Skip to main content Accessibility help
×
  • Cited by 87
Publisher:
Cambridge University Press
Online publication date:
May 2012
Print publication year:
2012
Online ISBN:
9781139026079

Book description

In the past several decades the classical Perron–Frobenius theory for nonnegative matrices has been extended to obtain remarkably precise and beautiful results for classes of nonlinear maps. This nonlinear Perron–Frobenius theory has found significant uses in computer science, mathematical biology, game theory and the study of dynamical systems. This is the first comprehensive and unified introduction to nonlinear Perron–Frobenius theory suitable for graduate students and researchers entering the field for the first time. It acquaints the reader with recent developments and provides a guide to challenging open problems. To enhance accessibility, the focus is on finite dimensional nonlinear Perron–Frobenius theory, but pointers are provided to infinite dimensional results. Prerequisites are little more than basic real analysis and topology.

Reviews

'In their introduction the authors state that 'the main purpose of this book is to give a systematic self-contained introduction to nonlinear Perron–Frobenius theory and to provide a guide to various challenging open problems'. They have achieved their aim excellently.'

Hans Schneider - University of Wisconsin, Madison

‘Undoubtedly, this remarkable book will be of interest to all specialists in nonlinear analysis and its applications. Certainly, any mathematical library ought to carry this book.’

Peter Zabreiko Source: Zentralblatt MATH

'This textbook is a carefully arranged journey through large parts of this beautiful theory, which has seen various contributions by the authors in the past. The material is accessible with little more than a basic knowledge of linear algebra, real analysis and some topology. The book is self-contained, all results are proven very rigorously, and where appropriate, the evolution of results is explained and framed in the historical context. I recommend this book very warmly and without any reservations to anyone interested in nonlinear Perron–Frobenius theory.'

Bjorn S. Ruffer Source: Mathematical Reviews

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
×

Contents

References
References
[1] M., Abate, Iteration Theory of Holomorphic Maps on Taut Manifolds. Research and Lecture Notes in Mathematics, Rende, Mediterranean Press, 1989.
[2] M.A., Akcoglu and U., Krengel, Nonlinear models of diffusion on a finite space. Probab. Theory Related Fields 76(4), (1987), 411–20.
[3] M., Akian and S., Gaubert, Spectral theorem for convex monotone homogeneous maps, and ergodic control. Nonlinear Anal. 52(2), (2003), 637–79.
[4] M., Akian, S., Gaubert, and B., Lemmens, Stability and convergence in discrete convex monotone dynamical systems. J. Fixed Point Theory Appl. 9(2), (2011), 295–325.
[5] M., Akian, S., Gaubert, B., Lemmens, and R.D., Nussbaum, Iteration of order preserving subhomogeneous maps on a cone. Math. Proc. Cambridge Philos. Soc. 140(1), (2006), 157–76.
[6] M., Akian, S., Gaubert, and R., Nussbaum, A Collatz–Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones, preprint. arXiv:1112.5968, 2012.
[7] M., Akian, S., Gaubert, and R., Nussbaum, Uniqueness of the fixed point of nonexpansive semidifferentiable maps, preprint. arXiv:1201.1536, 2012.
[8] P., Alexandroff and H., Hopf, Topologie.Berlin, Springer Verlag, 1935.
[9] J.C. Álvarez, Paiva, Hilbert's fourth problem in two dimensions. In MASS Selecta: Teaching and Learning Advanced Undergraduate Mathematics. S., Katok, A., Sossinsky, and S., Tabachnikov, eds., Providence, RI, American Mathematical Society, 2003, pp. 165–83.
[10] N., Aronszajn and P., Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces. Pacific J. Math. 6, (1956), 405Ɖ439.
[11] F., Baccelli, G., Cohen, G.J., Olsder, and J.P., Quadrat, Synchronization and Linearity: An Algebra for Discrete Event Systems. Wiley Series in Probability and Statistics, Chichester, Wiley, 1992.
[12] W., Ballmann, Lectures on Spaces of Nonpositive Curvature. DMV Seminar, 25, Basel, Birkhäuser Verlag, 1995.
[13] S., Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. Fund. Math. 3, (1922), 133–80.
[14] R., Bapat, D1 AD2 theorems for multidimensional matrices. Linear Algebra Appl. 48, (1982), 437–42.
[15] R.B., Bapat and T.E.S., Raghavan, Nonnegative Matrices and Applications. Encyclopedia of Mathematics and its Applications, 64, Cambridge, Cambridge University Press, 1997.
[16] G.P., Barker and R.E.L., Turner, Some observations on the spectra of conepreserving maps. Linear Algebra Appl. 6, (1973), 149–53.
[17] F.L., Bauer, An elementary proof of the Hopf inequality for positive operators. Numer. Math. 7, (1965), 331–7.
[18] A.F., Beardon, Iteration of contractions and analytic maps. J. Lond. Math. Soc. (2) 41(1), (1990), 141–50.
