`a`
Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Eigenvectors of homogeneous order-bounded order-preserving maps
Pages: 1073 - 1097, Issue 3, May 2017

doi:10.3934/dcdsb.2017053      Abstract        References        Full text (469.5K)           Related Articles

Horst R. Thieme - Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804, United States (email)

1 M. Akian, S. Gaubert and R. D. Nussbaum, A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones, arXiv:1112.5968 [math.FA], 2012.
2 H. Bauer, Probability Theory and Elements of Measure Theory, sec. ed., Academic Press, London, 1981.       
3 G. Birkhoff, Uniformly semi-primitive multiplicative processes, Trans. Amer. Math. Soc., 104 (1962), 37-51.       
4 E. Bohl, Eigenwertaufgaben bei monotonen Operatoren und Fehlerabschätzungen für Operatorgleichungen, Arch. Rat. Mech. Anal., 22 (1966), 313-332.       
5 E. Bohl, Monotonie: Lösbarkeit und Numerik bei Operatorgleichungen, Springer, Berlin Heidelberg, 1974.       
6 F. F. Bonsall, Linear operators in complete positive cones, Proc. London Math. Soc., 8 (1958), 53-75.       
7 R. S. Cantrell and C. Cosner, Effects of domain size on the persistence of populations in a diffusive food-chain model with Beddington-DeAngelis functional response, Natur. Resource Modeling, 14 (2001), 335-367.       
8 R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, John Wiley & Sons, Ltd., Chichester, 2003.       
9 R. S. Cantrell, C. Cosner and V. Hutson, Permanence in ecological systems with spatial heterogeneity, Proc. Royal Soc. Edinburgh Sect. A, 123 (1993), 533-559.       
10 R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math., 26 (1996), 1-35.       
11 L. Collatz, Einschließungssatz für die Eigenwerte von Integralgleichungen, Math. Z., 47 (1941), 395-398.       
12 L. Collatz, Einschließungssatz für die charakteristischen Zahlen von Matrizen, Math. Z., 48 (1942), 221-226.       
13 K. D. Deimling, Nonlinear Functional Analysis, Springer, Berlin Heidelberg, 1985.       
14 R. M. Dudley, Real Analysis and Probability, sec. ed., Cambridge University Press, Cambridge, 2002.       
15 J. H. M. Evers, S. C. Hille and A. Muntean, Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, J. Differential Equations, 259 (2015), 1068-1097.       
16 K.-H. Förster and B. Nagy, On the Collatz-Wielandt numbers and the local spectral radius of a nonnegative operator, Linear Algebra and its Applications, 120 (1980), 193-205.       
17 P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, J. Hyperbolic Differ. Equ., 7 (2010), 733-773.       
18 P. Gwiazda, A. Marciniak-Czochra and H. R. Thieme, Measures under the flat norm as ordered normed vector space (tentative title), in preparation.
19 K. P. Hadeler, R. Waldstätter and A. Wörz-Busekros, Models for pair formation in bisexual populations, J. Math. Biol., 26 (1988), 635-649.       
20 K. P. Hadeler, Pair formation in age-structured populations, Acta Appl. Math., 14 (1989), 91-102.       
21 S. C. Hille and D. T. H. Worm, Embedding of semigroups of Lipschitz maps into positive linear semigroups on ordered Banach spaces generated by measures, Integral Equations Operator Theory, 63 (2009), 351-371.       
22 W. Jin and H. R. Thieme, An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius, Disc. Cont. Dyn. Systems B, 21 (2016), 447-470.       
23 M. A. Krasnosel'skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.       
24 M. A. Krasnosel'skij, Je. A. Lifshits and A. V. Sobolev, Positive Linear Systems: The Method of Positive Operators, Heldermann Verlag, Berlin, 1989.       
25 M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin Heidelberg, 1984.       
26 U. Krause, Positive Dynamical Systems in Discrete Time. Theory, Models and Applications, De Gruyter, Berlin, 2015.       
