The weak semilinear Cauchy problem for ordinary differential equations
type: Item
instance of: scholarly article
zbMATH Open document ID: 1016.34059
author: Marianne Reichert
publication date: 2001-09-17T00:00:00Z
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 47J99
Mathematics Subject Classification ID: 47H99
zbMATH DE Number: 1647681
zbMATH Keywords: weak semilinear Cauchy problem
zbMATH Keywords: Volterra-Hammerstein and the Hammerstein integral equations
MaRDI profile type: Publication
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title: The weak semilinear Cauchy problem for ordinary differential equations
published in: Analysis (München)
review text: Let us consider the Cauchy problem NEWLINE\[NEWLINEx'=f(t,x),\qquad x(0)=x_0\in E, \tag{1}NEWLINE\]NEWLINE where \(t\in I\subset \mathbb{R}\) is a compact interval, \(E\) is a real Banach space, \(B=\{x\in E:\|x-x_0\|\leq b\}\), \(b>0\) and \(f: I\times B\to E\) is a bounded weakly-weakly continuous function, i.e., continuous with respect to the weak topologies of \(E\) and \(B\). The derivative in \((1)\) is considered in the weak sense, i.e., it is defined in the usual way but in the sense of weak topology. NEWLINENEWLINENEWLINEIn 1971, A. Szép proved that, under the above assumptions, problem \((1)\) has a weak solution defined on an interval \(J\subset I\) if \(E\) is reflexive. In 1978, this result was extended by \textit{E. Cramer, V. Lakshmikantham} and \textit{A. R. Mitchell} [Nonlinear Anal., Theory, Methods Appl. 2, 169-177 (1978; Zbl 0379.34041)] to the case when \(E\) is a sequentially complete Banach space; the additional assumption imposed on \(f\) was formulated in terms of the measure of weak noncompactness introduced by \textit{S. F. de Blasi} [Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Sér. 21(69), 259-262 (1977; Zbl 0365.46015)] in 1977. Kneser-type theorems, i.e., theorems which described the topological structure of weak solution sets to \((1)\) were proved by S. Szufla in 1978 for reflexive spaces and next by \textit{I. Kubiaczyk} and \textit{S. Szufla} [Publ. Inst. Math., Nouv. Sér. 32(46), 99-103 (1982; Zbl 0516.34058)] for nonreflexive ones. The existence and the topological structure of weak solutions to Volterra-Hammerstein and the Hammerstein integral equations in weakly sequentially complete Banach spaces have been investigated, e.g., by \textit{D. Bugajewski} [Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 34, 49-58 (1994; Zbl 0824.45014) and Commentat. Math. Univ. Carol. 35, No. 1, 35-41 (1994; Zbl 0816.45012)]. NEWLINENEWLINENEWLINEIn the reviewed paper, the author investigates the initial value problem NEWLINE\[NEWLINEx'(t)…
reviewed by: Dariusz Bugajewski
description: scientific article; zbMATH DE number 1647681