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May  2015, 20(3): 945-959. doi: 10.3934/dcdsb.2015.20.945

Belitskii--Lyubich conjecture for $C$-analytic dynamical systems

David Cheban 1,

1. 

State University of Moldova, Department of Mathematics and Informatics, A. Mateevich Street 60, MD–2009 Chişinău

Received  August 2013 Revised  September 2014 Published  January 2015

The aim of this paper is study the problem of global asymptotic stability of solutions for $\mathbb C$-analytical dynamical systems (both with continuous and discrete time). In particular we present some new results for the $C$-analytical version of Belitskii--Lyubich conjecture. Some applications of these results for periodic $\mathbb C$-analytical differential/difference equations and the equations with impulse are given.
Keywords: Belitskii--Lyubich conjecture., Global asymptotic stability, attractor, holomorphic dynamical systems.
Mathematics Subject Classification: 34C25, 34D05, 34D20, 34D23, 34D45, 34G20, 37B25, 37B55, 7L15, 37L30, 37L50, 39A13, 39A23, 39A30, 39A45, 39A11, 39C10, 39C5.
Citation: David Cheban. Belitskii--Lyubich conjecture for $C$-analytic dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 945-959. doi: 10.3934/dcdsb.2015.20.945
References:
[1]

R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, Measures of Noncompactness and Condensing Operators, Translated from the 1986 Russian original by A. Iacob, Operator Theory: Advances and Applications, 55, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0348-5727-7.  Google Scholar

[2]

G. R. Belitskii and Yu. I. Lyubich, Matrix norms and their applications, Operator Theory: Advances and Applications, 36, Birkhäuser Verlag, Basel, 1988. doi: 10.1007/978-3-0348-7400-7.  Google Scholar

[3]

D. N. Cheban, Global Attractors of Non-Autonomous Dissipstive Dynamical Systems, Interdisciplinary Mathematical Sciences, 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004. doi: 10.1142/9789812563088.  Google Scholar

[4]

D. N. Cheban, Markus-Yamabe conjecture, in Communications, The 14-th Conference of Applied and Industrial Mathematics, Chisinau, 2006, 107-110. Google Scholar

[5]

D. N. Cheban, Lyapunov Stability of Non-Autonomous Dynamical Systems, Nova Science Publishers Inc., New York, 2013. Google Scholar

[6]

D. N. Cheban and C. Mammana, Absolute asymptotic stability of discrete linear inclusions, Bul. Acad. Ştiinţe Repub. Mold. Mat., (2005), 43-68.  Google Scholar

[7]

G. Darbo, Punti uniti in transformazioni non-compacto, Rend. Sem. Mat. Univ. Padova, 24 (1955), 84-92.  Google Scholar

[8]

C. J. Earle and R. S. Hamilton, A fixed point theorem for holomorphic mappings, in Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., (1968)), American Mathematical Society, Rhode Island, 1970, 61-65.  Google Scholar

[9]

J. K. Hale, Asymptotic Behaviour of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

[10]

J. K. Hale and O. Lopes, Fixed point theorems and dissipative processes, Journal of Differential Equations, 13 (1973), 391-402. doi: 10.1016/0022-0396(73)90025-9.  Google Scholar

[11]

A. Halanay and D. Wexler, Teoria Calitativă a Sistemelor cu Impulsuri, Bucureşti, 1968. Google Scholar

[12]

L. A. Harris, Fixed points of holomorphic mappings for domains in Banach spaces, Abstract and Applied Analysis, (2003), 261-274. doi: 10.1155/S1085337503205042.  Google Scholar

[13]

G. S. Jones, The existence of critical points in generalized dynamical systems, in Seminaire on Differential Equations and Dynamical Systems, Lecture Notes in Math., Springer, Berlin, 1968, 7-19.  Google Scholar

[14]

O. A. Ladyzhenskaya, Finding minimal global attractors for Navier-Stokes equations and other partial differential equations, Uspekhi Mat. Nauk, 42 (1987), 25-60, 246; English translation: Russian Math. Surveys, 42 (1987), 27-73.  Google Scholar

[15]

J. Mujica, Complex Analysis in Banach Spaces. Holomorphic Functions and Domains of Holomorphy in Finite and Infinite Dimensions, North-Holland Mathematics Studies, 120, North-Holland Publishing Co., Amsterdam, 1986.  Google Scholar

[16]

Z. Nitecki, Differentiable Dynamics, The MIT Press, Cambridge, Mass.-London, 1971.  Google Scholar

[17]

V. I. Opoitsev, A converse of the contraction mapping principle, Uspehi Mat. Nauk, 31 (1976), 169-198.  Google Scholar

[18]

B. N. Sadovskii, About one fixed point principle, (in Russian) Functional Analysis and Applications, 1 (1967), 74-76; English translation: Functional Analysis and Its Applications, 1 (1967), 151-153.  Google Scholar

[19]