[19] A.F., Beardon, The dynamics of contractions. Ergodic Theory Dynam. Systems 17(6), (1997), 1257–66.
[20] A.F., Beardon, The Klein, Hilbert and Poincaré metrics of a domain. J. Comput. Appl. Math. 105(1–2), (1999), 155–62.
[21] R., Bellman, Dynamic Programming. Princeton, NJ, Princeton University Press, 1957.
[22] A., Berman and R.J., Plemmons, Nonnegative Matrices in the Mathematical Sciences. Computer Science and Applied Mathematics, New York, Academic Press, 1979.
[23] T., Bewley and E., Kohlberg, The asymptotic theory of stochastic games. Math. Oper. Res. 1(3), (1976), 197–208.
[24] R., Bhatia, Matrix Analysis. Graduate Texts in Mathematics, 169, New York, Springer Verlag, 1996.
[25] G., Birkhoff, Extensions of Jentzsch's theorem. Trans. Amer. Math. Soc. 85(1), (1957), 219–27.
[26] G., Birkhoff, Uniformly semi-primitive multiplicative processes. Trans. Amer. Math. Soc. 104, (1962), 37–51.
[27] A., Blokhuis and H.A., Wilbrink, Alternative proof of Sine's theorem on the size of a regular polygon in ℝk with the ℓ∞-metric. Discrete Comput. Geom. 7(4), (1992), 433–4.
[28] B., Bollobás, Combinatorics. Cambridge, Cambridge University Press, 1989.
[29] F.F., Bonsall, Sublinear functionals and ideals in partially ordered vector spaces. Proc. Lond. Math. Soc. 4, (1954). 402–18.
[30] F.F., Bonsall, Endomorphisms of partially ordered vector spaces. J. Lond. Math. Soc. 30, (1955), 133–44.
[31] F.F., Bonsall, Endomorphisms of partially ordered vector spaces without order unit. J. Lond. Math. Soc. 30, (1955), 145–53.
[32] F.F., Bonsall, Linear operators in complete positive cones. Proc. Lond. Math. Soc. 8(3), (1958), 53–75.
[33] J.M., Borwein and P.B., Borwein, The arithmetic-geometric mean and fast computation of elementary functions. SIAM Rev. 26(3), (1984), 351–66.
[34] J.M., Borwein and A.S., Lewis, Decomposition of multivariate functions. Canad. J. Math. 44(3), (1992), 463–82.
[35] J.M., Borwein, A.S., Lewis, and R.D., Nussbaum, Entropy minimization, DAD problems, and doubly stochastic kernels. J. Funct. Anal. 123(2), (1994), 264–307.
[36] T., Bousch and J., Mairesse, Finite-range topical functions and uniformly topical functions. Dyn. Syst. 21(1), (2006), 73–114.
[37] M.R., Bridson and A., Haefliger, Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften, 319, Berlin, Springer Verlag, 1999.
[38] R.A., Brualdi, The DAD theorem for arbitrary row sums. Proc. Amer. Math. Soc. 45(2), (1974), 189–94.
[39] R.A., Brualdi, S.V., Parter, and H., Schneider, The diagonal equivalence of a nonnegative matrix to a stochastic matrix. J. Math. Anal. Appl. 16, (1966), 31–50.
[40] R.E., Bruck, Jr., Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Trans. Amer. Math. Soc. 179, (1973), 251–62.
[41] A.D., Burbanks, R.D., Nussbaum, and C., Sparrow, Extension of order-preserving maps on a cone. Proc. Roy. Soc. Edinburgh Sect. A 133A, (2003), 35–59.
[42] H., Busemann, The Geometry of Geodesics. Pure and Applied Mathematics, 6, New York, Academic Press, 1955.
[43] P.J., Bushell, Hilbert's projective metric and positive contraction mappings in a Banach space. Arch. Ration. Mech. Anal. 52, (1973), 330–8.
[44] P.J., Bushell, On the projective contraction ratio for positive linear mappings. J. Lond. Math. Soc. 6, (1973), 256–8.
[45] P.J., Bushell, On solutions of the matrix equation T′ AT = A2. Linear Algebra Appl. 8, (1974), 465–9.
[46] P.J., Bushell, The Cayley-Hilbert metric and positive operators. Linear Algebra Appl. 84, (1986), 271–80.
[47] P., Butkovic, Max-linear Systems: Theory and Algorithms. Springer Monographs in Mathematics, Berlin, Springer Verlag, 2010.
[48] A., Całka, On a condition under which isometries have bounded orbits. Colloq. Math. 48, (1984), 219–27.
[49] B.C., Carlson, Algorithms involving arithmetic and geometric means. Amer. Math. Monthly 78, (1971), 496–505.
[50] I., Csiszár, I-divergence geometry of probability distributions and minimization problems. Ann. Probability 3, (1975), 146–58.
[51] J., Cochet-Terrasson, S., Gaubert, and J., Gunawardena, A constructive fixed-point theorem for min-max functions. Dynam. Stability Systems 14(4), (1999), 407–33.
