Weak asymptotic almost periodicity for semigroups of operators. (English) Zbl 0792.47044
Weak asymptotic almost periodicity of orbits and almost-orbits of a strongly continuous semigroup of operators (linear or nonlinear) is investigated. Under suitable conditions almost-orbits \(u\) are characterized as the sum of orbits and functions tending to zero at infinity. A further characterization is given in terms of weak to weak uniform equicontinuity of the semigroup on the range of \(u\).
These results are applied to the study of (weak) stability properties of semigroups and the problem of almost periodic motions for uniformly bounded \(C_ 0\)-semigroups of linear operators and for nonlinear contraction semigroups.
These results are applied to the study of (weak) stability properties of semigroups and the problem of almost periodic motions for uniformly bounded \(C_ 0\)-semigroups of linear operators and for nonlinear contraction semigroups.
Reviewer: J.Voigt (Dresden)
MSC:
| 47D06 | One-parameter semigroups and linear evolution equations |
| 47H20 | Semigroups of nonlinear operators |
| 43A60 | Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions |
| 46N99 | Miscellaneous applications of functional analysis |
Keywords:
asymptotic behaviour; weak asymptotic almost periodicity of orbits and almost-orbits; strongly continuous semigroup of operators; weak uniform equicontinuity; stability properties of semigroups; almost periodic motions for uniformly bounded \(C_ 0\)-semigroupsReferences:
| [1] | Amerio, L.; Prouse, G., Almost-Periodic Functions and Functional Equations (1971), Van Nostrand: Van Nostrand New York · Zbl 0215.15701 |
| [2] | Barbu, V., Nonlinear Semigroups and Differential Equations in Banach Spaces (1976), Noordhoff: Noordhoff Leyden · Zbl 0328.47035 |
| [3] | Bart, H.; Goldberg, S., Characterizations of almost periodic strongly continuous groups and semigroups, Math. Ann., 236, 105-116 (1978) · Zbl 0365.47019 |
| [4] | Bénilan, P., Solutions intégrales d’équations d’évolution dans un espace de Banach, C. R. Acad. Sci. Paris, 274, A47-A50 (1972) · Zbl 0246.47068 |
| [5] | Bochner, S., Abstrakte fastperiodische Funktionen, Acta Math., 61, 149-184 (1933) · JFM 59.0997.01 |
| [6] | Brézis, H., Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, (Zarantonello, E. H., Contributions to Nonlinear Functional Analysis (1971), Academic Press: Academic Press New York) · Zbl 0278.47033 |
| [7] | Bruck, R. E., Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Funct. Anal., 18, 15-26 (1975) · Zbl 0319.47041 |
| [8] | Bruck, R. E., On the weak asymptotic almost-periodicity of bounded solutions of \(u\)″ \(ϵ Au + f\) for monotone \(A\), J. Differential Equations, 37, 309-317 (1980) · Zbl 0439.34036 |
| [9] | Crandall, M. G.; Liggett, T. M., Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93, 265-298 (1971) · Zbl 0226.47038 |
| [10] | DeLeeuw, K.; Glicksberg, I., Applications of almost periodic compactifications, Acta Math., 105, 63-97 (1961) · Zbl 0104.05501 |
| [11] | DeLeeuw, K.; Glicksberg, I., Almost periodic functions on semigroups, Acta Math., 105, 99-140 (1961) · Zbl 0104.05601 |
| [12] | Eberlein, W. F., Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc., 67, 217-240 (1949) · Zbl 0034.06404 |
| [13] | Favard, J., Leçons sur les fonctions presque-périodiques, (Cahiers Scientifiques XIII (1933), Gauthier-Villars: Gauthier-Villars Paris) · JFM 59.0996.01 |
| [14] | Fink, A. M., Almost periodic differential equations, (Lecture Notes in Math., Vol. 377 (1974), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0325.34039 |
| [15] | Fitzgibbon, W. E., Weakly continuous nonlinear accretive operators in reflexive Banach spaces, (Proc. Amer. Math. Soc., 41 (1973)), 229-236 · Zbl 0283.34057 |
| [16] | Fréchet, M., Les fonctions asymptotiquement presque-périodiques continues, C. R. Acad. Sci. Paris, 213, 520-522 (1941) · JFM 67.0231.01 |
| [17] | Fréchet, M., Les fonctions asymptotiquement presque-périodiques, Rev. Sci., 79, 341-354 (1941) · JFM 67.1010.03 |
| [18] | Goldberg, S.; Irwin, P., Weakly almost periodic vector-valued functions, Dissertationes Math., 157 (1979) · Zbl 0414.43007 |
| [19] | Goldstein, J. A., Semigroups of linear operators and applications, (Oxford Math. Monographs (1985), Oxford Univ. Press: Oxford Univ. Press New York) · Zbl 0219.47037 |
| [20] | Gutman, S.; Pazy, A., An ergodic theorem for semigroups of contractions, (Proc. Amer. Math. Soc., 88 (1983)), 254-256 · Zbl 0517.47004 |
| [21] | Milnes, P., On vector-valued weakly almost periodic functions, J. London Math. Soc., 22, No. 2, 467-472 (1980) · Zbl 0456.43003 |
| [22] | Miyadera, I.; Kobayasi, K., On the asymptotic behavior of almost-orbits of nonlinear contraction semigroups in Banach spaces, Nonlinear Anal., 6, 349-365 (1982) · Zbl 0507.47041 |
| [23] | Pazy, A., On the asymptotic behavior of semigroups of nonlinear contractions in Hilbert space, J. Funct. Anal., 27, 292-307 (1978) · Zbl 0377.47045 |
| [24] | Pazy, A., Semigroups of linear operators and applications to partial differential equations, (Applied Math. Sci., Vol. 44 (1983), Springer-Verlag: Springer-Verlag New York) · Zbl 0516.47023 |
| [25] | Ruess, W. M.; Summers, W. H., Minimal sets of almost periodic motions, Math. Ann., 276, 145-158 (1986) · Zbl 0584.54038 |
| [26] | Ruess, W. M.; Summers, W. H., Asymptotic almost periodicity and motions of semigroups of operators, J. Linear Algebra Appl., 84, 335-351 (1986) · Zbl 0616.47047 |
| [27] | Ruess, W. M.; Summers, W. H., Compactness in spaces of vector valued continuous functions and asymptotic almost periodicity, Math. Nachr., 135, 7-33 (1988) · Zbl 0666.46007 |
| [28] | Ruess, W. M.; Summers, W. H., Weak almost periodicity and the strong ergodic limit theorem for contraction semigroups, Israel J. Math., 64, 139-157 (1988) · Zbl 0681.47036 |
| [29] | Ruess, W. M.; Summers, W. H., Integration of asymptotically almost periodic functions and weak asymptotic almost periodicity, Dissertationes Math., 279 (1989) · Zbl 0668.43005 |
| [30] | Ruess, W. M.; Summers, W. H., Weakly almost periodic semigroups of operators, Pacific J. Math., 143, 175-193 (1990) · Zbl 0655.47033 |
| [31] | W. M. Ruess and W. H. SummersProc. Amer. Math. Soc.; W. M. Ruess and W. H. SummersProc. Amer. Math. Soc. · Zbl 0789.47010 |
| [32] | Zaidman, S., Almost-periodic functions in abstract spaces, (Research Notes in Math., Vol. 126 (1985), Pitman: Pitman Boston) · Zbl 0648.42006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
