Stability and almost periodicity of solutions of ill-posed abstract Cauchy problems

deLaubenfels, R.; Phóng, Vũ

doi:10.1090/S0002-9939-97-03575-2

Stability and almost periodicity of solutions of ill-posed abstract Cauchy problems
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by R. deLaubenfels and Vũ Quôc Phóng

Proc. Amer. Math. Soc. 125 (1997), 235-241

DOI: https://doi.org/10.1090/S0002-9939-97-03575-2

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Abstract:

We give simple spectral sufficient conditions for a solution of the linear abstract Cauchy problem, on a Banach space, to be strongly stable or asymptotically almost periodic, without assuming that the associated operator generates a $C_{0}$-semigroup.

References

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc. 306 (1988), no. 2, 837–852. MR 933321, DOI 10.1090/S0002-9947-1988-0933321-3
Charles J. K. Batty and Vũ Quốc Phóng, Stability of individual elements under one-parameter semigroups, Trans. Amer. Math. Soc. 322 (1990), no. 2, 805–818. MR 1022866, DOI 10.1090/S0002-9947-1990-1022866-5
Ralph deLaubenfels, Existence families, functional calculi and evolution equations, Lecture Notes in Mathematics, vol. 1570, Springer-Verlag, Berlin, 1994. MR 1290783, DOI 10.1007/BFb0073401
Ralph deLaubenfels and Shmuel Kantorovitz, Laplace and Laplace-Stieltjes space, J. Funct. Anal. 116 (1993), no. 1, 1–61. MR 1237985, DOI 10.1006/jfan.1993.1103
K. de Leeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math. 105 (1961), 63–97. MR 131784, DOI 10.1007/BF02559535
Omar El-Mennaoui, Asymptotic behaviour of integrated semigroups, J. Comput. Appl. Math. 54 (1994), no. 3, 351–369. MR 1321079, DOI 10.1016/0377-0427(94)90256-9
J. Esterle, E. Strouse, and F. Zouakia, Stabilité asymptotique de certains semi-groupes d’opérateurs et idéaux primaires de $L^1(\textbf {R}^+)$, J. Operator Theory 28 (1992), no. 2, 203–227 (French). MR 1273043
Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
S. Huang, Characterizing spectra of closed operators through existence of slowly growing solutions of their Cauchy problems, Studia Math. 116 (1995), 23–41.
Shmuel Kantorovitz, The Hille-Yosida space of an arbitrary operator, J. Math. Anal. Appl. 136 (1988), no. 1, 107–111. MR 972586, DOI 10.1016/0022-247X(88)90118-7
Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), no. 3, 313–328. MR 859138, DOI 10.1016/0022-1236(86)90101-1
S. G. Kreĭn, G. I. Laptev, and G. A. Cvetkova, The Hadamard correctness of the Cauchy problem for an evolution equation, Dokl. Akad. Nauk SSSR 192 (1970), 980–983 (Russian). MR 0265728
B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations, Cambridge University Press, Cambridge-New York, 1982. Translated from the Russian by L. W. Longdon. MR 690064
Yu. I. Lyubich and Vũ Quốc Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math. 88 (1988), no. 1, 37–42. MR 932004, DOI 10.4064/sm-88-1-37-42
W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck, One-parameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184, Springer-Verlag, Berlin, 1986. MR 839450, DOI 10.1007/BFb0074922
Vu Kuok Fong and Yu. I. Lyubich, A spectral criterion for almost periodicity for one-parameter semigroups, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 47 (1987), 36–41 (Russian); English transl., J. Soviet Math. 48 (1990), no. 6, 644–647. MR 916441, DOI 10.1007/BF01094717
Vũ Quốc Phóng, Theorems of Katznelson-Tzafriri type for semigroups of operators, J. Funct. Anal. 103 (1992), no. 1, 74–84. MR 1144683, DOI 10.1016/0022-1236(92)90135-6
Vũ Quốc Phóng, On the spectrum, complete trajectories, and asymptotic stability of linear semi-dynamical systems, J. Differential Equations 105 (1993), no. 1, 30–45. MR 1237976, DOI 10.1006/jdeq.1993.1081