A NOTE ON SOME THEOREMS OF R. DATKO
The asymptotic behavior of the evolution families is a widely interesting topic in mathematics over time. In 1930, O. Perron was the first one who established the connection between the asymptotic behavior of the solution of the homogenous differential equation and the associated non-homogeneous equation, in finite dimensional spaces. Further, the result was extended for infinite dimensional spaces. The case of dynamical systems described by evolution processes was studied by C. Chicone and Y. Latushkin. One of the most remarkable results in the theory of stability of dynamical systems has been obtained by R. Datko in 1970 for the particular case of C0-semigroups. Practically, R. Datko defines a characterization for uniform exponential stability of the C0-semigroups. Later, it was proved that a similar characterization is also valid for two-parameter evolution families.
In this paper we obtain different versions of a well-known theorem of R. Datko for uniform and nonuniform exponential bounded evolution families. More precisely, we obtain theorems that characterize the nonuniform and uniform exponential stability of evolution families with uniform and nonuniform exponential growth. We show that, if we choose K dependent of t0 in the form of Datko's theorem used by C. Stoica and M. Megan, we obtain a result of nonuniform exponential stability, which is no longer possible in the original form of Datko's theorem.
In conclusion, we generalize the results initially obtained by Datko (1972) and Preda and Megan (1985), by presenting some sufficient conditions for the nonuniform exponential stability of evolution families with nonuniform exponential growth.
Barreira, L.,& Valls, C. (2010). Admissibility for nonuniform exponential contractions. Journal of Differential Equations,
Barreira, L.,& Valls, C. (2011). Nonuniform exponential dichotomies and admissibility. Discrete and Continuous Dynamical
Buse, C. (1994). On nonuniform exponential stability of evolutionary process. Rend. Sem. Mat. Univ. Pol. Torino, 52:395-406. Buse, C. (1997). Asymptotic stability of evolutors and normed function spaces. Rend. Sem. Mat. Univ. Pol. Torino, 55:109-122. Chicane, C.,& Latushkin, Y. (1999). Evolution Semigroups in Dynamical Systems and Differential Equations. Math. Surveys
Monography, Amer. Math. Soc., Providence, RI, 70.
Coppel, W.A. (1978). Dichotomies in Stability Theory, Leet. Notes Math., Springer-Verlag, New-York, 629.
Daleckij, J.L.,& Krein, M.G. (1974). Stability of Differential Equations in Banach Space. Amer. Math. Soc., Providence, RI. Datko, R. (1970). Extending a theorem of Liapunov to Hilbert spaces. J. Math. Anal. Appl., 32:610-616.
Datko, R. (1972). Uniform asymptotic stability of evolutionary processes in a Banach space. SIAM J. Math. Anal., 3:428-445.
Hartman, P. (1964). Ordinary Differential Equations. Wiley, New-York, London, Sydney.
Massera, J .L.,& Schaffer, J .J. ( 1958). Linear differential equations and functional analysis. The Annals of Mathematics, 67 :517-
Massera, J.L.,& Schaffer, J.J. (1966). Linear Differential Equations and Function Spaces. Academic Press, New York.
Megan, M., Sasu, A.L.,& Sasu, B. (2002). On uniform exponential stability of linear skew-product semiflows in Banach spaces. Bull. Belg. Math. Soc. Simon Stevin, 9: 143-154.
Megan, M., Sasu, A.L.,& Sasu, B. (2003). Perron conditions for uniform exponential expansiveness of linear skew-product flows. Monatsh. Math., 138:145-157.
Pazy, A. (1972). On the applicability of Liapunovs theorem in Hilbert spaces. SIAM J. Math. Anal. Appl., 3:291-294.
Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York. Perron, 0. (1930). Die stabilitatsfrage bei diferentialgeighungen. Math. Z., 32:703-728.
Preda, P.,& Megan, M. (1985). Exponential dichotomy of evolutionary processes in Banach spaces. Czechoslovak Math. J.,
Preda, C., Preda, P.,& Craciunescu, A. (2012). A version of a theorem of R. Datko for nonuniform exponential contractions. J.
Math. Anal. Appl., 385:572-581.
Rolewicz, S. (1986). On uniform N-equistability. J. Math. Anal. Appl., 115:434-441.
Stoica, C.,& Megan, M. (2010). On uniform exponential stability for skew-evolution semiflows on Banach spaces. Nonlinear
Analysis, 72: 1305-1313.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (Creative Commons Attribution License 3.0 - CC BY 3.0) that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
firstname.lastname@example.org, www.iseic.cz, ojs.journals.cz