Systems of essentially infinite-dimensional differential equations
type: Item
instance of: scholarly article
title: Systems of essentially infinite-dimensional differential equations
zbMATH Open document ID: 1273.35304
author: V. M. Statkevych
published in: Ukrainian Mathematical Journal
publication date: 2012-11-23T00:00:00Z
review text: Let \(F(Q, \mathbb{R}^n)\) be the Banach algebra of bounded functions on a set \(Q\) with values in \(\mathbb{R}^n\) and let \(X\) be a closed subalgebra of \(F(Q, \mathbb {R})\). For \(i \in \{1,\dots,n \}\) let \(S_i\) be a contraction \(C_0\)-semigroup in \(X\), \(S(t)u=(S_1(t)u_1, \dots,S_n(t)u_n)\) for \(u \in X^n\), such that there exists \(s >0\) for which \(S(s)=0,\; S_i(t)(uv)=S_i(t)u\,S_i(t)v\) for \(t \geq 0,\; u, v\in X\) and \(S_i(t)S_k(\tau)=S_k(\tau)S_i(t)\) for \(t, \tau \geq 0,\; i,k \in \{1,\dots,n\}\); the semigroup is generated by the operator \(S'(0)\), where \(S'(0)u=(S_1'(0)u_1,\dots,S_n'(0)u_n)\), and \(\text{dom}(S'(0))\) is dense in \(X^n\). Theorem 1. Let \(F: X^n \rightarrow X^n\) be such that there exists \(C>0\) for which \(|(F(u)-F(v))(x)| \leq C|(u-v)(x)|\) for every \(u, v \in X^n\) and \(x \in Q\). Then there is a unique \(u \in \text{dom}(S'(0)) \subset X^n\) such that \(S'(0)u=F(u)\). Theorem 2. If \(S_i=\Sigma\) and \(S_i'(0)=A\) for all \(i\in \{1,\dots,n\}\), \(a=(a_{i,k})_{i,k\in \{1,\dots,n\}}\), \(a_{i,k} \in X\), \(f \in X^n\), then \[ u\in \mathrm{dom}(S'(0)),\qquad u(x)=-\int_0^s \exp\left(\int_0^t(\Sigma(\tau)a)(x)d \tau\right) (S(t)f)(x)dt \] is the unique solution of \(S'(0)u+au=f\). Theorem 3. Let \(R>0\) and \(H\) a separable, infinite dimensional, real Hilbert space; let \(B_C(H)\) be the Banach space of self-adjoint bounded linear operators on \(H\); let \(Q_{n,d}\) be the set of the operators of \(B_C(H)\) whose rank does not exceed \(n \in \mathbb {N}\) and whose norm does not exceed \(d > 0\); let \(Z=\{u \in C^2(H,\mathbb{R})\), bounded on its support \(\subset \{x \in H: |x| \leq R \}\), such that \(u'': H \rightarrow B_C(H)\) is uniformly continuous and for every \(\epsilon > 0\) there exist \(K\) a compact subset of \(B_C(H)\), \(n\in \mathbb {N}\) and \(d > 0\) such that \(K + Q_{n,d}\) is an \(\varepsilon\)-net for \(\{u''(x): |x| \leq R\}\); let \(X\) be the closure of \(Z\) with respect to the sup-norm; …
reviewed by: Gianfranco Bottaro
Mathematics Subject Classification ID: 35R15
zbMATH DE Number: 6110039
zbMATH Keywords: essentially infinite-dimensional differential equations
MaRDI profile type: Publication
full work available at URL: s11253-012-0589-1
OpenAlex ID: W1968528226
cites work: Optimal methods for specifying information in the solution of integral equations with analytic kernels
cites work: Riquier problem for a nonlinear equation resolved with respect to the iterated Levi Laplacian
cites work: Nonlinear equations with essentially infinite-dimensional differential operators
DOI: 10.1007/S11253-012-0589-1
description: scientific article; zbMATH DE number 6110039