[52] L., Collatz, Einschliessungs satz für die charakteristischen Zahlen von Matrizen. Math. Z. 48, (1942), 221–6.
[53] D.A., Cox, The arithmetic-geometric mean of Gauss. Enseign. Math. (2) 30(3–4), (1984), 275–330.
[54] M.G., Crandall and L., Tartar, Some relations between nonexpansive and order preserving mappings. Proc. Amer. Math. Soc. 78(3), (1980), 385–90.
[55] C.M., Dafermos and M., Slemrod, Asymptotic behavior of nonlinear contraction semigroups. J. Funct. Anal. 13, (1973), 97–106.
[56] A., Denjoy, Sur l'itération des fonctions analytiques. C. R. Acad. Sc. Paris, Seriel 182, (1926), 255–7.
[57] W.F., Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation. Grundlehren der Mathematischen Wissenschaften, 207, New York, Springer Verlag, 1974.
[58] N., Dunford and J.T., Schwartz. Linear Operators. Wiley Classics Library, New York, John Wiley & Sons, 1988.
[59] B.C., Eaves, A.J., Hoffman, U.G., Rothblum, and H., Schneider, Line-sumsymmetric scalings of square nonnegative matrices. Math. Programming Stud. 25, (1985), 124–41.
[60] M., Edelstein, On non-expansive mappings of Banach spaces. Proc. Cambridge Philos. Soc. 60, (1964), 439–47.
[61] S.P., Eveson and R.D., Nussbaum, An elementary proof of the Birkhoff–Hopf theorem. Math. Proc. Cambridge Philos. Soc. 117, (1995), 31–55.
[62] S.P., Eveson and R.D., Nussbaum, Applications of the Birkhoff–Hopf theorem to the spectral theory of positive linear operators. Math. Proc. Cambridge Philos. Soc. 117, (1995), 491–512.
[63] J., Faraut and A., Korányi, Analysis on Symmetric Cones. Oxford, Clarendon Press, 1994.
[64] M., Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 17(1), (1923), 228–49.
[65] S.E., Fienberg, An iterative procedure for estimation in contingency tables. Ann. Math. Statist. 41, (1970), 907–17.
[66] T., Foertsch and A., Karlsson, Hilbert metrics and Minkowski norms. J. Geom. 83(1–2), (2005), 22–31.
[67] J., Franklin and J., Lorenz, On the scaling of multidimensional matrices. Linear Algebra Appl. 114/115, (1989), 717–35.
[68] H., Freudenthal and W., Hurewicz, Dehnungen, Verkürzungen, Isometrien. Fund. Math. 26 (1936), 120–2.
[69] S., Friedland and H., Schneider, The growth of powers of a nonnegative matrix. SIAM J. Algebraic Discrete Methods 1, (1980), 185–200.
[70] G., Frobenius, Über Matrizen aus positiven Elementen. S.-B. Preuss. Akad. Wiss (Berlin), (1908), 471–6.
[71] G., Frobenius, Über Matrizen aus positiven Elementen, II. S.-B. Preuss. Akad. Wiss (Berlin), (1908), 514–18.
[72] G., Frobenius, Über Matrizen aus nicht negativen Elementen. S.-B. Preuss. Akad. Wiss (Berlin), (1912), 456–77.
[73] F.R., Gantmacher, The Theory of Matrices, Vols. 1, 2. New York, Chelsea, 1959.
[74] S., Gaubert and J., Gunawardena, The Perron–Frobenius theory for homogeneous, monotone functions. Trans. Amer. Math. Soc. 356(12), (2004), 4931–50.
[75] S., Gaubert and G., Vigeral, A maximin characterization of the escape rate of nonexpansive mappings in metrically convex spaces, Math. Proc. Cambridge Phil. Soc., in press. arXiv: 1012.4765, 2010. DOI: 10.1017/S0305004111000673, 2011.
[76] K., Goebel and W.A., Kirk, Some problems in metric fixed point theory. J. Fixed Point Theory Appl. 4(1), (2008), 13–25.
[77] K., Goebel and S., Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Monographs and Textbooks in Pure and Applied Mathematics, 83, New York, Marcel Dekker, 1984.
[78] M., Gromov, Hyperbolic manifolds, groups and actions. In Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, NY, 1978), Annals of Mathematics Studies, 97, Princeton, NJ, Princeton University Press, 1981, pp. 183–213.
[79] J., Gunawardena, An introduction to idempotency. In Idempotency (Bristol, 1994), J., Gunawardena, ed., Publications of the Newton Institute, 11, Cambridge, Cambridge University Press, 1998, pp. 1–49.
[80] J., Gunawardena, From max-plus algebra to nonexpansive mappings: A nonlinear theory for discrete event systems. Theoret. Comput. Sci. 293(1), (2003), 141–67.
[81] J., Gunawardena and C., Walsh, Iterates of maps which are non-expansive in Hilbert's projective metric. Kybernetika (Prague) 39(2), (2003), 193–204.
