Existence of solutions and Galerkin approximations for nonlinear functional evolution equations
type: Item
instance of: scholarly article
title: Existence of solutions and Galerkin approximations for nonlinear functional evolution equations
zbMATH Open document ID: 0553.65040
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1982-01-01T00:00:00Z
review text: One examines the abstract nonlinear functional evolution equation: \[ (*)\quad \dot x(t)+A(t,x_ t)x(t)=0,\quad x_ 0=\Phi \] in a Banach space X, where \(x_ t=x(t+\theta)\) for \(\theta\in [-r,0]\) and \(x_ 0\in C=C([-r,0],x)\). The existence of a local solution of (*), uniformly continuous, X-valued and differentiable almost everywhere is proved under the conditions: \(X^*\) is uniformly convex. The domain at \(A_ 1(\cdot,\cdot,\cdot)\) with \(A_ 1(t,\psi,v)=A(t,\psi)v\) is the set \([0,T)\times C_ 0\times D\), where D is a subset of X and \(C_ 0\) consists of all \(f\in C\) with f(t)\(\in \bar D\cup M\), \(t\in [-r,0]\). Here \(M=\{\Phi(t);\) \(t\in [-r,0]\}\). For every \((t,\psi)\in [0,T]\times C_ 0\), \(A(t,\psi)v\) is m-accretive in v. For every \(t,s\in [0,T)\), \(\psi_ 1,\psi_ 2\in C_ 0\), \(v\in D\), \[ \| A(t,\psi_ 1)v-A(s,\psi_ 2)v\| \leq \ell (\| \psi_ 1\|_ C,\| \psi_ 2\|_ C,\| v\|)[| t-s| (1+\| A(s,\psi_ 2)v\|)+\| \psi_ 1-\psi_ 2\|_ C]. \] Sufficient conditions for the convergence of the Galerkin method solving (*) in the simplified case of \(A(t,x_ t)x(t)=A(t)x(t)-G(t,x_ t)\) with \(G(t,\Phi)\) satisfying Lipschitz type conditions with respect to both variables \(t\in [0,T]\) and \(\Phi\in C\), are also formulated.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 3881789
zbMATH Keywords: m-accretive operator
zbMATH Keywords: abstract nonlinear functional evolution equation
zbMATH Keywords: Banach space
zbMATH Keywords: local solution
zbMATH Keywords: convergence
zbMATH Keywords: Galerkin method
author: Athanassios G. Kartsatos
author: Mary E. Parrott
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178229153
OpenAlex ID: W1971509623
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DOI: 10.2748/TMJ/1178229153
description: scientific article; zbMATH DE number 3881789