Existence results for second-order nonlinear evolution inclusions
type: Item
instance of: scholarly article
title: Existence results for second-order nonlinear evolution inclusions
zbMATH Open document ID: 0922.34051
author: K. Appert
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1999-10-03T00:00:00Z
review text: The paper concerns a second-order evolution system: \[ x''(t)+A(t,x'(t))+Bx(t)\in F(t,x(t),x'(t)); \quad x(0)=x_0\in V; \quad x'(0)=x_1\in H. \] Here, \(V\subset H\subset V^*\) is a Gelfand triple, \(A:[0,T]\times V\to V^*\) is a nonlinear operator, \(B:V\to V^*\) is a bounded linear operator, and \(F:[0,T]\times H\times H\to H\) is a set-valued operator. Sufficient conditions are given for existence, local existence, and uniqueness of solutions. The proofs make use of a new Wirtinger inequality. The theory is illustrated through an example where \(H=L^2(\Omega)\), \(V=W^{m,2}_0(\Omega)\), and \(V^*=W^{-m,2}(\Omega)\), \(\Omega\) being a bounded domain in \(\mathbb{R}^n\) with smooth boundary.
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 34A60
zbMATH DE Number: 1283841
zbMATH Keywords: second-order evolution system
zbMATH Keywords: existence
zbMATH Keywords: local existence
zbMATH Keywords: uniqueness
zbMATH Keywords: solutions
zbMATH Keywords: Wirtinger inequality
reviewed by: Corneliu Ursescu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1283841