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Hilbert’s metric and positive contraction mappings in a Banach space. (English) Zbl 0275.46006


MSC:

46A40 Ordered topological linear spaces, vector lattices
51M10 Hyperbolic and elliptic geometries (general) and generalizations
47B60 Linear operators on ordered spaces
15B48 Positive matrices and their generalizations; cones of matrices
47H99 Nonlinear operators and their properties
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References:

[1] Bauer, F. L., An elementary proof of the Hopf inequality for positive operators. Numerische Math. 7, 331-337 (1965) · Zbl 0148.38103 · doi:10.1007/BF01436527
[2] Bauer, F. L., E. Deutsch, & J. Stoer, Abschätzungen für die Eigenwerte positiver linearer Operatoren. Linear Algebra and its Applications 2, 275-301 (1969) · Zbl 0199.45102 · doi:10.1016/0024-3795(69)90031-7
[3] Birkhoff, G., Extensions of Jentzch’s Theorem. Trans. Amer. Math. Soc. 85, 219-227 (1957) · Zbl 0079.13502
[4] Birkhoff, G., Uniformly semi-primitive multiplicative processes. Trans. Amer. Math. Soc. 104, 37-51 (1962) · Zbl 0101.33304 · doi:10.1090/S0002-9947-1962-0146100-6
[5] Birkhoff, G., Uniformly semi-primitive multiplicative processes II. J. Math. Mech. 14, 507-512 (1965) · Zbl 0131.33401
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[7] Birkhoff, G., & L. Kotin, Linear second order differential equations of positive type. J. d’Analyse Math. 18, 43-52 (1967) · Zbl 0148.05904 · doi:10.1007/BF02798033
[8] Birkhoff, G., & L. Kotin, Third order positive cyclic systems of linear differential equations. J. Diff. Eq. 5, 182-196 (1969) · Zbl 0169.11303 · doi:10.1016/0022-0396(69)90111-9
[9] Bushell, P. J., On systems of linear ordinary differential equations and the uniqueness of monotone solutions. J. London Math. Soc. (2) 5, 235-239 (1972) · Zbl 0251.34007 · doi:10.1112/jlms/s2-5.2.235
[10] Bushell, P. J., On the projective contraction ratio for positive linear mappings. J. London Math. Soc. (2) 6, 256-258 (1963) · Zbl 0255.47048 · doi:10.1112/jlms/s2-6.2.256
[11] Bushell, P. J., On solutions of the matrix equation T? AT= A 2. Linear Algebra and its Applications (to appear) · Zbl 0288.15019
[12] Bushell, P. J., On solutions in a cone of ordinary differential equations in an ordered Banach space (to appear) · Zbl 0317.34045
[13] Hilbert, D., Über die gerade Linie als kürzeste Verbindung zweier Punkte. Math. Ann. 46, 91-96 (1895) · JFM 26.0540.02 · doi:10.1007/BF02096204
[14] Karlin, S., Mathematical Methods and Theory in Games, Programming and Economics, Vol. I. Reading, Mass.: Addison-Wesley 1959 · Zbl 0139.12704
[15] Klein, F., Ueber die sogenannte Nicht-Euklidische Geometrie. Math. Ann. 4, 573-625 (1871) · JFM 03.0231.02 · doi:10.1007/BF02100583
[16] Krasnoselskii, M. A., Positive Solutions of Operator Equations. Groningen, The Netherlands: Noordhoff 1964
[17] Ostrowski, A. M., Positive matrices and functional analysis. Recent Advances in Matrix Theory. Madison: University of Wisconsin Press, 1964 · Zbl 0135.01504
[18] Samuelson, H., On the Perron-Frobenius Theorem. Mich. Math. J. 4, 57-59 (1957) · Zbl 0077.02303 · doi:10.1307/mmj/1028990177
[19] Thompson, A. C., On certain contraction mappings in a partially ordered vector space. Proc. Amer. Math. Soc. 14, 438-443 (1963) · Zbl 0147.34903
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