Journal of Dynamics and Differential Equations
, Volume 5, Issue 2, pp 279303
First online:
Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations
 P. PoláčikAffiliated withCenter for Dynamical Systems and Nonlinear Studies, Georgia Institute of TechnologyInstitute of Applied Mathematics, Comenius University
 , Ignác TereščákAffiliated withFaculty of Mathematics and Physics, Comenius University
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A vector bundle morphism of a vector bundle with strongly ordered Banach spaces as fibers is studied. It is assumed that the fiber maps of this morphism are compact and strongly positive. The existence of two complementary, dimensionone and codimensionone, continuous subbundles invariant under the morphism is established. Each fiber of the first bundle is spanned by a positive vector (that is, a nonzero vector lying in the order cone), while the fibers of the other bundle do not contain a positive vector. Moreover, the ratio between the norms of the components (given by the splitting of the bundle) of iterated images of any vector in the bundle approaches zero exponentially (if the positive component is in the denominator). This is an extension of the KreinRutman theorem which deals with one compact strongly positive map only. The existence of invariant bundles with the above properties appears to be very useful in the investigation of asymptotic behavior of trajectories of strongly monotone discretetime dynamical systems, as demonstrated by Poláčik and Tereščák (Arch. Ration. Math. Anal. 116, 339–360, 1991) and Hess and Poláčik (preprint). The present paper also contains some new results on typical asymptotic behavior in scalar periodic parabolic equations.
Key words
Vector bundle maps invariant subbundles exponential separation continuous separation positive operators strongly monotone dynamical systems periodic parabolic equations asymptotic behavior stable periodic solutions Title
 Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations
 Journal

Journal of Dynamics and Differential Equations
Volume 5, Issue 2 , pp 279303
 Cover Date
 199304
 DOI
 10.1007/BF01053163
 Print ISSN
 10407294
 Online ISSN
 15729222
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 Vector bundle maps
 invariant subbundles
 exponential separation
 continuous separation
 positive operators
 strongly monotone dynamical systems
 periodic parabolic equations
 asymptotic behavior
 stable periodic solutions
 Authors

 P. Poláčik ^{(1)} ^{(2)}
 Ignác Tereščák ^{(3)}
 Author Affiliations

 1. Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, 30332, Atlanta, Georgia
 2. Institute of Applied Mathematics, Comenius University, Mlynská dolina, 84215, Bratislava, Czechoslovakia
 3. Faculty of Mathematics and Physics, Comenius University, Mlynská dolina, 84215, Bratislava, Czechoslovakia