[82] G.H., Hardy, J.E., Littlewood, and G., Pólya, Inequalities. Cambridge Mathematical Library, Cambridge, Cambridge University Press, 1988.
[83] P., de la Harpe, On Hilbert's metric for simplices. In Geometric Group Theory, Vol. 1, Niblo, G.A., et al., eds., Lecture Note Series, 181, London Mathematical Society, 1993, pp. 97–119.
[84] T., Hawkins, Continued fractions and the origins of the Perron–Frobenius theorem. Arch. Hist. Exact Sci. 62(6), (2008), 655–717.
[85] M., Hervé, Quelques proprietés des applications analytiques d'une boule à m dimensions dans elle-même. J. Math. Pures Appl. 42, (1963), 117–47.
[86] D., Hilbert, Über die gerade Linie als kürzeste Verbindung zweier Punkte. Math. Ann. 46, (1895), 91–6
[87] M.W., Hirsch, Stability and convergence in strongly monotone dynamical systems. J. Reine Angew. Math. 383, (1988), 1–53.
[88] M.W., Hirsch, Positive equilibria and convergence in subhomogeneous monotone dynamics. In Comparison Methods and Stability Theory, X., Liu and D., Siegel, eds., Lecture Notes in Pure and Applied Mathematics, 162, New York, Marcel Dekker, 1994, pp. 169–88.
[89] M.W., Hirsch and H., Smith, Monotone dynamical systems. In Handbook of Differential Equations: Ordinary Differential Equations, Vol. II, Amsterdam, Elsevier, 2005, pp. 239–357.
[90] E., Hopf, An inequality for positive integral operators. J. Math. Mech 12, (1963), 683–92.
[91] E., Hopf, Remarks on my paper “An inequality for positive integral operators.”J. Math. Mech. 12, (1963), 889–92.
[92] R.A., Howard, Dynamic Programming and Markov Processes. New York, John Wiley & Sons, 1960.
[93] G., Jameson, Ordered Linear Spaces. Lecture Notes in Mathematics, 141, Berlin, Springer Verlag, 1970.
[94] P., Jordan, J., von Neumann, and E., Wigner, On an algebraic generalization of the quantum mechanical formalism. Ann. of Math. (2) 35(1), (1934), 29–64.
[95] B., Kalantari and L., Khachiyan, On the complexity of nonnegative-matrix scaling. Linear Algebra Appl. 240, (1996), 87–103.
[96] J., Kapeluszny, T., Kuczumow, and S., Reich, The Denjoy–Wolff theorem in the open unit ball of a strictly convex Banach space. Adv. Math. 143(1), (1999), 111–23.
[97] J., Kapeluszny, T., Kuczumow, and S., Reich, The Denjoy–Wolff theorem for condensing holomorphic mappings. J. Funct. Anal. 167(1), (1999), 79–93.
[98] S., Karlin and L., Nirenberg, On a theorem of P. Nowosad. J. Math. Anal. Appl. 17, (1967), 61–7.
[99] A., Karlsson, Nonexpanding maps and Busemann functions. Ergodic Theory Dynam. Systems 21(5), (2001), 1447–57.
[100] A., Karlsson, V., Metz, and G.A., Noskov, Horoballs in simplices and Minkowski spaces. Int. J. Math. Math. Sci. 2006, Article ID 23656, (2006), 20 pages.
[101] A., Karlsson and G.A., Noskov, The Hilbert metric and Gromov hyperbolicity. Enseign. Math. (2) 48(1–2), (2002), 73–89.
[102] T., Kato, Perturbation Theory for Linear Operators. Grundlehren der Mathematischen Wissenschaften, 132, Berlin, Springer Verlag, 1976.
[103] P.E., Kloeden and A.M., Rubinov, A generalization of the Perron–Frobenius theorem. Nonlinear Anal. 41(1–2), (2000), 97–115.
[104] E., Kohlberg, Invariant half-lines of nonexpansive piecewise-linear transformations. Math. Oper. Res. 5(3), (1980), 366–72.
[105] E., Kohlberg, The Perron–Frobenius theorem without additivity. J. Math. Econom. 10, (1982), 299–303.
[106] E., Kohlberg and A., Neyman, Asymptotic behavior of nonexpansive mappings in normed linear spaces. Israel J. Math. 38(4), (1981), 269–75.
[107] E., Kohlberg and J.W., Pratt, The contraction mapping approach to the Perron–Frobenius theory: Why Hilbert's metric?Math. Oper. Res. 7(2), (1982), 198–210.
[108] K., Koufany, Application of Hilbert's projective metric on symmetric cones. Acta Math. Sin. (Engl. Ser.) 22(5), (2006), 1467–72.
[109] M.A., Krasnosel'skii, Positive Solutions of Operator Equations. Groningen, Noordhoff, 1964.
[110] M.A., Krasnosel'skii, J. A., Lifshits, and A.V., Sobolev, Positive Linear Systems: The Method of Positive Operators. Sigma Series in Applied Mathematics, 5, Lemgo, Heldermann Verlag, 1989.