B. N. Sadovskii, Limit-compact and condensing operators, (in Russian) Uspehi Mat. Nauk, 27 (1972), 81-146; English translation: Russian Mathematical Surveys, 27 (1972), 85-155.  Google Scholar

[20]

V. E. Slyusarchuk, Counterexample to a conjecture on smooth mappings, Russian Mathematical Surveys, 53 (1998), 408-409. doi: 10.1070/rm1998v053n02ABEH000046.  Google Scholar

[21]

M.-H. Shih and J.-W. Wu, Asymptotic stability in the Schauder fixed point theorem, Stud. Math., 131 (1998), 143-148.  Google Scholar

show all references

References:
[1]

R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, Measures of Noncompactness and Condensing Operators, Translated from the 1986 Russian original by A. Iacob, Operator Theory: Advances and Applications, 55, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0348-5727-7.  Google Scholar

[2]

G. R. Belitskii and Yu. I. Lyubich, Matrix norms and their applications, Operator Theory: Advances and Applications, 36, Birkhäuser Verlag, Basel, 1988. doi: 10.1007/978-3-0348-7400-7.  Google Scholar

[3]

D. N. Cheban, Global Attractors of Non-Autonomous Dissipstive Dynamical Systems, Interdisciplinary Mathematical Sciences, 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004. doi: 10.1142/9789812563088.  Google Scholar

[4]

D. N. Cheban, Markus-Yamabe conjecture, in Communications, The 14-th Conference of Applied and Industrial Mathematics, Chisinau, 2006, 107-110. Google Scholar

[5]

D. N. Cheban, Lyapunov Stability of Non-Autonomous Dynamical Systems, Nova Science Publishers Inc., New York, 2013. Google Scholar

[6]

D. N. Cheban and C. Mammana, Absolute asymptotic stability of discrete linear inclusions, Bul. Acad. Ştiinţe Repub. Mold. Mat., (2005), 43-68.  Google Scholar

[7]

G. Darbo, Punti uniti in transformazioni non-compacto, Rend. Sem. Mat. Univ. Padova, 24 (1955), 84-92.  Google Scholar

[8]

C. J. Earle and R. S. Hamilton, A fixed point theorem for holomorphic mappings, in Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., (1968)), American Mathematical Society, Rhode Island, 1970, 61-65.  Google Scholar

[9]

J. K. Hale, Asymptotic Behaviour of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

[10]

J. K. Hale and O. Lopes, Fixed point theorems and dissipative processes, Journal of Differential Equations, 13 (1973), 391-402. doi: 10.1016/0022-0396(73)90025-9.  Google Scholar

[11]

A. Halanay and D. Wexler, Teoria Calitativă a Sistemelor cu Impulsuri, Bucureşti, 1968. Google Scholar

[12]

L. A. Harris, Fixed points of holomorphic mappings for domains in Banach spaces, Abstract and Applied Analysis, (2003), 261-274. doi: 10.1155/S1085337503205042.  Google Scholar

[13]

G. S. Jones, The existence of critical points in generalized dynamical systems, in Seminaire on Differential Equations and Dynamical Systems, Lecture Notes in Math., Springer, Berlin, 1968, 7-19.  Google Scholar

[14]

O. A. Ladyzhenskaya, Finding minimal global attractors for Navier-Stokes equations and other partial differential equations, Uspekhi Mat. Nauk, 42 (1987), 25-60, 246; English translation: Russian Math. Surveys, 42 (1987), 27-73.  Google Scholar

[15]

J. Mujica, Complex Analysis in Banach Spaces. Holomorphic Functions and Domains of Holomorphy in Finite and Infinite Dimensions, North-Holland Mathematics Studies, 120, North-Holland Publishing Co., Amsterdam, 1986.  Google Scholar

[16]

Z. Nitecki, Differentiable Dynamics, The MIT Press, Cambridge, Mass.-London, 1971.  Google Scholar

[17]

V. I. Opoitsev, A converse of the contraction mapping principle, Uspehi Mat. Nauk, 31 (1976), 169-198.  Google Scholar

[18]

B. N. Sadovskii, About one fixed point principle, (in Russian) Functional Analysis and Applications, 1 (1967), 74-76; English translation: Functional Analysis and Its Applications, 1 (1967), 151-153.  Google Scholar

[19]

B. N. Sadovskii, Limit-compact and condensing operators, (in Russian) Uspehi Mat. Nauk, 27 (1972), 81-146; English translation: Russian Mathematical Surveys, 27 (1972), 85-155.  Google Scholar

[20]

V. E. Slyusarchuk, Counterexample to a conjecture on smooth mappings, Russian Mathematical Surveys, 53 (1998), 408-409. doi: 10.1070/rm1998v053n02ABEH000046.  Google Scholar

[21]

M.-H. Shih and J.-W. Wu, Asymptotic stability in the Schauder fixed point theorem, Stud. Math., 131 (1998), 143-148.  Google Scholar

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