Functional differential equations of delay type and nonlinear evolution in \(L_ p\)-spaces
type: Item
instance of: scholarly article
title: Functional differential equations of delay type and nonlinear evolution in \(L_ p\)-spaces
zbMATH Open document ID: 0637.34055
DOI: 10.1016/0022-247X(87)90294-0
author: Mohd. Faheem
author: M. Rama Mohana Rao
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: Nonlinear nonautonomous functional differential equations with \(L_ p\)- initial functions are studied by means of the theory of evolution operators in \(L_ p\)-spaces. The equations have the form \[ \dot x(t)=L(t,x_ t)+f(t,x_ t),\quad t\in [a,b], \] with initial conditions \[ x_ a=\phi \in L_ p([-r,0];R^ n),\quad x(a)=\eta \in R^ n, \] \[ L(t,\phi)=\sum^{m}_{j=1}A_ j(t)\phi (-r_ j)+\int^{0}_{- r}D(t,\theta)\phi (\theta)du. \] The function \(f: [a,b]\times L_ p\to R^ n\) is assumed to be Lipschitz in the second variable and to satisfy another condition with respect to the first variable. Evolution operators for the linear and nonlinear equations are constructed and it is shown that the evolution operator in \(L_ p\times R^ n\) determines the solution of the initial value problem in the exact sense. A variation of parameters formula is developed for the nonlinear equation.
Mathematics Subject Classification ID: 34K05
zbMATH DE Number: 4038293
zbMATH Keywords: Nonlinear nonautonomous functional differential equations
zbMATH Keywords: evolution operators
zbMATH Keywords: \(L_ p\)-spaces
zbMATH Keywords: variation of parameters formula
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4038293