[111] M.A., Krasnosl'skii and A.V., Sobolev, Spectral clearance of a focussing operator. Funct. Anal. Appl. 17, (1983), 58–9.
[112] U., Krause, A nonlinear extension of the Birkhoff-Jentzsch theorem. J. Math. Anal. Appl. 114(2), (1986), 552–68.
[113] U., Krause, Relative stability for ascending and positively homogeneous operators on Banach spaces. J. Math. Anal. Appl. 188(1), (1994), 182–202.
[114] U., Krause and R.D., Nussbaum, A limit set trichotomy for self-mappings of normal cones in Banach spaces. Nonlinear Anal. 20(7), (1993), 855–70.
[115] U., Krause and P., Ranft, A limit set trichotomy for monotone nonlinear dynamical systems. Nonlinear Anal. 19(4), (1992), 375–92.
[116] M.G., Kreĭn and D., Milman, On the extreme points of regularly convex sets. Studia Math. 9, (1940), 133–8.
[117] M.G., Kreĭn and M.A., Rutman, Linear operators leaving invariant a cone in a Banach space. Amer. Math. Soc. Translation 1950 26, (1950), 199–325.
[118] E., Landau, Handbuch der Lehre von der Verteilung der Primzahlen, I, 2nd ed. New York, Chelsea, 1953.
[119] B., Lemmens, Nonexpansive mappings on Hilbert's metric spaces. Topol. Methods Nonlinear Anal. 38(1), (2011), 45–58.
[120] B., Lemmens and R., Nussbaum, Continuity of the cone spectral radius. Proc. Amer. Math. Soc.,in press. arXiv: 1107.4532, 2011.
[121] B., Lemmens, R.D., Nussbaum, and S.M. Verduyn, Lunel, Lower and upper bounds for ω-limit sets of nonexpansive maps. Indag. Math. (N.S.) 12(2), (2001), 191–211.
[122] B., Lemmens and M., Scheutzow, A characterization of the periods of periodic points of 1-norm nonexpansive maps. Selecta Math. (N.S.) 9(4), (2003), 557–78.
[123] B., Lemmens and M., Scheutzow, On the dynamics of sup-norm nonexpansive maps. Ergodic Theory Dynam. Systems 25(3), (2005), 861–71.
[124] B., Lemmens and C., Sparrow, A note on periodic points of order preserving subhomogeneous maps. Proc. Amer. Math. Soc. 134(5), (2006), 1513–17.
[125] B., Lemmens, C., Sparrow, and M., Scheutzow, Transitive actions of finite abelian groups of sup-norm isometries. European J. Combin. 28(4), (2007), 1163–79.
[126] B., Lemmens and O.W., van Gaans, Periods of order-preserving nonexpansive maps on strictly convex normed spaces. J. Nonlinear Convex Anal. 4(3), (2003), 353–63.
[127] B., Lemmens and O.W., van Gaans, On non-expansive maps dynamics on strictly convex Banach spaces. Israel J. Math. 171, (2009), 425–42.
[128] Y., Lim, Hilbert's projective metric on Lorentz cones and Birkhoff formula for Lorentzian compressions. Linear Algebra Appl. 423(2–3), (2007), 246–54.
[129] N., Linial, A., Samorodnitsky, and A., Wigderson, A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents. Combinatorica 20(4), (2000), 545–68.
[130] B., Lins, Asymptotic Behavior and Denjoy–Wolff Theorems for Hilbert Metric Nonexpansive Maps. PhD. thesis, Rutgers University, New Brunswick, NJ, 2007.
[131] B., Lins, A Denjoy–Wolff theorem for Hilbert metric nonexpansive maps on a polyhedral domain. Math. Proc. Cambridge. Philos. Soc. 143(1), (2007), 157–64.
[132] B., Lins, Asymptotic behavior of nonexpansive mappings in finite-dimensional normed spaces. Proc. Amer. Math. Soc. 137(7), (2009), 2387–92.
[133] B., Lins and R., Nussbaum, Iterated linear maps on a cone and Denjoy–Wolff theorems. Linear Algebra Appl. 416(2–3), (2006), 615–26.
[134] B., Lins and R., Nussbaum, Denjoy–Wolff theorems, Hilbert metric nonexpansive maps and reproduction-decimation operators. J. Funct. Anal. 254(9), (2008), 2365–86.
[135] K., Löwner, Über monotone Matrixfunktionen. Math. Z. 38, (1934), 177–216.
[136] D., Lubell, A short proof of Sperner's lemma. J. Combin. Theory 1, (1966), 299.
[137] R., Lyons and R.D., Nussbaum, On transitive and commutative finite groups of isometries. In Fixed Point Theory and Applications, K.-K., Tan, ed., Singapore, World Scientific, 1992, pp. 189–228.
[138] J., Mallet-Paret and R.D., Nussbaum, Eigenvalues for a class of homogeneous cone maps arising from max-plus operators. Discrete Contin. Dyn. Syst. 8 (2002), 519–63.
