Positive periodic solutions of parabolic evolution problems: a translation along trajectories approach
type: Item
instance of: scholarly article
title: Positive periodic solutions of parabolic evolution problems: a translation along trajectories approach
zbMATH Open document ID: 1222.47132
author: Aleksander Ćwiszewski
published in: Central European Journal of Mathematics
publication date: 2011-05-27T00:00:00Z
review text: Using the translation along trajectories method in infinite dimensions [\textit{R. Bader} and \textit{W. Kryszewski}, Nonlinear Anal., Theory Methods Appl. 54, No.~4, A, 707--754 (2003; Zbl 1034.34072)] with subsequent application of averaging and degree theories, the author derives existence criteria of periodic solutions to the abstract periodic evolution problem of the form \[ \dot{u}(t)=-Au(t)+F(t,u(t)),\quad t>0;\;u(t)\in M,\;t\geq 0;\;u(0)=u(\tau), \] with a linear operator \(A:D(A)\to E\), so that \(-A\) generates a \(C_0\)-semigroup of compact linear contractions on a Banach space \(E\), \(M\subset E\) is a closed convex cone and \(F: [0,+\infty)\times M\to E\) is a continuous mapping satisfying the tangency condition \(F(t, \overline{u})\in T_M(\overline{u})\) for all \(\overline{u}\in M\), \(t \geq 0\), where \(T_M(\overline{u})\) is the tangent cone to \(M\) at \(\overline{u}\). Aiming at applications to partial differential equations and systems, the author considers the problem in a more general case, when \(F\) is tangent to \(M\), i.e., \(F\) may take values outside \(M\), and without the assumption that nonlinear perturbations are positive.
reviewed by: Irina V. Konopleva
Mathematics Subject Classification ID: 47J35
Mathematics Subject Classification ID: 47J15
Mathematics Subject Classification ID: 37L05
Mathematics Subject Classification ID: 47H11
zbMATH DE Number: 5900634
zbMATH Keywords: semigroup
zbMATH Keywords: evolution equation
zbMATH Keywords: topological degree
zbMATH Keywords: periodic solution
Wikidata QID: Q96744681
MaRDI profile type: Publication
full work available at URL: s11533-011-0010-6
OpenAlex ID: W2019388163
cites work: Q3915354
cites work: On the solution sets of differential inclusions and the periodic problem in Banach spaces.
cites work: Flow invariance for perturbed nonlinear evolution equations
cites work: Q4167193
cites work: The generalized degree for compact perturbations of m-accretive operators and applications
cites work: Topological degree methods for perturbations of operators generating compact \(C_0\) semigroups
cites work: Constrained topological degree and positive solutions of fully nonlinear boundary value problems
cites work: On a generalized Poincaré-Hopf formula in infinite dimensions
cites work: Q4399897
cites work: The Leray-Schauder index and the fixed point theory for arbitrary ANRs
cites work: Q4421936
cites work: On the eigenvalue problem for perturbations of nonlinear accretive and monotone operators in banach spaces
cites work: Geometric theory of semilinear parabolic equations
cites work: Q3998030
cites work: Condensing multivalued maps and semilinear differential inclusions in Banach spaces
cites work: Q4877749
cites work: Topological structure of solution sets of differential inclusions: the constrained case
cites work: Periodic solutions of the boundary value problem for the nonlinear heat equation
cites work: The fixed point index for local condensing maps
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Q4332942
DOI: 10.2478/S11533-011-0010-6
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description: scientific article; zbMATH DE number 5900634