Banach function spaces and exponential instability of evolution families

Mihail Megan; Adina Luminiţa Sasu; Bogdan Sasu

Archivum Mathematicum (2003)

  • Volume: 039, Issue: 4, page 277-286
  • ISSN: 0044-8753

Abstract

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In this paper we give necessary and sufficient conditions for uniform exponential instability of evolution families in Banach spaces, in terms of Banach function spaces. Versions of some well-known theorems due to Datko, Neerven, Rolewicz and Zabczyk, are obtained for the case of uniform exponential instability of evolution families.

How to cite

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Megan, Mihail, Sasu, Adina Luminiţa, and Sasu, Bogdan. "Banach function spaces and exponential instability of evolution families." Archivum Mathematicum 039.4 (2003): 277-286. <http://eudml.org/doc/249138>.

@article{Megan2003,
abstract = {In this paper we give necessary and sufficient conditions for uniform exponential instability of evolution families in Banach spaces, in terms of Banach function spaces. Versions of some well-known theorems due to Datko, Neerven, Rolewicz and Zabczyk, are obtained for the case of uniform exponential instability of evolution families.},
author = {Megan, Mihail, Sasu, Adina Luminiţa, Sasu, Bogdan},
journal = {Archivum Mathematicum},
keywords = {evolution family; uniform exponential instability; Banach function spaces; uniform exponential instability},
language = {eng},
number = {4},
pages = {277-286},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Banach function spaces and exponential instability of evolution families},
url = {http://eudml.org/doc/249138},
volume = {039},
year = {2003},
}

TY - JOUR
AU - Megan, Mihail
AU - Sasu, Adina Luminiţa
AU - Sasu, Bogdan
TI - Banach function spaces and exponential instability of evolution families
JO - Archivum Mathematicum
PY - 2003
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 039
IS - 4
SP - 277
EP - 286
AB - In this paper we give necessary and sufficient conditions for uniform exponential instability of evolution families in Banach spaces, in terms of Banach function spaces. Versions of some well-known theorems due to Datko, Neerven, Rolewicz and Zabczyk, are obtained for the case of uniform exponential instability of evolution families.
LA - eng
KW - evolution family; uniform exponential instability; Banach function spaces; uniform exponential instability
UR - http://eudml.org/doc/249138
ER -

References

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  1. Chow S. N., Leiva H., Existence and roughness of the exponential dichotomy for linear skew-product semiflows in Banach space, J. Differential Equations 120 (1995), 429–477. (1995) MR1347351
  2. Chicone C., Latushkin Y., Evolution Semigroups in Dynamical Systems and Differential Equations, Math. Surveys Monogr. 70, Amer. Math. Soc., 1999. (1999) Zbl0970.47027MR1707332
  3. Daleckii J. L., Krein M. G., Stability of Solutions of Differential Equations in Banach Spaces, Transl. Math. Monogr. 43, Amer. Math. Soc., Providence, R.I., 1974. (1974) MR0352639
  4. Datko R., Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal. 3 (1972), 428–445. (1972) Zbl0241.34071MR0320465
  5. Meyer-Nieberg P., Banach Lattices, Springer Verlag, Berlin, Heidelberg, New York, 1991. (1991) Zbl0743.46015MR1128093
  6. Megan M., Sasu B., Sasu A. L., On uniform exponential stability of evolution families, Riv. Mat. Univ. Parma 4 (2001), 27–43. Zbl1003.34045MR1878009
  7. Megan M., Sasu A. L., Sasu B., Nonuniform exponential instability of evolution operators in Banach spaces, Glas. Mat. Ser. III 56 (2001), 287–295. MR1884449
  8. Megan M., Sasu B., Sasu A. L., On nonuniform exponential dichotomy of evolution operators in Banach spaces, Integral Equations Operator Theory 44 (2002), 71–78. Zbl1034.34056MR1913424
  9. Megan M., Sasu A. L., Sasu B., On uniform exponential stability of linear skew- -product semiflows in Banach spaces, Bull. Belg. Math. Soc. Simon Stevin 9 (2002), 143–154. Zbl1032.34046MR1905653
  10. Megan M., Sasu A. L., Sasu B., Discrete admissibility and exponential dichotomy for evolution families, Discrete Contin. Dynam. Systems 9 (2003), 383–397. Zbl1032.34048MR1952381
  11. Megan M., Sasu A. L., Sasu B., Theorems of Perron type for uniform exponential dichotomy of linear skew-product semiflows, Bull. Belg. Mat. Soc. Simon Stevin 10 (2003), 1–21. Zbl1045.34022MR2032321
  12. Megan M., Sasu A. L., Sasu B., Perron conditions for uniform exponential expansiveness of linear skew-product flows, Monatsh. Math. 138 (2003), 145–157. Zbl1023.34043MR1964462
  13. Megan M., Sasu B., Sasu A. L., Exponential expansiveness and complete admissibility for evolution families, Czech. Math. J. 53 (2003). Zbl1080.34546MR2086730
  14. Megan M., Sasu A. L., Sasu B., Perron conditions for pointwise and global exponential dichotomy of linear skew-product flows, accepted for publication in Integral Equations Operator Theory. Zbl1064.34035MR2105960
  15. Megan M., Sasu A. L., Sasu B., Theorems of Perron type for uniform exponential stability of linear skew-product semiflows, accepted for publication in Dynam. Contin. Discrete Impuls. Systems. Zbl1079.34047
  16. van Minh N., Räbiger F., Schnaubelt R., Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Integral Equations Operator Theory 32 (1998), 332–353. (1998) Zbl0977.34056MR1652689
  17. van Neerven J. M. A. M., Exponential stability of operators and operator semigroups, J. Funct. Anal. 130 (1995), 293–309. (1995) Zbl0832.47034MR1335382
  18. van Neerven J. M. A. M., The Asymptotic Behaviour of Semigroups of Linear Operators, Operator Theory Adv. Appl. 88, Birkhäuser, Bassel, 1996. (1996) Zbl0905.47001MR1409370
  19. Rolewicz S., On uniform N - equistability, J. Math. Anal. Appl. 115 (1986), 434–441. (1986) Zbl0597.34064MR0836237
  20. Zabczyk J., Remarks on the control of discrete-time distributed parameter systems, SIAM J. Control Optim. 12 (1994), 721–735. (1994) MR0410506

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