[139] J., Mallet-Paret and R.D., Nussbaum, Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index. J. Fixed Point Theory Appl. 7(1), (2010), 103–43.
[140] P., Martus, Asymptotic Properties of Nonstationary Operator Sequences in the Nonlinear Case. PhD. thesis, Friedrich-Alexander Universität, Erlangen-Nürnberg, Germany, 1989.
[141] J.-P., Massias, Majoration explicite de l'ordre maximum d'un élément du groupe symétrique. Ann. Fac. Sci. Toulouse Math. (5) 6(3–4), 269–81, 1984.
[142] P., Mellon, A general Wolff theorem for arbitrary Banach spaces. Math. Proc. R. Ir. Acad. 104A(2), (2004), 127–42.
[143] M.V., Menon, Reduction of a matrix with positive elements to a doubly stochastic matrix. Proc. Amer. Math. Soc. 18, (1967), 244–7.
[144] M.V., Menon, Some spectral properties of an operator associated with a pair of nonnegative matrices. Trans. Amer. Math. Soc. 132, (1968), 369–75.
[145] M.V., Menon and H., Schneider, The spectrum of a nonlinear operator associated with a matrix. Linear Algebra Appl. 2, (1969), 321–34.
[146] L.D., Meshalkin, Generalization of Sperner's theorem on the number of subsets of a finite set (in Russian). Teor. Veroyatn. Primen. 8, (1963), 219–20.
[147] W., Miller, The maximum order of an element of a? nite symmetric group. Amer. Math. Monthly 94(6), (1987), 497–506.
[148] H., Minc, Nonnegative Matrices. Wiley-Interscience Series in Discrete Mathematics and Optimization, New York, John Wiley & Sons, 1988.
[149] M., Misiurewicz, Rigid sets in finite-dimensional l1-spaces. Technical Report 45, Mathematica Göttingensis Schriftenreihe des Sonderforschungsbereichs Geometrie und Analysis, Göttingen, Mathematisches Institut, George-August-Universität Göttingen, 1987.
[150] M., Morishima. Equilibrium, Stability and Growth. Oxford, Clarendon Press, 1964
[151] B., de Sz.Nagy, Sur les lattis linéaires de dimension finie. Comment. Math. Helv. 17, (1945), 209–13.
[152] A., Neyman, Stochastic games and nonexpansive maps. In Stochastic Games and Applications (Stony Brook, NY, 1999), A., Neyman and S., Sorin, eds., NATO Science Series C: Mathematical and Physical Sciences, 570, Dordrecht, Kluwer Academic Publishers, 2003, pp. 397–415.
[153] P., Nowosad, On the integral equation kf = 1/f arising in a problem in communication. J. Math. Anal. Appl. 14, (1966), 484–92.
[154] R.D., Nussbaum, A generalization of the Ascoli theorem and an application to functional differential equations. J. Math. Anal. Appl. 35, (1971), 600–10.
[155] R.D., Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem. In Fixed Point Theory, E., Fadell and G., Fournier, eds., Lecture Notes in Mathematics, 886, Berlin, Springer Verlag, 1981, pp. 309–331.
[156] R.D., Nussbaum, Convexity and log convexity for the spectral radius. Linear Algebra Appl. 73, (1986), 59–122.
[157] R.D., Nussbaum, Iterated nonlinear maps and Hilbert's projective metric: A summary. In Dynamics of Infinite-Dimensional Systems (Lisbon, 1986), NATO Advanced Science Institutes Series F: Computer and Systems Science, 37, Berlin, Springer Verlag, 1987, pp. 231–248.
[158] R.D., Nussbaum, Hilbert's projective metric and iterated nonlinear maps. Mem. Amer. Math. Soc. 391, (1988), 1–137.
[159] R.D., Nussbaum, Iterated nonlinear maps and Hilbert's projective metric, II, Mem. Amer. Math. Soc. 401, (1989), 1–118.
[160] R. D., Nussbaum, Omega limit sets of nonexpansive maps: Finiteness and cardinality estimates. Differential Integral Equations 3(3), (1990), 523–40.
[161] R.D., Nussbaum, Estimates of the periods of periodic points of nonexpansive operators. Israel J. Math. 76(3), (1991), 345–80.
[162] R.D., Nussbaum, Convergence of iterates of a nonlinear operator arising in statistical mechanics. Nonlinearity 4(4), (1991), 1223–40.
[163] R.D., Nussbaum, Entropy minimization, Hilbert's projective metric, and scaling integral kernels. J. Funct. Anal. 115(1), (1993), 45–99.
[164] R.D., Nussbaum, Finsler structures for the part metric and Hilbert's projective metric and applications to ordinary differential equations. Differential Integral Equations 7(5–6), (1994), 1649–1707.
[165] R.D., Nussbaum, Lattice isomorphisms and iterates of nonexpansive maps. Nonlinear Anal. 22(8), (1994), 945–70.
[166] R.D., Nussbaum, Eigenvectors of order-preserving linear operators. J. Lond. Math. Soc. 58, (1998), 480–96.
