On solutions of differential inclusions in homogeneous spaces of functions
type: Item
instance of: scholarly article
title: On solutions of differential inclusions in homogeneous spaces of functions
zbMATH Open document ID: 1113.34045
author: Pietro Zecca
author: Anatoly G. Baskakov
author: Valerii V. Obukhovskij
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-11-15T00:00:00Z
review text: This paper is concerned with the following class of semilinear differential inclusions \[ x'(t)\in Ax(t)+f(t),\;t\in\mathbb R, \] where the multivalued linear operator \(A\in ML(E)\) (\(ML\) is the collection of all closed linear multivalued maps in \(E\)) and \(f\) belong to the homogeneous space \({\mathcal U}(\mathbb R,E)\). The theory of multivalued linear operators [see \textit{R. Cross}, Multivalued linear operators. Pure and Applied Mathematics, 213. New York, NY: Marcel Dekker (1998; Zbl 0911.47002)]. The fundamental ideas for the first result are based on the spectrum properties of \(U(1)\subset \mathbb C\backslash S\) (\(U(.)\) is a generalized \(C_0\)-semigroup and \(S\) is the unit circle), for the second result the authors prove that the conditions of Theorem 16 are necessary for the existence and uniqueness of mild solutions of the above problem in each specially homogeneous space, \(C(\mathbb R,E),\;L^p(\mathbb R,E), S_p(\mathbb R,E), 1\leq p<\infty, AP(\mathbb R,E), APS_p(\mathbb R,E).\) Also, the authors prove the existence and uniqueness of \(T\)-periodic mild solutions, where \(A\in ML(E)\) is a sectorial operator. An example of applications to feedback control systems governed by differential equations of Sobolev type in a Banach space is presented.
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 5073025
zbMATH Keywords: multivalued linear operator
zbMATH Keywords: generalized \(C_0\)-semigroup
zbMATH Keywords: sectorial operator
zbMATH Keywords: feedback control
reviewed by: Abdelghani Ouahab
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.12.058
OpenAlex ID: W2073563148
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DOI: 10.1016/J.JMAA.2005.12.058
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description: scientific article; zbMATH DE number 5073025