27 M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space (Russian), Uspehi Mat. Nauk (N.S.), 3 (1948), 3-95, English Translation, AMS Translation, 1950 (1950), 128pp.       
28 B. Lemmens, B. Lins, R. D. Nussbaum and M. Wortel, Denjoy-Wolff theorems for Hilbert's and Thompson's metric spaces, J. Analyse Math., (to appear), arXiv:1410.1056v2[math.DS]
29 B. Lemmens and R. D. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge University Press, Cambridge, 2012.       
30 B. Lemmens and R. D. Nussbaum, Continuity of the cone spectral radius, Proc. Amer. Math. Soc., 141 (2013), 2741-2754.       
31 J. Mallet-Paret and R. D. Nussbaum, Eigenvalues for a class of homogeneous cone maps arising from max-plus operators, Discr. Cont. Dyn. Sys. (DCDS-A), 8 (2002), 519-562.       
32 J. Mallet-Paret and R. D. Nussbaum, Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fixed Point Theory and Appl., 7 (2010), 103-143.       
33 J. Mallet-Paret and R. D. Nussbaum, Asymptotic fixed point theory and the beer barrel theorem, J. Fixed Point Theory Appl., 4 (2008), 203-245.       
34 H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic Theories and the Boltzmann Equation, (Montecatini, 1981), Lecture Notes in Math., Springer, Berlin, 1048 (1984), 60-110.       
35 R. D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem, Fixed Point Theory, (E. Fadell and G. Fournier, eds.), Springer, Berlin New York, 886 (1981), 309-330.       
36 R. D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc., Providence, 75 (1988), iv+137 pp.       
37 R. D. Nussbaum, Eigenvectors of order-preserving linear operators, J. London Math. Soc., 58 (1998), 480-496.       
38 H. H. Schaefer, Positive Transformationen in lokalkonvexen halbgeordneten Vektorräumen, Math. Ann., 129 (1955), 323-329.       
39 H. H Schaefer, Halbgeordnete lokalkonvexe Vektorräume. II, Math. Ann., 138 (1959), 259-286.       
40 H. H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966.       
41 H. R. Thieme, Eigenvectors and eigenfunctionals of homogeneous order-preserving maps, arXiv:1302.3905v1[math.FA], 2013.
42 H. R. Thieme, Comparison of spectral radii and Collatz-Wielandt numbers for homogeneous maps, and other applications of the monotone companion norm on ordered normed vector spaces, arXiv:1406.6657v2[math.FA], 2014 (revised 2016).
43 H. R. Thieme, Spectral radii and Collatz-Wielandt numbers for homogeneous order-preserving maps and the monotone companion norm, Ordered Structures and Applications, (tentative title), Positivity VII (Zaanen Centennial Conference) (M. de Jeu, B. de Pagter, O. van Gaans, M. Veraar, eds.), Birkhäuser, (2016), 415-467.
44 H. R. Thieme, Eigenfunctionals of homogeneous order-preserving maps with applications to sexually reproducing populations, J. Dynamics and Differential Equations, 28 (2016), 1115-1144.       
45 A. C. Thompson, On the eigenvectors of some not-necessarily-linear transformations, Proc. London Math. Soc., 15 (1965), 577-598.       
46 A. J. Tromba, The beer barrel theorem, a new proof of the asymptotic conjecture in fixed point theory, Functional Differential Equations and Approximations of Fixed Points, (H.-O. Peitgen, H.-O. Walther, eds.), 484-488, Lecture Notes in Math. 730, Springer, Berlin Heidelberg 1979.       
47 H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Z., 52 (1950), 642-648.       
48 K. Yosida, Functional Analysis, Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 123 Springer-Verlag New York Inc., New York, 1968.       
49 P. P. Zabreiko, M. A. Krasnosel'skii and Yu. V. Pokornyi, On a class of linear positive operators, Functional Analysis and Its Applications, 5 (1971), 272-279.

Go to top