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I. U. Bronshtein's conjecture for monotone nonautonomous dynamical systems
State University of Moldova, Faculty of Mathematics and Informatics, Department of Mathematics, A. Mateevich Street 60, Chișinău, MD 2009, Moldova |
In this paper we study the problem of Levitan/Bohr almost periodicity of solutions for dissipative differential equations (Bronshtein's conjecture for Bohr almost periodic case). We give a positive answer to this conjecture for monotone Levitan/Bohr almost periodic systems of differential/difference equations.
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
V. M. Bebutov,
On the shift dynamical systems on the space of continuous functions, Bull. of Inst. of Math. of Moscow University, 2 (1940), 1-65.
|
[3] |
H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, 1947. ii+114 pp. |
[4] |
I. U. Bronsteyn, Extensions of Minimal Transformation Group, Translated from the Russian. Martinus Nijhoff Publishers, The Hague, 1979. |
[5] |
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.
doi: 10.1016/j.jde.2008.04.001. |
[6] |
D. N. Cheban, On the comparability of the points of dynamical systems with regard to the character of recurrence property in the limit, Mathematical Sciences, issue N1, Kishinev, "Shtiintsa", 1977, 66–71. |
[7] |
D. N. Cheban,
Global pullback atttactors of c-analytic nonautonomous dynamical systems, Stochastics and Dynamics, 1 (2001), 511-535.
doi: 10.1142/S0219493701000254. |
[8] |
D. N. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, Hindawi Publishing Corporation, New York, 2009, ix+186 pp.
doi: 10.1155/9789774540998. |
[9] |
D. N. Cheban, Global Attractors of Nonautonomous Dynamical and Control Systems, 2nd Edition, Interdisciplinary Mathematical Sciences, vol.18, River Edge, NJ: World Scientific, 2015, xxv+589 pp.
doi: 10.1142/9297. |
[10] |
D. Cheban and Z. Liu, Bohr/Levitan Almost Periodic, Almost Automorphic, Recurrent and Poisson Stable Solutions of Monotone Differential Equations, SCIENCE CHINA Mathematics, Accepted 2018, 28 pages. |
[11] |
B. P. Demidovich, Lectures on Mathematical Theory of Stability, Moscow, "Nauka", 1967. (in Russian) |
[12] |
A. M. Fink, Almost Periodic Differential Equations, Lecture notes in Mathematics, Vol. 377, Springer-Verlag, 1974. |
[13] |
A. M. Fink and P. O. Fredericson,
Ultimate boundedness does not imply almost periodicity, Journal of Differential Equations, 9 (1971), 280-284.
doi: 10.1016/0022-0396(71)90081-7. |
[14] |
F. Flandoli and B. Schmalfuß,
Random Attractors for the Stochastic 3D Navier–Stokes Equation with Multiplicative White Noise, Stochastics and Stochastics Reports, 59 (1996), 21-45.
doi: 10.1080/17442509608834083. |
[15] |
M. Hirsch and H. Smith, Monotone dynamical systems, Handbook of Differential Equations: Ordinary Differential Equations, Vol. Ⅱ, 239–357, Elsevier B. V., Amsterdam, 2005. |
[16] |
D. Husemoller, Fibre Bundles, Third edition. Graduate Texts in Mathematics, 20. Springer-Verlag, New York, 1994. xx+353 pp.
doi: 10.1007/978-1-4757-2261-1. |
[17] |
J. Jiang and X. Zhao,
Convergence in monotone and uniformly stable skew-product semiflows with applications, J. Reine Angew. Math., 589 (2005), 21-55.
doi: 10.1515/crll.2005.2005.589.21. |
[18] |
B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Moscow State University Press, Moscow, 1978.
![]() ![]() |
[19] |
V. A. Pliss, Nonlocal Problems in the Theory of Oscilations, Nauka, Moscow, 1964. (in Russian) [English translation: Nonlocal Problems in the Theory of Oscilations, Academic Press, 1966,367 p.] |
[20] |
G. R. Sell,
Non-autonomous differential equations and topological dynamics, I. The basic theory, Trans. Amer. Math. Soc., 127 (1967), 241-262.
doi: 10.2307/1994645. |
[21] |
G. R. Sell,
Non-autonomous differential equations and topological dynamics, II. Limiting equations, Trans. Amer. Math. Soc., 127 (1967), 263-283.
doi: 10.1090/S0002-9947-1967-0212314-4. |
[22] |
G. R. Sell, Lectures on Topological Dynamics and Differential Equations, volume 2 of Van Nostrand Reinhold Math. Studies, Van Nostrand–Reinbold, London, 1971. |
[23] |
B. A. Shcherbakov,
Classification of Poisson-stable motions, Pseudo-recurrent motions, Dokl. Akad. Nauk SSSR, 146 (1962), 322-324.
