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

Preda, Ciprian ; Sipos, Ciprian

Canadian mathematical bulletin, Tome 54 (2011) no. 3, pp. 527-537

Résumé

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 $\mathcal{U}\,=\,{{\{U(t,\,s)\}}_{t\ge s\ge 0}}$ 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 ${{l}^{\Phi }}(\mathbb{N},\,X)$ admits a solution in the same ${{l}^{\Phi }}(\mathbb{N},\,X)$ . The technique of proof effectively eliminates the continuity hypothesis on the evolution family (i.e., we do not assume that $U(\,\cdot \,,\,s)x$ or $U(t,\,\cdot \,)x$ is continuous on $[s,\,\infty )$ , and respectively $[0,\,t])$ . Thus, some known results given by Coffman and Schaffer, Perron, and Ta Li are extended.

Détail

DOI : 10.4153/CMB-2010-105-1

Mots-clés : 34D05, 47D06, 93D20, evolution families, exponential dichotomy, Orlicz sequence spaces, admissibility

Bibliographie

[1] [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. 90–119. Google Scholar

[2] [2] Chicone, C. and Latushkin, Y., Evolution semigroups in dynamical systems and differential equations. Mathematical Surveys andMonographs, 70, American Mathematical Society, Providence, RI, 1999. Google Scholar

[3] [3] Coffman, C. V. and Schäffer, J. J., Dichotomies for linear difference equations. Math. Ann. 172(1967), 139–166. doi:10.1007/BF01350095 Google Scholar

[4] [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. Google Scholar

[5] [5] Henry, D., Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. Google Scholar

[6] [6] LaSalle, J. P., The stability and control of discrete processes. Applied Mathematical Sciences, 62, Springer-Verlag, New York, 1986. Google Scholar

[7] [7] Latushkin, Y. and Montgomery-Smith, S., Evolutionary semigroups and Lyapunov theorems in Banach spaces. J. Funct. Anal. 127(1995), no. 1, 173–197. doi:10.1006/jfan.1995.1007 Google Scholar

[8] [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, 472–500. doi:10.1007/BF01203919 Google Scholar

[9] [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. Google Scholar

[10] [10] Megan, M. and Preda, P., Admissibility and dichotomies for linear discrete-time systems in Banach spaces. An. Univ. Timişoara Ştiint Mat. 26(1988), 45–54. Google Scholar

[11] [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, 332–353. doi:10.1007/BF01203774 Google Scholar

[12] [12] van Minh, N., On the proof of characterizations of the exponential dichotomy. Proc. Amer. Math. Soc. 127(1999), no. 3, 779–782. doi:10.1090/S0002-9939-99-04640-7 Google Scholar

[13] [13] Perron, O., Die Stabilitätsfrage bei Differentialgeighungen. Math. Z. 32(1930), no. 1, 703–728. doi:10.1007/BF01194662 Google Scholar

[14] [14] Pinto, M., Discrete dichotomies. Advances in difference equations. Comput. Math. Appl. 28(1994), no. 1–3, 259–270. doi:10.1016/0898-1221(94)00114-6 Google Scholar

[15] [15] Preda, C., A discrete Perron-Ta Li type theorem for the dichotomy of evolution operators. J. Math. Anal. Appl. 332(2007), no. 1, 727–734. doi:10.1016/j.jmaa.2006.10.056 Google Scholar

[16] [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, 65–70. Google Scholar

[17] [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, 307–315. doi:10.1016/S0167-6911(84)80042-0 Google Scholar

[18] [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, 397–408. doi:10.1016/j.jmaa.2005.04.047 Google Scholar

[19] [19] Schnaubelt, R., Sufficient conditions for exponential stability and dichotomy of evolution equations. Forum Math. 11(1999), no. 5, 543–566. doi:10.1515/form.1999.013 Google Scholar

[20] [20] Li, Ta, Die Stabilitätsfrage bei Differenzgleichungen. Acta Math. 63(1934), 99–141. doi:10.1007/BF02547352 Google Scholar