American Institute of Mathematical Sciences

March  2019, 24(3): 1095-1113. doi: 10.3934/dcdsb.2019008

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

* Corresponding author: David Cheban

Received  November 2017 Revised  January 2018 Published  January 2019

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.

Citation: David Cheban. I. U. Bronshtein's conjecture for monotone nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1095-1113. doi: 10.3934/dcdsb.2019008
References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar [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.   Google Scholar [3] H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, 1947. ii+114 pp.  Google Scholar [4] I. U. Bronsteyn, Extensions of Minimal Transformation Group, Translated from the Russian. Martinus Nijhoff Publishers, The Hague, 1979.  Google Scholar [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.  Google Scholar [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. Google Scholar [7] D. N. Cheban, Global pullback atttactors of c-analytic nonautonomous dynamical systems, Stochastics and Dynamics, 1 (2001), 511-535.  doi: 10.1142/S0219493701000254.  Google Scholar [8] D. N. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, Hindawi Publishing Corporation, New York, 2009, ix+186 pp. doi: 10.1155/9789774540998.  Google Scholar [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.  Google Scholar [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. Google Scholar [11] B. P. Demidovich, Lectures on Mathematical Theory of Stability, Moscow, "Nauka", 1967. (in Russian)  Google Scholar [12] A. M. Fink, Almost Periodic Differential Equations, Lecture notes in Mathematics, Vol. 377, Springer-Verlag, 1974.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Moscow State University Press, Moscow, 1978.   Google Scholar [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.]  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] G. R. Sell, Lectures on Topological Dynamics and Differential Equations, volume 2 of Van Nostrand Reinhold Math. Studies, Van Nostrand–Reinbold, London, 1971.  Google Scholar [23] B. A. Shcherbakov, Classification of Poisson-stable motions, Pseudo-recurrent motions, Dokl. Akad. Nauk SSSR, 146 (1962), 322-324.   Google Scholar [24] B. A. Shcherbakov, On classes of Poisson stability of motion, Pseudorecurrent motions, Bul. Akad. Stiince RSS Moldoven., 1963 (1963), 58-72.   Google Scholar [25] B. A. Shcherbakov, Constituent classes of Poisson-stable motions, Sibirsk. Mat. Zh., 5 (1964), 1397-1417.   Google Scholar [26] B. A. Shcherbakov, Topologic Dynamics and Poisson Stability of Solutions of Differential Equations, Ştiinţa, Chişinău, 1972,231 p.(in Russian)  Google Scholar [27] B. A. Shcherbakov, The compatible recurrence of the bounded solutions of first order differential equations, Differencial'nye Uravnenija, 10 (1974), 270-275.   Google Scholar [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.   Google Scholar [29] B. A. Shcherbakov, Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations, Ştiinţa, Chişinău, 1985,147 pp. (in Russian)  Google Scholar [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]  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] T. Yoshizawa, Lyapunov's function and boundedness of solutions, Funkcialaj Ekvacioj, 2 (1959), 95-142.   Google Scholar [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.   Google Scholar

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References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar [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.   Google Scholar [3] H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, 1947. ii+114 pp.  Google Scholar [4] I. U. Bronsteyn, Extensions of Minimal Transformation Group, Translated from the Russian. Martinus Nijhoff Publishers, The Hague, 1979.  Google Scholar [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.  Google Scholar [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. Google Scholar [7] D. N. Cheban, Global pullback atttactors of c-analytic nonautonomous dynamical systems, Stochastics and Dynamics, 1 (2001), 511-535.  doi: 10.1142/S0219493701000254.  Google Scholar [8] D. N. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, Hindawi Publishing Corporation, New York, 2009, ix+186 pp. doi: 10.1155/9789774540998.  Google Scholar [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.  Google Scholar [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. Google Scholar [11] B. P. Demidovich, Lectures on Mathematical Theory of Stability, Moscow, "Nauka", 1967. (in Russian)  Google Scholar [12] A. M. Fink, Almost Periodic Differential Equations, Lecture notes in Mathematics, Vol. 377, Springer-Verlag, 1974.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Moscow State University Press, Moscow, 1978.   Google Scholar [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.]  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] G. R. Sell, Lectures on Topological Dynamics and Differential Equations, volume 2 of Van Nostrand Reinhold Math. Studies, Van Nostrand–Reinbold, London, 1971.  Google Scholar [23] B. A. Shcherbakov, Classification of Poisson-stable motions, Pseudo-recurrent motions, Dokl. Akad. Nauk SSSR, 146 (1962), 322-324.   Google Scholar [24] B. A. Shcherbakov, On classes of Poisson stability of motion, Pseudorecurrent motions, Bul. Akad. Stiince RSS Moldoven., 1963 (1963), 58-72.   Google Scholar [25] B. A. Shcherbakov, Constituent classes of Poisson-stable motions, Sibirsk. Mat. Zh., 5 (1964), 1397-1417.   Google Scholar [26] B. A. Shcherbakov, Topologic Dynamics and Poisson Stability of Solutions of Differential Equations, Ştiinţa, Chişinău, 1972,231 p.(in Russian)  Google Scholar [27] B. A. Shcherbakov, The compatible recurrence of the bounded solutions of first order differential equations, Differencial'nye Uravnenija, 10 (1974), 270-275.   Google Scholar [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.   Google Scholar [29] B. A. Shcherbakov, Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations, Ştiinţa, Chişinău, 1985,147 pp. (in Russian)  Google Scholar [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]  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] T. Yoshizawa, Lyapunov's function and boundedness of solutions, Funkcialaj Ekvacioj, 2 (1959), 95-142.   Google Scholar [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.   Google Scholar
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