A remark on nonlinear cosine functions
type: Item
instance of: scholarly article
title: A remark on nonlinear cosine functions
zbMATH Open document ID: 0609.47075
DOI: 10.1007/BF02573173
author: Shinnosuke Oharu
author: Jerome A. Goldstein
published in: Semigroup Forum
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134849
review text: Well-posed linear initial value problems of the form \(d^ 2u(t)/dt^ 2=Au(t)\), \(u(0)=x\), \((du/dt)(0)=0\) have unique solutions given by \(u(t)=C(t)x\) where the cosine function C satisfies \[ \begin{cases} C(t+s)+C(t- s)= 2C(t)C(s) \quad (t,s\in {\mathbb{R}})\\ C(0)=I.\end{cases} \tag{*} \] Linear cosine function theory is analogous to \((C_ 0)\) semigroup theory, and while a theory of one parameter nonlinear semigroups has been developed, a nonlinear cosine function theory does not exist. Our paper is devoted to the construction of some examples of nonlinear cosine functions, i.e. nonlinear operators \(C(t)\), on a Banach space, satisfying (*).
Mathematics Subject Classification ID: 47H20
zbMATH DE Number: 3986019
zbMATH Keywords: one parameter nonlinear semigroups
zbMATH Keywords: nonlinear cosine function theory
MaRDI profile type: Publication
cites work: Q3291821
cites work: Q4094029
cites work: Q3906660
cites work: Q3721264
cites work: Affine semigroups on Banach spaces
cites work: One-parameter semigroups of positive operators
cites work: Semigroups of linear operators and applications to partial differential equations
description: scientific article; zbMATH DE number 3986019