[167] R.D., Nussbaum, Periodic points of positive linear operators and Perron–Frobenius operators. Integral Equations Operator Theory 39, (2001), 41–97.
[168] R.D., Nussbaum, Fixed Point Theorems and Denjoy–Wolff Theorems for Hilbert's Projective Metric in Infinite Dimensions. Topol. Methods Nonlinear Anal. 29(2), (2007), 199–250.
[169] R.D., Nussbaum and M., Scheutzow, Admissible arrays and a generalization of Perron-Frobenius theory. J. Lond. Math. Soc. 58(2), (1998), 526–44.
[170] R.D., Nussbaum, M., Scheutzow, and S.M. Verduyn, Lunel, Periodic points of nonexpansive maps and nonlinear generalizations of the Perron–Frobenius theory. Selecta Math. (N.S.) 4(1), (1998), 1–41.
[171] R.D., Nussbaum and S.M. Verduyn, Lunel, Generalizations of the Perron–Frobenius theorem for nonlinear maps. Mem. Amer. Math. Soc. 138(659), (1999), 1–98.
[172] R.D., Nussbaum and S.M. Verduyn, Lunel, Asymptotic estimates for the periods of periodic points of non-expansive maps. Ergodic Theory Dynam. Systems 23, (2003), 1199–1226.
[173] R.D., Nussbaum and B.J., Walsh, Approximations by polynomials with nonnegative coefficients and the spectral theory of positive operators. Trans. Amer. Math. Soc. 350, (1998), 2367–91.
[174] Y., Oshime, An extension of Morishima's nonlinear Perron–Frobenius theorem. J. Math. Kyoto Univ. 23(4), (1983), 803–30.
[175] A.M., Ostrowski, On positive matrices. Math. Ann. 150, (1963), 276–84.
[176] A.M., Ostrowski, Positive matrices and functional analysis. In Recent Advances in Matrix Theory, Madison, WI, University of Wisconsin Press, 1964, pp. 81–101.
[177] A., Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature. IRMA Lectures in Mathematics and Theoretical Physics, 6, Zürich, European Mathematical Society Publishing House, 2005.
[178] B.N., Parlett and T.L., Landis, Methods for scaling to doubly stochastic form. Linear Algebra Appl. 48, (1982), 53–79.
[179] O., Perron, Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus. Math. Ann. 64, (1907), 1–76.
[180] O., Perron, Zur Theorie der Über Matrizen. Math. Ann. 64, (1907), 248–63.
[181] B.B., Phadke, A triangular world with hexagonal circles. Geom. Dedicata 3, (1975), 511–20.
[182] P., Poláčik and I., Tereščak, Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems. Arch. Ration. Mech. Anal. 116(4), (1992), 339–60.
[183] A.J.B., Potter, Applications of Hilbert's projective metric to certain classes of non- homogeneous operators. Quart. J. Math. Oxford Ser. (2) 28, (1977), 93–9.
[184] H., Rademacher, Über partielle und totale Differenzierbarkeit von Funktionenmehrerer Variabeln und über die Transformation der Doppelintegrale. Math. Ann. 79, 340–59.
[185] C.E., Rickart, General Theory of Banach Algebras. University Series in Higher Mathematics, Princeton, NJ, van Nostrand, 1960
[186] R.T., Rockafellar, Convex Analysis. Princeton Landmarks in Mathematics, Princeton, NJ, Princeton University Press, 1997.
[187] R.T., Rockafellar and R. J.-B., Wets, Variational Analysis. Grundlehren der Mathematischen Wissenschaften, 317, Berlin, Springer Verlag, 1998.
[188] U.G., Rothblum, Generalized scalings satisfying linear equations. Linear Algebra Appl. 114/115, (1989), 765–83.
[189] U.G., Rothblum, H., Schneider, and M.H., Schneider, Scaling matrices to prescribed row and column maxima. SIAM J. Matrix Anal. Appl. 15(1), (1994), 1–14.
[190] W., Rudin, Functional Analysis. International Series in Pure and Applied Mathematics, New York, McGraw-Hill, 1991.
[191] W., Rudin, Real and Complex Analysis. Series in Higher Mathematics, New York, McGraw-Hill, 1966.
[192] H., Samelson, On the Perron–Frobenius theorem. Michigan Math. J. 4, (1957), 57–9.
[193] H.H., Schaefer, Positive Transformationen in lokalkonvexen halbgeordneten Vektorräumen. Math. Ann. 129, (1955), 323–9.
[194] H.H., Schaefer, On nonlinear positive operators. Pacific J. Math. 9(3), (1959), 847–60.
[195] H.H., Schaefer, Some spectral properties of positive linear operators. Pacific J. Math. 10, (1960), 1009–19.
[196] H.H., Schaefer, Banach Lattices and Positive Operators. Grundlehren der Mathematischen Wissenschaften, 215, New York, Springer Verlag, 1974.
[197] M., Scheutzow, Periods of nonexpansive operators on finite l1-spaces. European J. Combin. 9, (1988), 73–8.
