Volterra integro-differential equations with accretive nonlinearity
type: Item
instance of: scholarly article
title: Volterra integro-differential equations with accretive nonlinearity
zbMATH Open document ID: 0575.45013
DOI: 10.1016/0022-0396(85)90120-2
published in: Journal of Differential Equations
publication date: 1985-01-01T00:00:00Z
review text: The topic of this paper is the initial value problem \[ (1)\quad d/dt(\gamma (u(t)-x)+\int^{t}_{0}k(t-s)(u(s)-x)ds)+A(u(t))\ni G(u)(t),\quad t\in R^+,\quad u(0)=x, \] which is set in a real Banach space X. Here \(\gamma\geq 0\), \(k: R^+\to R\) is a locally integrable, nonnegative, and nonincreasing, A is an m-accretive operator in X with domain D(A), \(x\in \overline{D(A)}\), and G is a mapping of \(C(R^+;X)\) when \(\gamma >0\), or of \(L^ 1_{loc}(R^+;X)\) when \(\gamma =0\) into \(L^ 1_{loc}(R^+;X)\) satisfying a Lipschitz condition. The author discusses the existence, uniqueness, regularity and continuous dependence upon the data of solutions of (1). The basic idea of the approach is to replace the operator L: D(L)\(\subset L^ 1_{loc}(R^+;X)\to L^ 1_{loc}(R^+;X)\), \(L(v)(t)=d/dt(v(t)+\int^{t}_{0}k(t-s)v(s)ds)\), \(D(L)=\{v\in L^ 1_{loc}(R^+;X)/t\to \gamma v(t)+\int^{t}_{0}k(t-s)v(s)ds\) is locally absolutely continuous and a.e. differentiable on \(R^+\), and takes the value 0 at \(t=0\}\) by its Yosida approximation \(L_{\lambda}\), \(\lambda >0\), and then to pass to the limit as \(\lambda \to 0^+\).
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45J05
zbMATH DE Number: 3919058
zbMATH Keywords: accretive nonlinearity
zbMATH Keywords: initial value problem
zbMATH Keywords: Banach space
zbMATH Keywords: m-accretive operator
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: regularity
zbMATH Keywords: Yosida approximation
author: Gustaf Gripenberg
reviewed by: Sergiu Aizicovici
MaRDI profile type: Publication
full work available at URL: 0022-0396(85)90120-2
OpenAlex ID: W2057533632
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description: scientific article; zbMATH DE number 3919058