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Almost periodic solutions of first- and second-order Cauchy problems. (English) Zbl 0879.34046
Let \(S(t)\) be the shift group on the space \(B\cup C(\mathbb{R},X)\) of all bounded uniformly continuous functions \(x: \mathbb{R}\to X\), where \(X\) is a complete \(B\)-space and let \(\overline S(t)\) be the induced group on \(B\cup C(\mathbb{R}, X)/AP(\mathbb{R},X)\) with generator \(\overline B\). The authors use spectral properties of bounded groups to reformulate and prove the known Kadet’s result, namely it holds \(c_0 \not \subset X\) iff \(\overline B\) has no point spectrum, or, in other situations, if an ergodicity condition holds. Section 3 contains a spectral characterisation of almost periodic functions and the results are used in Section 4 to prove almost periodicity of solutions of some first- and second-order inhomogeneous Cauchy problems. The paper closes with an investigation of the case where the imaginary spectrum of the operator consists only of poles.

MSC:
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
47A10 Spectrum, resolvent
34G10 Linear differential equations in abstract spaces
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[1] Arendt, W.; Batty, C.J.K., Tauberian theorems and stability of one-parameter semigroups, Trans. amer. math. soc., 306, 783-804, (1988) · Zbl 0652.47022
[2] W. Arendt, C. J. K. Batty, Asymptotically almost periodic solutions of inhomogeneous Cauchy problems on the half-line · Zbl 0952.34048
[3] Arendt, W.; Prüss, J., Vector-valued Tauberian theorems and asymptotic behavior of linear Volterra equations, SIAM J. math. anal., 23, 412-448, (1992) · Zbl 0765.45009
[4] Allan, G.R.; Ransford, T.J., Power-dominated elements in a Banach algebra, Studia math., 94, 63-79, (1989) · Zbl 0705.46021
[5] Arveson, W., The harmonic analysis of automorphism groups, Proc. sympos. pure math., 38, 199-269, (1982)
[6] B. Basit, Harmonic analysis and asymptotic behavior of solutions of the abstract Cauchy problem · Zbl 0868.47027
[7] Basit, B., Some problems concerning different types of vector-valued almost periodic functions, Dissertationes math. (rozprawy mat), 338, (1995) · Zbl 0828.43004
[8] C. J. K. Batty, J. van Neerven, F. Räbiger, Local spectra and individual stability of bounded C_0 -semigroups, Trans. Amer. Math. Soc. · Zbl 0893.47026
[9] C. J. K. Batty, J. van Neerven, F. Räbiger, Tauberian theorems and stability of solutions of the Cauchy problem, Trans. Amer. Math. Soc.
[10] E. B. Davies, 1980, One-Parameter Semigroups, Academic Press, London · Zbl 0457.47030
[11] Fink, A.M., Springer lecture notes, 377, (1974)
[12] Korevaar, J., On Newman’s quick way to the prime number theorem, Math. intelligencer, 4, 108-115, (1982) · Zbl 0496.10027
[13] B. M. Levitan, V. V. Zhikov, 1982, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge · Zbl 0499.43005
[14] Nagel, R., Lecture notes in mathematics, 1184, (1984)
[15] J. van Neerven, 1996, The Asymptotic behaviour of Semigroups of Linear Operators, Birkhäuser, Basel
[16] J. Prüss, 1993, Evolutionary Integral Equations and Applications, Birkhäuser, Basel
[17] Ruess, W.M.; Vũ Quôc Phóng, Asymptotically almost periodic solutions of evolution equations in Banach spaces, J. differential equations, 122, 282-301, (1995) · Zbl 0837.34067
[18] Vũ Quôc Phóng, On the spectrum, complete trajectories, and asymptotic stability of linear semi-dynamical systems, J. differential equations, 105, 30-45, (1993) · Zbl 0786.34068
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