Nonlinear evolution operators in Banach spaces
type: Item
instance of: scholarly article
title: Nonlinear evolution operators in Banach spaces
zbMATH Open document ID: 0567.47047
author: Kazuo Kobayasi
author: Yoshikazu Kobayashi
author: Shinnosuke Oharu
published in: Osaka Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Let X be a real Banach space, and \[ (DE)_ s\quad u'(t)\in A_ tu(t),\quad s<t<T, \] with initial conditions \(u(s)=x\), \(x\in \overline{D(A_ s)}\); wherein \(0\leq s<T\), u(\(\cdot)\) stands for an X- valued unknown function on the interval [s,T], u'(\(\cdot)\) represents a derivative of u, and \(\{A_ t: t\in [0,T]\}\) is a given family of nonlinear, possibly multivalued operators in X. From the text: ''The present paper is devoted to the study of two problems. The first aim of this paper is to introduce two notions of generalized solutions to \((DE)_ s\) such that the associated solution operators form an evolution operator. Firstly we consider the case in which approximate differential or difference equations for \((DE)_{s'}\) can be derived and there exist limits of solutions of those approximate equations.'' ''On the other hand,... we employ a certain type of t-dependence of the operator \(A_ t\) in order to extend the notion of Benilan's integral solution to the time-dependent case under consideration, and then show that given an initial-value \(x\in D(A_ s)\), the associated weak solution is uniquely determined in the class of such extended integral solutions.'' ''The next problem is to discuss the generation of an evolution operator that provides generalized solutions of the initial value problem (for \((DE)_ s)\). In 1967, \textit{T. Kato} [J. Math. Soc. Jap. 19, 508-520 (1967; Zbl 0163.383)] gave a general theory of nonlinear evolution operators in Banach spaces with uniformly convex duals and, in 1972, \textit{M. G. Crandall} and \textit{A. Pazy} [Isr. J. Math. 11, 57-94 (1972; Zbl 0249.34049)] established a generation theory in general Banach spaces.'' The authors' second aim is to discuss the construction of evolution operators which furnish the integral solutions of \((DE)_ s\) and extend the results of Crandall and Pazy [ibid.], \textit{L. C. Evans} [Isr. J. Math. 26, No.1, 1-42 (1977; Zbl 0349.34043)] and others.
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 47H06
zbMATH DE Number: 3905297
zbMATH Keywords: multivalued operators
zbMATH Keywords: generalized solutions
zbMATH Keywords: evolution operator
zbMATH Keywords: approximate differential or difference equations
zbMATH Keywords: Benilan's integral solution
zbMATH Keywords: time-dependent case
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3905297