[198] M., Scheutzow, Corrections to periods of nonexpansive operators on finite l1- spaces. European J. Combin. 12(2), (1991), 183.
[199] M.H., Schneider, Matrix scaling, entropy minimization, and conjugate duality, I: Existence conditions. Linear Algebra Appl. 114/115, (1989), 785–813.
[200] M.H., Schneider, Matrix scaling, entropy minimization, and conjugate duality, II: The dual problem. Math. Programming (Ser. B) 48(1), (1990), 103–24.
[201] A., Schrijver, Theory of Linear and Integer Programming. Chichester, John Wiley & Sons, 1986.
[202] E., Seneta, Nonnegative Matrices and Markov Chains, 2nd edn. Springer Series in Statistics, New York, Springer Verlag, 1981.
[203] L.S., Shapley, Stochastic games. Proc. Natl. Acad. Sci. USA 39, (1953), 1095–1100.
[204] R., Sine, A nonlinear Perron–Frobenius theorem. Proc. Amer. Math. Soc. 109(2), (1990), 331–6.
[205] R., Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Statist. 35, (1964), 876–9.
[206] R., Sinkhorn and P., Knopp, Concerning nonnegative matrices and doubly stochastic matrices. Pacific J. Math. 21. (1967), 343–8.
[207] H.L., Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs, 41, Providence, RI, American Mathematical Society, 1995.
[208] S., Sorin, A First Course on Zero-Sum Repeated Games. Mathématiques et Applications, 37, Berlin, Springer Verlag, 2002.
[209] E., Sperner, Ein Satz über Untermengen einer endlichen Menge. Math. Z. 27, (1928), 544–8.
[210] S., Straszewicz, Über exponierte Punkte abgeschlossener Punktmengen. Fund. Math. 24, (1935), 139–43.
[211] P., Takác, Asymptotic behavior of discrete-time semigroups of sublinear, strongly increasing mappings with applications to biology. Nonlinear Anal. 14(1), (1990), 35–42.
[212] P., Takác, Convergence in the part metric for discrete dynamical systems in ordered topological cones. Nonlinear Anal. 26(11), (1996), 1753–77.
[213] B.-S., Tam, On the distinguished eigenvalues of a cone-preserving map. Linear Algebra Appl. 131, (1990), 17–37.
[214] B.-S., Tam, A cone-theoretic approach to the spectral theory of positive linear operators: The finite-dimensional case. Taiwanese J. Math. 5(2), (2001), 207–77.
[215] A.E., Taylor, Introduction to Functional Analysis. London, John Wiley & Sons, 1958.
[216] A.C., Thompson, Generalizations of the Perron–Frobenius Theorem to Operators Mapping a Cone into Itself. PhD. thesis, Newcastle upon Tyne, UK, 1963.
[217] A.C., Thompson, On certain contraction mappings in a partially ordered vector space. Proc. Amer. Math. Soc. 14, (1963), 438–43.
[218] A.C., Thompson, On the eigenvalues of some not-necessarily-linear transformations. Proc. Lond. Math. Soc. 15(3), (1965), 577–98.
[219] J.S., Vandergraft, Spectral properties of matrices which have invariant cones. SIAM J. Appl. Math. 16, (1968), 1208–22.
[220] C., Walsh, The horofunction boundary of finite-dimensional normed spaces. Math. Proc. Cambridge Philos. Soc. 142(3), (2007), 497–507.
[221] C., Walsh, The horofunction boundary of the Hilbert geometry. Adv. Geom. 8(4), (2008), 503–29.
[222] D., Weller, Hilbert's Metric, Part Metric and Self-Mappings of a Cone. PhD. thesis, Universität Bremen, Germany, 1987.
[223] J.H., Wells and L.R., Williams, Embeddings and Extensions in Analysis. Ergebnisse Mathematik und ihrer Grenzgebiete, 84, Berlin, Springer Verlag, 1975.
[224] P., Whittle, Optimization Over Time. Vol. I. Dynamic Programming and Stochastic Control. Wiley Series in Probability and Statistics, Chichester, John Wiley & Sons, 1982.
[225] H., Wielandt, Unzerlegbare, nicht negative Matrizen. Math. Z. 52, (1950), 642–8.
[226] J., Wolff, Sur l'itération des fonctions bornées. C. R. Acad. Sc. Paris, Serie 1, 182, (1926), 200–1.
[227] J., Wolff, Sur une généralisation d'un théoreme de Schwartz. C. R. Acad. Sc. Paris, Serie 1, 183, (1926), 500–2.
[228] K., Wysocki, Behavior of directions of solutions of differential equations. Differential Integral Equations 5(2), (1992), 281–305.
[229] K., Yamamoto, Logarithmic order of free distributive lattice. J. Math. Soc. Japan 6, (1954), 343–53.
[230] P.P., Zabreiko, M.A., Krasnosel'skii, and Yu.V., Pokornyi, On a class of positive linear operators. Funct. Anal. Appl. 5, (1972), 57–70.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.