|
[24] |
B. A. Shcherbakov,
On classes of Poisson stability of motion, Pseudorecurrent motions, Bul. Akad. Stiince RSS Moldoven., 1963 (1963), 58-72.
|
[25] |
B. A. Shcherbakov,
Constituent classes of Poisson-stable motions, Sibirsk. Mat. Zh., 5 (1964), 1397-1417.
|
[26] |
B. A. Shcherbakov, Topologic Dynamics and Poisson Stability of Solutions of Differential Equations, Ştiinţa, Chişinău, 1972,231 p.(in Russian) |
[27] |
B. A. Shcherbakov,
The compatible recurrence of the bounded solutions of first order differential equations, Differencial'nye Uravnenija, 10 (1974), 270-275.
|
[28] |
B. A. Shcherbakov,
The comparability of the motions of dynamical systems with regard to the nature of their recurrence, Differencial?nye Uravnenija, 11 (1975), 1246-1255.
|
[29] |
B. A. Shcherbakov, Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations, Ştiinţa, Chişinău, 1985,147 pp. (in Russian) |
[30] |
K. S. Sibirsky, Introduction to Topological Dynamics, Kishinev, RIA AN MSSR, 1970,144 p. (in Russian). [English translation: Introduction to Topological Dynamics. Noordhoff, Leyden, 1975] |
[31] |
H. L. Smith,
Monotone semiflows generated by functional differential equations, Journal of Differential Equations, 66 (1987), 420-442.
doi: 10.1016/0022-0396(87)90027-1. |
[32] |
H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Series: Mathematical surveys and monographs, Volume 41., American Mathematical Society. Providence, Rhode Island, 1995, x+174 p. |
[33] |
Y. Wang and X.-Q. Zhao, Global convergence in monotone and uniformly stable re- current skew-product semiflows, Infinite Dimensional Dynamical Systems, 391–406, Fields Inst. Commun., 64, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4523-4_15. |
[34] |
T. Yoshizawa,
Note on the Boundedness and the Ultimate Boundedness of Solutions $x'=f(t,x)$, Memoirs of the College of Science. University of Kyoto, Serie A, 29 (1955), 275-291.
doi: 10.1215/kjm/1250777188. |
[35] |
T. Yoshizawa,
Lyapunov's function and boundedness of solutions, Funkcialaj Ekvacioj, 2 (1959), 95-142.
|
[36] |
V. V. Zhikov,
On Problem of Existence of Almost Periodic Solutions of Differential and Operator Equations, Nauchnye Trudy VVPI, Matematika, 8 (1969), 94-188.
|
show all references
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
V. M. Bebutov,
On the shift dynamical systems on the space of continuous functions, Bull. of Inst. of Math. of Moscow University, 2 (1940), 1-65.
|
[3] |
H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, 1947. ii+114 pp. |
[4] |
I. U. Bronsteyn, Extensions of Minimal Transformation Group, Translated from the Russian. Martinus Nijhoff Publishers, The Hague, 1979. |
[5] |
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.
doi: 10.1016/j.jde.2008.04.001. |
[6] |
D. N. Cheban, On the comparability of the points of dynamical systems with regard to the character of recurrence property in the limit, Mathematical Sciences, issue N1, Kishinev, "Shtiintsa", 1977, 66–71. |
[7] |
D. N. Cheban,
Global pullback atttactors of c-analytic nonautonomous dynamical systems, Stochastics and Dynamics, 1 (2001), 511-535.
doi: 10.1142/S0219493701000254. |
[8] |
D. N. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, Hindawi Publishing Corporation, New York, 2009, ix+186 pp.
doi: 10.1155/9789774540998. |
[9] |
D. N. Cheban, Global Attractors of Nonautonomous Dynamical and Control Systems, 2nd Edition, Interdisciplinary Mathematical Sciences, vol.18, River Edge, NJ: World Scientific, 2015, xxv+589 pp.
doi: 10.1142/9297. |
[10] |
D. Cheban and Z. Liu, Bohr/Levitan Almost Periodic, Almost Automorphic, Recurrent and Poisson Stable Solutions of Monotone Differential Equations, SCIENCE CHINA Mathematics, Accepted 2018, 28 pages. |
[11] |
B. P. Demidovich, Lectures on Mathematical Theory of Stability, Moscow, "Nauka", 1967. (in Russian) |
[12] |
A. M. Fink, Almost Periodic Differential Equations, Lecture notes in Mathematics, Vol. 377, Springer-Verlag, 1974. |
[13] |
A. M. Fink and P. O. Fredericson,
Ultimate boundedness does not imply almost periodicity, Journal of Differential Equations, 9 (1971), 280-284.
