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May  2017, 22(3): 1073-1097. doi: 10.3934/dcdsb.2017053

Eigenvectors of homogeneous order-bounded order-preserving maps

Horst R. Thieme

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA

Received  September 2015 Revised  May 2016 Published  January 2017

The existence of eigenvectors associated with the cone spectral radius is shown for homogenous, order-preserving, continuous maps that have compact and order-bounded powers (iterates). The order-boundedness makes it possible to show the existence of eigenvectors for perturbations of the maps using Hilbert's projective metric, while the power compactness or similar compactness properties together with a uniform continuity condition let the eigenvectors of the perturbations converge to an eigenvector of the original map.

Keywords: Ordered normed vector space, homogeneous map, cone spectral radius, Collatz-Wielandt numbers, Krein-Rutman theorem, nonlinear eigenvectors, power compactness, population models with mating.
Mathematics Subject Classification: Primary: 47H07, 47J10; Secondary: 47N6.
Citation: Horst R. Thieme. Eigenvectors of homogeneous order-bounded order-preserving maps. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1073-1097. doi: 10.3934/dcdsb.2017053
References:
[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. Google Scholar

[2]

H. Bauer, Probability Theory and Elements of Measure Theory, sec. ed. , Academic Press, London, 1981. Google Scholar

[3]

G. Birkhoff, Uniformly semi-primitive multiplicative processes, Trans. Amer. Math. Soc., 104 (1962), 37-51.  doi: 10.1090/S0002-9947-1962-0146100-6.  Google Scholar

[4]

E. Bohl, Eigenwertaufgaben bei monotonen Operatoren und Fehlerabschätzungen für Operatorgleichungen, Arch. Rat. Mech. Anal., 22 (1966), 313-332.   Google Scholar

[5]

E. Bohl, Monotonie: Lösbarkeit und Numerik bei Operatorgleichungen, Springer, Berlin Heidelberg, 1974. Google Scholar

[6]

F. F. Bonsall, Linear operators in complete positive cones, Proc. London Math. Soc., 8 (1958), 53-75.   Google Scholar

[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.  doi: 10.1111/j.1939-7445.2001.tb00062.x.  Google Scholar

[8]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, John Wiley & Sons, Ltd. , Chichester, 2003. Google Scholar

[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.  doi: 10.1017/S0308210500025877.  Google Scholar

[10]

R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math., 26 (1996), 1-35.  doi: 10.1216/rmjm/1181072101.  Google Scholar

[11]

L. Collatz, Einschließungssatz für die Eigenwerte von Integralgleichungen, Math. Z., 47 (1941), 395-398.   Google Scholar

[12]

L. Collatz, Einschließungssatz für die charakteristischen Zahlen von Matrizen, Math. Z., 48 (1942), 221-226.  doi: 10.1007/BF01180013.  Google Scholar

[13]

K. D. Deimling, Nonlinear Functional Analysis, Springer, Berlin Heidelberg, 1985. Google Scholar

[14]

R. M. Dudley, Real Analysis and Probability, sec. ed. , Cambridge University Press, Cambridge, 2002. Google Scholar

[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.  doi: 10.1016/j.jde.2015.02.037.  Google Scholar

[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.  doi: 10.1016/0024-3795(89)90378-9.  Google Scholar

[17]

P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, J. Hyperbolic Differ. Equ., 7 (2010), 733-773.  doi: 10.1142/S021989161000227X.  Google Scholar

[18]

P. Gwiazda, A. Marciniak-Czochra and H. R. Thieme, Measures under the flat norm as ordered normed vector space (tentative title), in preparation. Google Scholar

[19]

K. P. Hadeler, R. Waldstätter and A. Wörz-Busekros, Models for pair formation in bisexual populations, J. Math. Biol., 26 (1988), 635-649.  doi: 10.1007/BF00276145.  Google Scholar

[20]

K. P. Hadeler, Pair formation in age-structured populations, Acta Appl. Math., 14 (1989), 91-102.  doi: 10.1007/BF00046676.  Google Scholar

[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.  doi: 10.1007/s00020-008-1652-z.  Google Scholar

[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.  doi: 10.3934/dcdsb.2016.21.447.  Google Scholar

[23]

