Existence and asymptotic stability of periodic solution for evolution equations with delays
type: Item
instance of: scholarly article
title: Existence and asymptotic stability of periodic solution for evolution equations with delays
zbMATH Open document ID: 1233.34028
author: Yongxiang Li
published in: Journal of Functional Analysis
publication date: 2011-07-18T00:00:00Z
review text: The author studies the existence and asymptotic stability of a time periodic solution for the evolution equation \[ u'(t)+ Au(t)= F(t, u(t)), u(t-\tau_1), u(t-\tau_2),\dots, u(t-\tau_n)),\;t\in\mathbb{R}, \] where \(A: D(A)\subset H\to H\) is a positive-definite selfadjoint operator on a Hilbert space \(H\), \(F: \mathbb{R}\times H^{n+1}\to H\) is a nonlinear mapping which is \(\omega\)-periodic in \(t\), and \(\tau_1,\tau_2,\dots, \tau_n\) are positive constants. Essential conditions on the nonlinearity are given, which guarantee that the equation has co-periodic solutions or an asymptotically stable co-periodic solution. The author improves and extends some previous results on the given topic by a different method. In this way, some milder conditions can be used, and the specific condition \(n\leq 3\) can be deleted. The results are based on analytic semigroups and on an integral inequality of Bellman type with delays. At the end, periodic solutions of delay parabolic equations are treated.
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 35R10
zbMATH DE Number: 5925492
zbMATH Keywords: evolution equations with delays
zbMATH Keywords: time periodic solutions
zbMATH Keywords: existence and uniqueness
zbMATH Keywords: asymptotic stability
zbMATH Keywords: analytic semigroups
reviewed by: Miklavž Mastinšek
MaRDI profile type: Publication
full work available at URL: j.jfa.2011.05.001
OpenAlex ID: W2040897169
cites work: Q3915354
cites work: Periodic solutions of semilinear equations of evolution of compact type
cites work: Periodic solutions of abstract differential equations with infinite delay
cites work: Introduction to functional differential equations
cites work: Nonlinear evolution equations - global behavior of solutions
cites work: Geometric theory of semilinear parabolic equations
cites work: Q3153998
cites work: Existence and uniqueness of periodic solution for a class of semilinear evolution equations
cites work: Bounded and periodic solutions of finite delay evolution equations
cites work: Periodic solutions of infinite delay evolution equations
cites work: Bounded and periodic solutions of infinite delay evolution equations.
cites work: Schaefer type theorem and periodic solutions of evolution equations
cites work: On the existence of periodic solutions to nonlinear abstract parabolic equations
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Infinite-dimensional dynamical systems in mechanics and physics.
cites work: Theory and applications of partial functional differential equations
cites work: Existence of periodic solutions of semilinear evolution equations with time lags
cites work: The existence and attractivity of time periodic solutions for evolution equations with delays
DOI: 10.1016/J.JFA.2011.05.001
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description: scientific article; zbMATH DE number 5925492