doi: 10.1016/0022-0396(71)90081-7. |
[14] |
F. Flandoli and B. Schmalfuß,
Random Attractors for the Stochastic 3D Navier–Stokes Equation with Multiplicative White Noise, Stochastics and Stochastics Reports, 59 (1996), 21-45.
doi: 10.1080/17442509608834083. |
[15] |
M. Hirsch and H. Smith, Monotone dynamical systems, Handbook of Differential Equations: Ordinary Differential Equations, Vol. Ⅱ, 239–357, Elsevier B. V., Amsterdam, 2005. |
[16] |
D. Husemoller, Fibre Bundles, Third edition. Graduate Texts in Mathematics, 20. Springer-Verlag, New York, 1994. xx+353 pp.
doi: 10.1007/978-1-4757-2261-1. |
[17] |
J. Jiang and X. Zhao,
Convergence in monotone and uniformly stable skew-product semiflows with applications, J. Reine Angew. Math., 589 (2005), 21-55.
doi: 10.1515/crll.2005.2005.589.21. |
[18] |
B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Moscow State University Press, Moscow, 1978.
![]() ![]() |
[19] |
V. A. Pliss, Nonlocal Problems in the Theory of Oscilations, Nauka, Moscow, 1964. (in Russian) [English translation: Nonlocal Problems in the Theory of Oscilations, Academic Press, 1966,367 p.] |
[20] |
G. R. Sell,
Non-autonomous differential equations and topological dynamics, I. The basic theory, Trans. Amer. Math. Soc., 127 (1967), 241-262.
doi: 10.2307/1994645. |
[21] |
G. R. Sell,
Non-autonomous differential equations and topological dynamics, II. Limiting equations, Trans. Amer. Math. Soc., 127 (1967), 263-283.
doi: 10.1090/S0002-9947-1967-0212314-4. |
[22] |
G. R. Sell, Lectures on Topological Dynamics and Differential Equations, volume 2 of Van Nostrand Reinhold Math. Studies, Van Nostrand–Reinbold, London, 1971. |
[23] |
B. A. Shcherbakov,
Classification of Poisson-stable motions, Pseudo-recurrent motions, Dokl. Akad. Nauk SSSR, 146 (1962), 322-324.
|
[24] |
B. A. Shcherbakov,
On classes of Poisson stability of motion, Pseudorecurrent motions, Bul. Akad. Stiince RSS Moldoven., 1963 (1963), 58-72.
|
[25] |
B. A. Shcherbakov,
Constituent classes of Poisson-stable motions, Sibirsk. Mat. Zh., 5 (1964), 1397-1417.
|
[26] |
B. A. Shcherbakov, Topologic Dynamics and Poisson Stability of Solutions of Differential Equations, Ştiinţa, Chişinău, 1972,231 p.(in Russian) |
[27] |
B. A. Shcherbakov,
The compatible recurrence of the bounded solutions of first order differential equations, Differencial'nye Uravnenija, 10 (1974), 270-275.
|
[28] |
B. A. Shcherbakov,
The comparability of the motions of dynamical systems with regard to the nature of their recurrence, Differencial?nye Uravnenija, 11 (1975), 1246-1255.
|
[29] |
B. A. Shcherbakov, Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations, Ştiinţa, Chişinău, 1985,147 pp. (in Russian) |
[30] |
K. S. Sibirsky, Introduction to Topological Dynamics, Kishinev, RIA AN MSSR, 1970,144 p. (in Russian). [English translation: Introduction to Topological Dynamics. Noordhoff, Leyden, 1975] |
[31] |
H. L. Smith,
Monotone semiflows generated by functional differential equations, Journal of Differential Equations, 66 (1987), 420-442.
doi: 10.1016/0022-0396(87)90027-1. |
[32] |
H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Series: Mathematical surveys and monographs, Volume 41., American Mathematical Society. Providence, Rhode Island, 1995, x+174 p. |
[33] |
Y. Wang and X.-Q. Zhao, Global convergence in monotone and uniformly stable re- current skew-product semiflows, Infinite Dimensional Dynamical Systems, 391–406, Fields Inst. Commun., 64, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4523-4_15. |
[34] |
T. Yoshizawa,
Note on the Boundedness and the Ultimate Boundedness of Solutions $x'=f(t,x)$, Memoirs of the College of Science. University of Kyoto, Serie A, 29 (1955), 275-291.
doi: 10.1215/kjm/1250777188. |
[35] |
T. Yoshizawa,
Lyapunov's function and boundedness of solutions, Funkcialaj Ekvacioj, 2 (1959), 95-142.
|
[36] |
V. V. Zhikov,
On Problem of Existence of Almost Periodic Solutions of Differential and Operator Equations, Nauchnye Trudy VVPI, Matematika, 8 (1969), 94-188.
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