M. A. Krasnosel'skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. Google Scholar

[24]

M. A. Krasnosel'skij, Je. A. Lifshits and A. V. Sobolev, Positive Linear Systems: The Method of Positive Operators, Heldermann Verlag, Berlin, 1989. Google Scholar

[25]

M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin Heidelberg, 1984. Google Scholar

[26]

U. Krause, Positive Dynamical Systems in Discrete Time. Theory, Models and Applications, De Gruyter, Berlin, 2015. Google Scholar

[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. Google Scholar

[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] Google Scholar

[29] B. Lemmens and R. D. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge, Cambridge University Press, 2012.  doi: 10.1017/CBO9781139026079.  Google Scholar
[30]

B. Lemmens and R. D. Nussbaum, Continuity of the cone spectral radius, Proc. Amer. Math. Soc., 141 (2013), 2741-2754.  doi: 10.1090/S0002-9939-2013-11520-0.  Google Scholar

[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.  doi: 10.3934/dcds.2002.8.519.  Google Scholar

[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.  doi: 10.1007/s11784-010-0010-3.  Google Scholar

[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.  doi: 10.1007/s11784-008-0095-0.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

R. D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc. , Providence, 75 (1988), ⅳ+137 pp. Google Scholar

[37]

R. D. Nussbaum, Eigenvectors of order-preserving linear operators, J. London Math. Soc., 58 (1998), 480-496.  doi: 10.1112/S0024610798006425.  Google Scholar

[38]

H. H. Schaefer, Positive Transformationen in lokalkonvexen halbgeordneten Vektorräumen, Math. Ann., 129 (1955), 323-329.  doi: 10.1007/BF01362375.  Google Scholar

[39]

H. H Schaefer, Halbgeordnete lokalkonvexe Vektorräume. Ⅱ, Math. Ann., 138 (1959), 259-286.  doi: 10.1007/BF01342907.  Google Scholar

[40]

H. H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966. Google Scholar

[41]

H. R. Thieme, Eigenvectors and eigenfunctionals of homogeneous order-preserving maps, arXiv: 1302.3905v1[math. FA], 2013. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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.  doi: 10.1007/s10884-015-9463-9.  Google Scholar

[45]

A. C. Thompson, On the eigenvectors of some not-necessarily-linear transformations, Proc. London Math. Soc., 15 (1965), 577-598.   Google Scholar

[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. Google Scholar

[47]

H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Z., 52 (1950), 642-648.  doi: 10.1007/BF02230720.  Google Scholar

[48]

K. Yosida, Functional Analysis, Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 123 Springer-Verlag New York Inc. , New York, 1968. Google Scholar

[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.   Google Scholar

show all references

References:
[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. Google Scholar

[2]

H. Bauer, Probability Theory and Elements of Measure Theory, sec. ed. , Academic Press, London, 1981. Google Scholar

[3]

G. Birkhoff, Uniformly semi-primitive multiplicative processes, Trans. Amer. Math. Soc., 104 (1962), 37-51.  doi: 10.1090/S0002-9947-1962-0146100-6.  Google Scholar

[4]

E. Bohl, Eigenwertaufgaben bei monotonen Operatoren und Fehlerabschätzungen für Operatorgleichungen, Arch. Rat. Mech. Anal., 22 (1966), 313-332.   Google Scholar

[5]

E. Bohl, Monotonie: Lösbarkeit und Numerik bei Operatorgleichungen, Springer, Berlin Heidelberg, 1974. Google Scholar

[6]

F. F. Bonsall, Linear operators in complete positive cones, Proc. London Math. Soc., 8 (1958), 53-75.   Google Scholar

[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.  doi: 10.1111/j.1939-7445.2001.tb00062.x.  Google Scholar

[8]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, John Wiley & Sons, Ltd. , Chichester, 2003. Google Scholar

[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.  doi: 10.1017/S0308210500025877.  Google Scholar

[10]

R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math., 26 (1996), 1-35.  doi: 10.1216/rmjm/1181072101.  Google Scholar

[11]

L. Collatz, Einschließungssatz für die Eigenwerte von Integralgleichungen, Math. Z., 47 (1941), 395-398.   Google Scholar

[12]

