2006
DOI: 10.1007/s00209-005-0919-1
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Periodic solutions of second order differential equations in Banach spaces

Abstract: Abstract. We use operator-valued Fourier multiplier theorems to study second order differential equations in Banach spaces. We establish maximal regularity results in L p and C s for strong solutions of a complete second order equation. In the second part, we study mild solutions for the second order problem. Two types of mild solutions are considered. When the operator A involved is the generator of a strongly continuous cosine function, we give characterizations in terms of Fourier multipliers and spectral p… Show more

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Cited by 41 publications
(30 citation statements)
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“…We note that in the special case when G = F = 0, L p -maximal regularity and B s p,q -maximal regularity of (P 2 ) have been studied by Arendt and Bu [1,2], and F s p,q -maximal regularity of (P 2 ) has been studied by Bu and Kim [7]. Maximal regularity of second order equations with periodic condition has been also studied by Keyantuo and Lizama [9] (see also [6]). …”
Section: Introductionmentioning
confidence: 93%
“…We note that in the special case when G = F = 0, L p -maximal regularity and B s p,q -maximal regularity of (P 2 ) have been studied by Arendt and Bu [1,2], and F s p,q -maximal regularity of (P 2 ) has been studied by Bu and Kim [7]. Maximal regularity of second order equations with periodic condition has been also studied by Keyantuo and Lizama [9] (see also [6]). …”
Section: Introductionmentioning
confidence: 93%
“…Explicitly it is due to Berkson and Gillespie [11] and Clément et al [17]. It plays a fundamental role in recent works by Clément-Da Prato [16], Weis [39,40], Arendt-Bu [4,5], Keyantuo-Lizama [31,32,33] and Ashyralyev et al [7,8].…”
Section: Introductionmentioning
confidence: 99%
“…We remark that the concept of R-boundedness was introduced by Bourgain [7] and play a fundamental role in recent works by Clément-Da Prato [8], Clément et al [9], Weis [18,19], Arendt-Bu [3,4] and Keyantuo-Lizama [13,14,15].…”
Section: Introductionmentioning
confidence: 99%