Bounded solutions of linear and weakly nonlinear differential equations in a Banach space with an unbounded operator in the linear part
type: Item
instance of: scholarly article
title: Bounded solutions of linear and weakly nonlinear differential equations in a Banach space with an unbounded operator in the linear part
zbMATH Open document ID: 1275.34084
DOI: 10.1134/S0012266112060055
published in: Differential Equations
publication date: 2012-11-14T00:00:00Z
review text: The paper presents necessary and sufficient conditions for the existence, on the whole real line, of a weak bounded solution of the following weakly nonlinear differential equation in a Banach space \(\mathbb{B}\) \[ \frac{dx}{dt}=A(t)x(t)+f(t), \;t\in J\subset \mathbb{R}, \] where \(f\) is a Banach-valued continuous and bounded operator and the operator \(A(t)\) is a closed operator with dense domain in \(\mathbb{B}\), where the domain is independent of \(t\). The approach is based on the assumption that the homogeneous equation \[ \frac{dx}{dt}=A(t)x(t), \;t\in J\subset \mathbb{R} \] admits an exponential dichotomy on the half-lines.
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34D09
Mathematics Subject Classification ID: 34G20
zbMATH DE Number: 6105658
zbMATH Keywords: weak bounded solution
zbMATH Keywords: weakly nonlinear differential equations
zbMATH Keywords: generalized inverse equation
author: O. O. Pokutnyj
reviewed by: Adrian Petruşel
MaRDI profile type: Publication
cites work: Exponential dichotomies and transversal homoclinic points
cites work: Q4674795
cites work: Q3049314
cites work: Q4258125
cites work: Fredholm differential operators with unbounded coefficients
cites work: Evolution equations: Dichotomies and the Fredholm alternative for bounded solutions
cites work: Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations
cites work: Bounded solutions of linear differential equations in a Banach space
cites work: Geometric theory of semilinear parabolic equations
cites work: Q3665685
cites work: Q3663911
description: scientific article; zbMATH DE number 6105658