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Periodic solutions of second order differential equations in Banach spaces

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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 properties of the cosine function. The results obtained are applied to elliptic partial differential operators.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Natural Sciences, University of Puerto Rico, PO Box 23355, San Juan, PR, 00931, USA

    Valentin Keyantuo

  2. Departamento de Matemática, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307-Correo 2, Santiago, Chile

    Carlos Lizama

Authors
  1. Valentin Keyantuo
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  2. Carlos Lizama
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Correspondence to Valentin Keyantuo.

Additional information

The first author is supported in part by Convenio de Cooperación Internacional (CONICYT) Grant # 7010675 and the second author is partially financed by FONDECYT Grant # 1010675

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Keyantuo, V., Lizama, C. Periodic solutions of second order differential equations in Banach spaces. Math. Z. 253, 489–514 (2006). https://doi.org/10.1007/s00209-005-0919-1

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