1985, 147 p. (in Russian)

    work page 1985

  26. [26]

    S., A fixed-point theorem

    Shulman V. S., A fixed-point theorem. Funktsional. Anal. i Prilozhen. 13 (1979), no. 1, pp.88–89. (in Russian) [English translation: Funct. Anal. Appl., 13:1 (1979), pp.74–75]

    work page 1979

  27. [27]

    K. S. Sibirsky, Introduction to Topological Dynamics. Kishinev, RIA AN MSSR, 1970, 144 p. (in Russian). [English translationn: Introduction to To pological Dynamics. Noordhoff, Leyden, 1975]

    work page 1970

  28. [28]

    Song, Periodic and almost periodic solutions of func tional difference equations with finite delays

    Y. Song, Periodic and almost periodic solutions of func tional difference equations with finite delays. Advances in Difference Equationss, V.2007, ID 68023 (doi: 10.1155/2007/68023)

    work page doi:10.1155/2007/68023 2007

  29. [29]

    Kodai Math

    W ataru Takahashi, A Convexity in Metric Space and Nonex pansive Mappings, I. Kodai Math. Sem. Rep., 22 (1970), pp.142-149

    work page 1970

  30. [30]

    V. V. Zhikov, The existence of Levitan almost-periodic solutions of linear systems (second complement to Favard’s classical theory). Mat. Zametki, Vol. 9, No.4, 1971, pp.409–414. [English translation: Math. Notes, 9 (1971), pp.235–238.]

    work page 1971

  31. [31]

    Zhikov, B.M

    V.V. Zhikov, B.M. Levitan, Favard’s theory. Uspekhi Mat. Nauk, 32 (1977), pp.123-171 (in Russian). [English translation: Russian Math. Surveys, 32 (1977), pp.129-180.] (D. Cheban) State University of Moldova, F aculty of Mathematics and Info rmatics, Department of Mathematics, A. Mateevich Street 60, MD–2009 C his ¸in˘au, Moldova E-mail address , D. Cheba...

    work page 1977