Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations
 P. Poláčik,
 Ignác Tereščák
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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.
 Alikakos, N. D., Fusco, G. (1991) A dynamical systems proof of the KreinRutman theorem and an extension of the Perron theorem. Proc. Roy. Soc. Edinburgh 117A: pp. 209214
 Amann, H. (1985) Global existence for semilinear parabolic systems. J. reine angew. Math. 366: pp. 4789
 Amann, H. (1987) On abstract parabolic fundamental solutions. J. Math. Soc. Jap. 39: pp. 93116
 Brunovský, P., Poláčik, P., Sandstede, B. (1992) Convergence in general periodic parabolic equations in one space dimension. Nonlin. Anal. 18: pp. 209215
 Dancer, E. N., Hess, P. (1991) Stability of fixed points for orderpreserving discretetime dynamical systems. J. reine angew. Math. 419: pp. 125139
 Dancer, E. N., and Hess, P. Stable subharmonic solutions in periodic reactiondiffusion equations (preprint).
 Deimling, K. (1985) Nonlinear Functional Analysis. SpringerVerlag, BerlinHeidelbergNew York
 Henry, D. (1981) Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 89. SpringerVerlag, New York
 Hess, P. (1987) Spatial homogenity of stable solutions of some periodicparabolic problems with Neumann boundary conditions. J. Diff. Eq. 68: pp. 320331
 Hess, P. (1991) PeriodicParabolic Boundary Value Problems and Positivity. Pitman Research Notes in Mathematics 247. Longman Scientific and Technical, New York
 Hess, P., and Poláčik, P. Boundedness of prime periods of stable cycles and convergence to fixed points in discrete monotone dynamical systems (preprint).
 Hirsch, M. W. (1988) Stability and convergence in strongly monotone dynamical systems. J. reine angew. Math. 383: pp. 158
 Matano, H. (1979) Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ. Res. Inst. Math. Sci. 15: pp. 401454
 Mierczyński, J. Flows on ordered bundles (preprint).
 Poláčik, P. (1989) Convergence in smooth strongly monotone flows defined by semilinear parabolic equations. J. Diff. Eq. 79: pp. 89110
 Poláčik, P. (1989) Domains of attraction of equilibria and monotonicity properties of convergent trajectories in semilinear parabolic systems admitting strong comparison principle. J. reine angew. Math. 400: pp. 3256
 Poláčik, P. Dynamics of scalar semilinear parabolic equations (preprint).
 Poláčik, P., Tereščák, I. (1991) Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discretetime dynamical systems. Arch. Ration. Mech. Anal. 116: pp. 339360
 Protter, M. H., Weinberger, H. F. (1967) Maximum Principles in Differential Equations. PrenticeHall, Englewood Cliffs, NJ
 Ruelle, D. (1979) Analycity properties of characteristic exponents of random matrix products. Ada. Math. 32: pp. 6880
 Smith, H. L., Thieme, H. R. (1990) Quasiconvergence and stability for strongly orderpreserving semiflows. SIAM J. Math. Anal. 21: pp. 673692
 Smith, H. L., Thieme, H. R. (1991) Convergence for strongly orderpreserving semiflows. SIAM J. Math. Anal. 22: pp. 10811101
 Takáč, P. (1992) Asymptotic behavior of strongly monotone timeperiodic dynamical processes with symmetry. J. Diff. Eq. 100: pp. 355378
 Takáč, P. (1992b). Linearly stable subharmonic orbits in strongly monotone timeperiodic dynamical systems.Proc. Am. Math. Soc. (in press).
 Takáč, P. A construction of stable subharmonic orbits in monotone timeperiodic dynamical systems (preprint).
 Temam, R. (1988) InfiniteDimensional Dynamical Systems in Mechanics and Physics. Appl. Math. Sci. 68. SpringerVerlag, New York
 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
 19930401
 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