Belitskii--Lyubich conjecture for C-analytic dynamical systems

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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

Abstract
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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 and 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.
[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.
[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.
[4]	D. N. Cheban, Markus-Yamabe conjecture, in Communications, The 14-th Conference of Applied and Industrial Mathematics, Chisinau, 2006, 107-110.
[5]	D. N. Cheban, Lyapunov Stability of Non-Autonomous Dynamical Systems, Nova Science Publishers Inc., New York, 2013.
[6]	D. N. Cheban and C. Mammana, Absolute asymptotic stability of discrete linear inclusions, Bul. Acad. Ştiinţe Repub. Mold. Mat., (2005), 43-68.
[7]	G. Darbo, Punti uniti in transformazioni non-compacto, Rend. Sem. Mat. Univ. Padova, 24 (1955), 84-92.
[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.
[9]	J. K. Hale, Asymptotic Behaviour of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.
[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.
[11]	A. Halanay and D. Wexler, Teoria Calitativă a Sistemelor cu Impulsuri, Bucureşti, 1968.
[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.
[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.
[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.
[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.
[16]	Z. Nitecki, Differentiable Dynamics, The MIT Press, Cambridge, Mass.-London, 1971.
[17]	V. I. Opoitsev, A converse of the contraction mapping principle, Uspehi Mat. Nauk, 31 (1976), 169-198.
[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.
[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.
[20]	V. E. Slyusarchuk, Counterexample to a conjecture on smooth mappings, Russian Mathematical Surveys, 53 (1998), 408-409. doi: 10.1070/rm1998v053n02ABEH000046.
[21]	M.-H. Shih and J.-W. Wu, Asymptotic stability in the Schauder fixed point theorem, Stud. Math., 131 (1998), 143-148.

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.
[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.
[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.
[4]	D. N. Cheban, Markus-Yamabe conjecture, in Communications, The 14-th Conference of Applied and Industrial Mathematics, Chisinau, 2006, 107-110.
[5]	D. N. Cheban, Lyapunov Stability of Non-Autonomous Dynamical Systems, Nova Science Publishers Inc., New York, 2013.
[6]	D. N. Cheban and C. Mammana, Absolute asymptotic stability of discrete linear inclusions, Bul. Acad. Ştiinţe Repub. Mold. Mat., (2005), 43-68.
[7]	G. Darbo, Punti uniti in transformazioni non-compacto, Rend. Sem. Mat. Univ. Padova, 24 (1955), 84-92.
[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.
[9]	J. K. Hale, Asymptotic Behaviour of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.
[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.
[11]	A. Halanay and D. Wexler, Teoria Calitativă a Sistemelor cu Impulsuri, Bucureşti, 1968.
[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.
[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.
[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.
[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.
[16]	Z. Nitecki, Differentiable Dynamics, The MIT Press, Cambridge, Mass.-London, 1971.
[17]	V. I. Opoitsev, A converse of the contraction mapping principle, Uspehi Mat. Nauk, 31 (1976), 169-198.
[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.
[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.
[20]	V. E. Slyusarchuk, Counterexample to a conjecture on smooth mappings, Russian Mathematical Surveys, 53 (1998), 408-409. doi: 10.1070/rm1998v053n02ABEH000046.
[21]	M.-H. Shih and J.-W. Wu, Asymptotic stability in the Schauder fixed point theorem, Stud. Math., 131 (1998), 143-148.

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