Glasnik Matematicki, Vol. 51, No. 2 (2016), 379-390.

PERRON TYPE THEOREMS FOR SKEW-EVOLUTION SEMIFLOWS

Ciprian Preda, Sebastian Rămneanţu and Raluca Mureşan

Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timişoara, 300115 Timişoara, Romania
e-mail: ramneantusebastian@yahoo.com

Department of Computer Science, Faculty of Mathematics and Computer Science, West University of Timişoara, 300115 Timişoara, Romania
e-mail: raluca.muresan@e-uvt.ro

Abstract.   In the present paper we obtain some results for the asymptotic behavior of a large class of evolution families. Our approach uses the admissibility method initiated by O. Perron in the 1930's but the test functions that we choose are different from those employed in the case of differential systems.

2010 Mathematics Subject Classification.   34D05, 34D09.

Key words and phrases.   Linear skew-evolution semiflow, evolution cocycle, uniform exponential stability, nonuniform exponential stability, Perron method.

DOI: 10.3336/gm.51.2.07

References:

1. L. Barreira, D. Dragičević and C. Valls, Characterization of strong exponential dichotomies, Bull. Braz. Math. Soc. (N.S.) 46 (2015), 81-103.
   MathSciNet     CrossRef

2. L. Barreira, D. Dragičević and C. Valls, Strong and weak (L^p,L^q)-admissibility, Bull. Sci. Math. 138 (2014), 721-741.
   MathSciNet     CrossRef

3. L. Barreira, D. Dragičević and C. Valls, Exponential dichotomies with respect to a sequence of norms and admissibility, Internat. J. Math. 25 (2014), 1450024, 20 pp.
   MathSciNet     CrossRef

4. C. Chicone and Y. Latushkin, Evolution semigroups in dynamical systems and differential equations, AMS, Providence, 1999.
   MathSciNet     CrossRef

5. S.-N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces, J. Differential Equations 120 (1995), 429-477.
   MathSciNet     CrossRef

6. S.-N. Chow and H. Leiva, Two definitions of exponential dichotomy for skew-product semiflow in Banach spaces, Proc. Amer. Math. Soc. 124 (1996), 1071-1081.
   MathSciNet     CrossRef

7. W. A. Coppel, Dichotomies in stability theory, Springer-Verlag, Berlin-New York, 1978.
   MathSciNet    

8. J. L. Daleckij and M. G. Kreĭn, Stability of differential equations in Banach space, Amer. Math. Soc., Providence, Rhode Island, 1974.
   MathSciNet    

9. P. Hartman, Ordinary differential equations, Wiley, New-York, London, Sydney, 1964.
   MathSciNet    

10. Yu. D. Latushkin and A. M. Stëpin, Linear skew-product flows and semigroups of weighted composition operators, in Lyapunov exponents, Springer-Verlag, New-York, 1991.
    MathSciNet     CrossRef

11. Y. Latushkin and R. Schnaubelt, Evolution semigroups, translation algebra and exponential dichotomy of cocycles, J. Differential Equations 159 (1999), 321-369.
    MathSciNet     CrossRef

12. J. L. Massera and J. J. Schäffer, Linear differential equations and functional analysis. I, Ann. Math. (2) 67 (1958), 517-573.
    MathSciNet     CrossRef

13. J. L. Massera and J. J. Schäffer, Linear differential equations and function spaces, Academic Press, New-York, 1966.
    MathSciNet    

14. M. Megan and C. Stoica, Exponential instability of skew-evolution semiflows in Banach spaces, Stud. Univ. Babeş-Bolyai Math. 53 (2008), 17-24.
    MathSciNet    

15. O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), 703-728.
    MathSciNet     CrossRef

16. C. Preda, P. Preda and A. Petre, On the asymptotic behavior of an exponentially bounded, strongly continuous cocycle over a semiflow, Commun. Pure Appl. Anal. 8 (2009), 1637-1645. CrossRef

17. P. Preda, A. Pogan and C. Preda, Schäffer spaces and uniform exponential stability of linear skew-product semiflows, J. Diff. Eq. 212(2005), 191-207.
    MathSciNet     CrossRef

18. M. Reghiş, On nonuniform asymptotic stability, Prikl Math. Meh. 27 (1963), 231-243 (Russian) [English transl. J. Appl. Math. Mech. 27(1963), 344-362].
    MathSciNet     CrossRef

19. M. Rasmussen, Dichotomy spectra and Morse decompositions of linear nonautonomous differential equations, J. Differential Equations 246 (2009), 2242-2263.
    MathSciNet     CrossRef

20. R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations 113 (1994), 17-67.
    MathSciNet     CrossRef

21. G. R. Sell and Y. You, Dynamics of evolutionary equations, Springer Verlag, New-York, 2002.
    MathSciNet     CrossRef