Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
polynomial expansiveness; evolution family
Summary:
The paper investigates the interaction between the notions of expansiveness and admissibility. We consider a polynomially bounded discrete evolution family and define an admissibility notion via the solvability of an associated difference equation. Using the admissibility of weighted Lebesgue spaces of sequences, we give a characterization of discrete evolution families which are polynomially expansive and also those which are expansive in the ordinary sense. Thereafter, we apply the main results in order to infer continuous-time characterizations for the notions of expansiveness through the solvability of an associated integral equation.
References:
[1] Barreira, L., Valls, C.: Polynomial growth rates. Nonlinear Anal., Theory Methods Appl. 71 (2009), 5208-5219. DOI 10.1016/j.na.2009.04.005 | MR 2560190 | Zbl 1181.34046
[2] Bătăran, F., Preda, C., Preda, P.: Extending some results of L. Barreira and C. Valls to the case of linear skew-product semiflows. Result. Math. 72 (2017), 965-978. DOI 10.1007/s00025-017-0666-8 | MR 3684470 | Zbl 1375.37091
[3] Coffman, C. V., Schäffer, J. J.: Dichotomies for linear difference equations. Math. Ann. 172 (1967), 139-166. DOI 10.1007/BF01350095 | MR 0214946 | Zbl 0189.40303
[4] Daletskij, Ju. L., Krejn, M. G.: Stability of Solutions of Differential Equations in Banach Space. Translations of Mathematical Monographs 43. American Mathematical Society, Providence (1974). DOI 10.1090/mmono/043 | MR 0352639 | Zbl 0286.34094
[5] Hai, P. V.: On the polynomial stability of evolution families. Appl. Anal. 95 (2016), 1239-1255. DOI 10.1080/00036811.2015.1058364 | MR 3479001 | Zbl 1343.34143
[6] Hai, P. V.: Polynomial stability of evolution cocycles and Banach function spaces. Bull. Belg. Math. Soc.---Simon Stevin 26 (2019), 299-314. DOI 10.36045/bbms/1561687567 | MR 3975830 | Zbl 07094830
[7] Levitan, B. M., Zhikov, V. V.: Almost Periodic Functions and Differential Equations. Cambridge University Press, Cambridge (1982). MR 0690064 | Zbl 0499.43005
[8] Li, T.: Die Stabilitätsfrage bei Differenzengleichungen. Acta Math. 63 (1934), 99-141 German \99999JFM99999 60.0397.03. DOI 10.1007/BF02547352 | MR 1555392
[9] Massera, J. L., Schäffer, J. J.: Linear Differential Equations and Function Spaces. Pure and Applied Mathematics 21. Academic Press, New York (1966). MR 0212324 | Zbl 0243.34107
[10] Megan, M., Sasu, A. L., Sasu, B.: Theorems of Perron type for uniform exponential dichotomy of linear skew-product semiflows. Bull. Belg. Math. Soc.---Simon Stevin 10 (2003), 1-21. DOI 10.36045/bbms/1047309409 | MR 2032321 | Zbl 1045.34022
[11] Megan, M., Sasu, B., Sasu, A. L.: Exponential expansiveness and complete admissibility for evolution families. Czech. Math. J. 54 (2004), 739-749. DOI 10.1007/s10587-004-6422-8 | MR 2086730 | Zbl 1080.34546
[12] Minh, N. V., Räbiger, F., Schnaubelt, R.: Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line. Integral Equations Oper. Theory 32 (1998), 332-353. DOI 10.1007/BF01203774 | MR 1652689 | Zbl 0977.34056
[13] Perron, O.: Die Stabilitätsfrage bei Differentialgleichungen. Math. Z. 32 (1930), 703-728 German \99999JFM99999 56.1040.01. DOI 10.1007/BF01194662 | MR 1545194
[14] Popa, I.-L., Ceauşu, T., Megan, M.: Nonuniform power instability and Lyapunov sequences. Appl. Math. Comput. 247 (2014), 969-975. DOI 10.1016/j.amc.2014.09.051 | MR 3270899 | Zbl 1338.34101
[15] Popa, I.-L., Megan, M., Ceauşu, T.: Exponential dichotomies for linear discrete-time systems in Banach spaces. Appl. Anal. Discrete Math. 6 (2012), 140-155. DOI 10.2298/AADM120319008P | MR 2952610 | Zbl 1289.39030
[16] Preda, C., Preda, P., Petre, A.-P.: On the uniform exponential stability of linear skew-product three-parameter semiflows. Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 54 (2011), 269-279. MR 2856303 | Zbl 1274.34174
[17] Sasu, B.: New criteria for exponential expansiveness of variational difference equations. J. Math. Anal. Appl. 327 (2007), 287-297. DOI 10.1016/j.jmaa.2006.04.024 | MR 2277412 | Zbl 1115.39005
[18] Slyusarchuk, V. E.: Instability of difference equations with respect to the first approximation. Differ. Uravn. 22 (1986), 722-723 Russian. MR 0843238 | Zbl 0606.39003
Partner of
EuDML logo