Skip to main content
×
×
Home

On the Dichotomy of the Evolution Families: A Discrete-Argument Approach

  • Ciprian Preda (a1) and Ciprian Sipos (a2)
Abstract

We establish a discrete-time criteria guaranteeing the existence of an exponential dichotomy in the continuous-time behavior of an abstract evolution family. We prove that an evolution family acting on a Banach space X is uniformly exponentially dichotomic (with respect to its continuous-time behavior) if and only if the corresponding difference equation with the inhomogeneous term from a vector-valued Orlicz sequence space l Φ(ℕ, X) admits a solution in the same l Φ(ℕ, X). The technique of proof effectively eliminates the continuity hypothesis on the evolution family (i.e., we do not assume that U( · , s)x or U(t, · )x is continuous on [s, ∞), and respectively [0, t]). Thus, some known results given by Coffman and Schaffer, Perron, and Ta Li are extended.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On the Dichotomy of the Evolution Families: A Discrete-Argument Approach
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On the Dichotomy of the Evolution Families: A Discrete-Argument Approach
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On the Dichotomy of the Evolution Families: A Discrete-Argument Approach
      Available formats
      ×
Copyright
References
Hide All
[1] Ben-Artzi, A. and Gohberg, I., Dichotomy of systems and invertibility of linear ordinary differential operators. In: Time-variant systems and interpolation, Oper. Theory Adv. Appl., 56, Birkhäuser, Basel, 1992, pp. 90119.
[2] Chicone, C. and Latushkin, Y., Evolution semigroups in dynamical systems and differential equations. Mathematical Surveys andMonographs, 70, American Mathematical Society, Providence, RI, 1999.
[3] Coffman, C. V. and Schäffer, J. J., Dichotomies for linear difference equations. Math. Ann. 172(1967), 139166. doi:10.1007/BF01350095
[4] Daleckii, J. L, Kreĭn, M. G., Stability of solutions of differential equations in Banach space. Translations of Mathematical Monographs, 43, American Mathematical Society, Providence, RI, 1974.
[5] Henry, D., Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
[6] LaSalle, J. P., The stability and control of discrete processes. Applied Mathematical Sciences, 62, Springer-Verlag, New York, 1986.
[7] Latushkin, Y. and Montgomery-Smith, S., Evolutionary semigroups and Lyapunov theorems in Banach spaces. J. Funct. Anal. 127(1995), no. 1, 173197. doi:10.1006/jfan.1995.1007
[8] Latushkin, Y. and Randolph, T., Dichotomy of differential equations on Banach spaces and an algebra of weighted translation operators. Integral Equations Operator Theory 23(1995), no. 4, 472500. doi:10.1007/BF01203919
[9] Massera, J. L. and Schäffer, J. J., Linear differential equations and function spaces. Pure and Applied Mathematics, 21, Academic Press, New York-London, 1966.
[10] Megan, M. and Preda, P., Admissibility and dichotomies for linear discrete-time systems in Banach spaces. An. Univ. Timişoara Ştiint Mat. 26(1988), 4554.
[11] van Minh, N., Räbiger, F., and Schnaubelt, R., Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line. Integral Equations Operator Theory 32(1998), no. 3, 332353. doi:10.1007/BF01203774
[12] van Minh, N., On the proof of characterizations of the exponential dichotomy. Proc. Amer. Math. Soc. 127(1999), no. 3, 779782. doi:10.1090/S0002-9939-99-04640-7
[13] Perron, O., Die Stabilitätsfrage bei Differentialgeighungen. Math. Z. 32(1930), no. 1, 703728. doi:10.1007/BF01194662
[14] Pinto, M., Discrete dichotomies. Advances in difference equations. Comput. Math. Appl. 28(1994), no. 1–3, 259270. doi:10.1016/0898-1221(94)00114-6
[15] Preda, C., A discrete Perron-Ta Li type theorem for the dichotomy of evolution operators. J. Math. Anal. Appl. 332(2007), no. 1, 727734. doi:10.1016/j.jmaa.2006.10.056
[16] Preda, P., On a Perron condition for evolutionary processes in Banach spaces. Bull. Math. Soc. Sci. Math. R.S. Roumanie (N.S.) 32(80)(1988), no. 1, 6570.
[17] Przyluski, K. M. and Rolewicz, S., On stability of linear time-varying infinite-dimensional discrete-time systems. Systems Control Lett. 4(1984), no. 5, 307315. doi:10.1016/S0167-6911(84)80042-0
[18] Sasu, B. and Sasu, A. L., Exponential dichotomy and (lp, lq )-admissibility on the half-line. J. Math. Anal. Appl. 316(2006), no. 2, 397408. doi:10.1016/j.jmaa.2005.04.047
[19] Schnaubelt, R., Sufficient conditions for exponential stability and dichotomy of evolution equations. Forum Math. 11(1999), no. 5, 543566. doi:10.1515/form.1999.013
[20] Li, Ta, Die Stabilitätsfrage bei Differenzgleichungen. Acta Math. 63(1934), 99141. doi:10.1007/BF02547352
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed