A remark on the mild solutions of non-local evolution equations
type: Item
instance of: scholarly article
title: A remark on the mild solutions of non-local evolution equations
zbMATH Open document ID: 1015.37045
author: J. Liu
published in: Semigroup Forum
publication date: 2003-03-18T00:00:00Z
review text: This paper is devoted to study the nonlocal evaluation equation \[ \begin{cases} u'(t)= Au(t)+ f(t,u(t))\\ u(0)+ g(u)= u_0,\end{cases}\qquad 0\leq t\leq T,\tag{1} \] where \(g: C([0, T],X)\to X\) is a continuous function, and \(X\) is a general Banach space. To study (1), a very common approach is to define a map \(F: C([0,T],X)\to C([0,T], X)\) by \[ F(u)(t)= T(t)[u_0- g(u)]+ \int^t_0 T(t- s) f(s,u(s)) ds, \qquad 0\leq t\leq T,\tag{2} \] and prove that \(F\) has a fixed point, which is called a mild solution of (1). Here \(T(\cdot)\) is the corresponding semigroup generated by (1). The author addresses to the following question: Can the map \(F\) defined by (2) be a compact operator? In the case \(u_0= 0\), \(f(\cdot,\cdot)= 0\), this question becomes: Can the map defined by \[ [T(\cdot)g](u):= T(\cdot)(g(u)),\quad u\in C([0,T], X) \] be compact? The author shows that, the answer is no in general.
Mathematics Subject Classification ID: 37L05
Mathematics Subject Classification ID: 34G20
zbMATH DE Number: 1883938
zbMATH Keywords: semigroup approach
zbMATH Keywords: nonlocal evaluation equation
zbMATH Keywords: mild solution
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: s002330010158
OpenAlex ID: W2039819702
DOI: 10.1007/S002330010158
description: scientific article; zbMATH DE number 1883938