Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the \(\alpha \)-norm
type: Item
instance of: scholarly article
title: Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the \(\alpha \)-norm
zbMATH Open document ID: 1185.34112
author: Jung-Chan Chang
author: Hsiang Liu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-07-10T00:00:00Z
review text: The authors consider the existence of solutions of the neutral partial differential equation with nonlocal conditions: \[ d/dt\bigl(x(t)+ F(t, x(t))\bigr)= -Ax(t)+ G(t, x(t)),\quad t\geq 0, \] \[ x(0)+ g(x)= x_0\in X, \] where \(-A\) generates an analytic compact semigroup on a Banach space \(X\) and where the functions \(F\), \(G\) and \(g\) satisfy specific continuity and measurability constraints. Fractional powers of \(-A\) are used. By the fixed point theorem of Sadovskii existence of a mild solution is obtained. When \(X\) is a reflexive Banach space and \(G\) is Lipschitz continuous, a strong solution is obtained. The results are applied to a class of partial differential equations with nonlocal conditions.
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 5577443
zbMATH Keywords: analytic compact semigroup
zbMATH Keywords: nonlocal condition
zbMATH Keywords: mild solution
zbMATH Keywords: strong solution
zbMATH Keywords: \(\alpha \)-norm
reviewed by: Miklavž Mastinšek
MaRDI profile type: Publication
full work available at URL: j.na.2009.02.035
OpenAlex ID: W2056339675
cites work: A class of functional equations of neutral type
cites work: Existence of solutions of a delay differential equation with nonlocal condition
cites work: Uniqueness of solutions of parabolic semilinear nonlocal-boundary problems
cites work: Theorem about the existence and uniqueness of a solution of a nonlocal abstract cauchy problem in a banach space
cites work: Existence of solutions of a semilinear functional-differential evolution nonlocal problem
cites work: Existence of solutions for neutral functional differential evolution equations with nonlocal conditions.
cites work: Existence and regularity in the \(\alpha\)-norm for some neutral partial differential equations with nonlocal conditions
cites work: Nonlocal Cauchy problems for semilinear evolution equations
cites work: Semilinear integrodifferential equations with nonlocal cauchy problem
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Existence for a class of partial differential equations with nonlocal conditions
cites work: Q5551837
cites work: Q3966611
cites work: Differentiability of nonlinear semigroups
cites work: Existence, Stability, and Compactness in the α-Norm for Partial Functional Differential Equations
cites work: Q5620934
DOI: 10.1016/J.NA.2009.02.035
description: scientific article; zbMATH DE number 5577443