, Volume 5, Issue 2, pp 279-303

Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations

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Abstract

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, dimension-one and codimension-one, 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 Krein-Rutman 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 discrete-time 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.