On local integrated C-cosine function and weak solution of second order abstract Cauchy problem
type: Item
instance of: scholarly article
title: On local integrated C-cosine function and weak solution of second order abstract Cauchy problem
zbMATH Open document ID: 1238.47026
author: Chung-Cheng Kuo
published in: Taiwanese Journal of Mathematics
publication date: 2011-06-21T00:00:00Z
review text: Let \(X\) be a Banach space over \({\mathbb F}\) with norm \(\|\cdot\|\) and \(X^*\) its dual space. Denote by \(B(X)\) the set of all bounded linear operators from \(X\) into itself. For each \(0<T_0\leq \infty\), it is considered the following Cauchy problem \[ ACP_2(A,f,x,y):\quad \begin{cases} u^{\prime\prime}(t)=Au(t)+f(t), \;t\in (0,T_0), \\ u(0)=x, \;u^{\prime}(0)=y, \end{cases} \] where \(x,y\in X\) are given, \(A:D(A)\subset X\rightarrow X\) is a closed linear operator and \(f\) is an \(X\)-valued function defined on a subset of \({\mathbb R}\) containing \((0,T_0)\). The author shows that \(A\) generates a nondegenerate local \(1\)-times (respectively, \(0\)-times) integrated \(C\)-cosine function on \(X\) if and only if \(ACP_2(A,C_g,(\cdot),0,C_x)\) has a unique weak solution in \(C([0,T_0),X)\) (respectively, in \(C^1([0,T_0),X)\)) for all \(x\in X\) and \(g\in L^1_{\text{loc}}([0,T_0),X)\) if and only if \(ACP_2(A,0,0,C_x)\) has a unique weak solution in \(C([0,T_0),X)\) (respectively, in \(C^1([0,T_0),X)\) for all \(x\in X\).
reviewed by: Silvia-Otilia Corduneanu
Mathematics Subject Classification ID: 47D60
Mathematics Subject Classification ID: 47D62
Mathematics Subject Classification ID: 34G10
zbMATH DE Number: 5910201
zbMATH Keywords: second order abstract Cauchy problem
zbMATH Keywords: local integrated C-cosine function
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500406030
OpenAlex ID: W4252392352
DOI: 10.11650/TWJM/1500406030
description: scientific article; zbMATH DE number 5910201