Semilinear evolution equations of second order via maximal regularity
type: Item
instance of: scholarly article
title: Semilinear evolution equations of second order via maximal regularity
zbMATH Open document ID: 1155.47056
DOI: 10.1155/2008/316207
author: Claudio Cuevas
author: Carlos Lizama
published in: Advances in Difference Equations
publication date: 2008-08-15T00:00:00Z
full work available at URL: https://eudml.org/doc/55092
review text: Let \(X\) be a complex Banach space and assume that \(A:X\rightarrow X\) is a bounded linear operator. This paper is concerned with existence and stability results for the semilinear discrete problem \[ \Delta^2x_n-Ax_n=f(n,x_n,\Delta x_n),\quad n\geq 1, \] where \(x_0=x_1=0\) and \(f:{\mathbb Z}_+\times X\times X\rightarrow X\) is a Lipschitz function such that \(f(\cdot ,0,0)\in l_1({\mathbb Z}_+,X)\). The main result of the present paper establishes the existence of bounded solutions whose second discrete derivative is in \(l_p\) (\(1<p<\infty\)). The proof combines the Banach contraction principle with maximal regularity arguments. By means of the discrete Gronwall inequality, the authors also deduce \textit{a priori} estimates for the solutions \(x_n\) and their discrete derivatives \(\Delta x_n\) and \(\Delta^2x_n\).
reviewed by: Vicenţiu D. Rădulescu
Mathematics Subject Classification ID: 47J35
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 49M25
zbMATH DE Number: 5312521
zbMATH Keywords: difference equation
zbMATH Keywords: semilinear evolution equation
zbMATH Keywords: discrete maximal regularity
MaRDI profile type: Publication
OpenAlex ID: W2085945176
cites work: Asymptotic Behavior of the Solutions of the Second Order Difference Equation
cites work: A survey: stability and boundedness of Volterra difference equations
cites work: Quasilinear evolutionary equations and continuous interpolation spaces.
cites work: Analyticity and discrete maximal regularity on \(\ell_p\)-spaces
cites work: Discrete time analytic semigroups and the geometry of Banach spaces
cites work: Maximal regularity of evolution equations on discrete time scales
cites work: Stability of the crank-nicolson scheme and maximal regularity for parabolic equations in spaces
cites work: MaximalLp regularity for parabolic difference equations
cites work: Maximal regularity of discrete second order Cauchy problems in Banach spaces
cites work: Best possible Gronwall inequalities
cites work: The operator-valued Marcinkiewicz multiplier theorem and maximal regularity
cites work: Operator–valued Fourier multiplier theorems on Besov spaces
cites work: Some remarks on Banach spaces in which martingale difference sequences are unconditional
cites work: A note on time regularity for discrete time heat kernels
cites work: Existence and regularity results for an integral equation with infinite delay in a Banach space
cites work: Operator-valued Fourier multiplier theorems and maximal \(L_p\)-regularity
cites work: OPERATOR-VALUED FOURIER MULTIPLIERS ON PERIODIC BESOV SPACES AND APPLICATIONS
cites work: Fourier Multipliers and Integro‐Differential Equations in Banach Spaces
cites work: Periodic solutions of second order differential equations in Banach spaces
cites work: Asymptotic properties of solutions to nonautonomous Volterra difference systems with infinite delay
description: scientific article; zbMATH DE number 5312521