L. Collatz, Einschließungssatz für die charakteristischen Zahlen von Matrizen, Math. Z., 48 (1942), 221-226.  doi: 10.1007/BF01180013.  Google Scholar

[13]

K. D. Deimling, Nonlinear Functional Analysis, Springer, Berlin Heidelberg, 1985. Google Scholar

[14]

R. M. Dudley, Real Analysis and Probability, sec. ed. , Cambridge University Press, Cambridge, 2002. Google Scholar

[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.  doi: 10.1016/j.jde.2015.02.037.  Google Scholar

[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.  doi: 10.1016/0024-3795(89)90378-9.  Google Scholar

[17]

P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, J. Hyperbolic Differ. Equ., 7 (2010), 733-773.  doi: 10.1142/S021989161000227X.  Google Scholar

[18]

P. Gwiazda, A. Marciniak-Czochra and H. R. Thieme, Measures under the flat norm as ordered normed vector space (tentative title), in preparation. Google Scholar

[19]

K. P. Hadeler, R. Waldstätter and A. Wörz-Busekros, Models for pair formation in bisexual populations, J. Math. Biol., 26 (1988), 635-649.  doi: 10.1007/BF00276145.  Google Scholar

[20]

K. P. Hadeler, Pair formation in age-structured populations, Acta Appl. Math., 14 (1989), 91-102.  doi: 10.1007/BF00046676.  Google Scholar

[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.  doi: 10.1007/s00020-008-1652-z.  Google Scholar

[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.  doi: 10.3934/dcdsb.2016.21.447.  Google Scholar

[23]

M. A. Krasnosel'skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. Google Scholar

[24]

M. A. Krasnosel'skij, Je. A. Lifshits and A. V. Sobolev, Positive Linear Systems: The Method of Positive Operators, Heldermann Verlag, Berlin, 1989. Google Scholar

[25]

M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin Heidelberg, 1984. Google Scholar

[26]

U. Krause, Positive Dynamical Systems in Discrete Time. Theory, Models and Applications, De Gruyter, Berlin, 2015. Google Scholar

[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. Google Scholar

[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] Google Scholar

[29] B. Lemmens and R. D. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge, Cambridge University Press, 2012.  doi: 10.1017/CBO9781139026079.  Google Scholar
[30]

B. Lemmens and R. D. Nussbaum, Continuity of the cone spectral radius, Proc. Amer. Math. Soc., 141 (2013), 2741-2754.  doi: 10.1090/S0002-9939-2013-11520-0.  Google Scholar

[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.  doi: 10.3934/dcds.2002.8.519.  Google Scholar

[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.  doi: 10.1007/s11784-010-0010-3.  Google Scholar

[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.  doi: 10.1007/s11784-008-0095-0.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

R. D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc. , Providence, 75 (1988), ⅳ+137 pp. Google Scholar

[37]

R. D. Nussbaum, Eigenvectors of order-preserving linear operators, J. London Math. Soc., 58 (1998), 480-496.  doi: 10.1112/S0024610798006425.  Google Scholar

[38]

H. H. Schaefer, Positive Transformationen in lokalkonvexen halbgeordneten Vektorräumen, Math. Ann., 129 (1955), 323-329.  doi: 10.1007/BF01362375.  Google Scholar

[39]

H. H Schaefer, Halbgeordnete lokalkonvexe Vektorräume. Ⅱ, Math. Ann., 138 (1959), 259-286.  doi: 10.1007/BF01342907.  Google Scholar

[40]

H. H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966. Google Scholar

[41]

H. R. Thieme, Eigenvectors and eigenfunctionals of homogeneous order-preserving maps, arXiv: 1302.3905v1[math. FA], 2013. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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.  doi: 10.1007/s10884-015-9463-9.  Google Scholar

[45]

A. C. Thompson, On the eigenvectors of some not-necessarily-linear transformations, Proc. London Math. Soc., 15 (1965), 577-598.   Google Scholar

[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. Google Scholar

[47]

H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Z., 52 (1950), 642-648.  doi: 10.1007/BF02230720.  Google Scholar

[48]

K. Yosida, Functional Analysis, Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 123 Springer-Verlag New York Inc. , New York, 1968. Google Scholar

[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.   Google Scholar

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