# American Institute of Mathematical Sciences

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

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

 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.
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.

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##### 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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