Abstract
A famous open question in metric Fixed Point Theory is whether every Banach space which is isomorphic to the Hilbert space ℓ2 has the fixed point property for nonexpansive mappings. We give an overview about the state of the advances towards their solution.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alspach D.E.: A fixed point free nonexpansive map. Proc. Am. Math. Soc. 82(3), 423–424, (1981)
Amir D., Franchetti C.: The radius ratio and convexity properties in normed linear spaces. Trans. Am. Math. Soc. 282, 275–291 (1984)
Ayerbe Toledano, J.; Domínguez Benavides, T.; López Acedo, G.: Measures of Noncompactness in Metric Fixed Point Theory, vol. 99. Birkhäuser, Op. Th., Basel (1997)
Baillon J.B., Schöneberg R.: Asymptotic normal structure and fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 81, 257–264 (1981)
Belluce L.P., Kirk W.A., Steiner E.F.: Normal structure in Banach spaces. Pac. J. Math. 26, 433–440 (1968)
Brown D.R.: P-convexity and B-convexity in Banach spaces. Trans. Am. Math. Soc. 187, 77–81 (1974)
Büber, T.; Kirk, W.A.: Constructive aspects of Fixed Point Theory for nonexpansive mappings. World Congress of Nonlinear Analysts ’92, vols. I–IV (Tampa, FL, 1992), pp. 2115–2125. de Gruyer, Berlin
Bynum W.L.: Normal structure coefficients for normal structure for Banach spaces. Pac. J. Math. 86, 427–436 (1980)
Domínguez Benavides T.: Stability of the fixed point property for nonexpansive mappings. Houston J. Math. 22, 835–849 (1996)
Domínguez Benavides T., Japón Pineda M.A.: Stability of the fixed point property for nonexpansive mappings in some classes of spaces. Comm. Appl. Nonlinear Anal. 5, 37–46 (1998)
Dowling P.N., Lennard C.J.: Every nonreflexive subspace of L1[0,1] fails the fixed point property. Proc. Am. Math. Soc. 125, 443–446 (1997)
Dowling P.N., Randrianantoanina B, Turett B.: The fixed point property via dual space properties. J. Funct. Anal. 255, 768–775 (2008)
Fetter, H.; Gamboa de Buen, B.: (r, k, l)—somewhat uniformly noncreasy Banach spaces. International Conference on Fixed Point Theory and Applications, pp. 71–80. Yokohama Publications, Yokohama (2004)
Fetter, H.; Gamboa de Buen, B.: Properties WORTH and WORTH*, (1 + δ) embeddings in Banach spaces with 1-unconditional basis and wFPP. Fixed Point Theory Appl. Art. ID 342691 (2010)
Fetter H., Gamboade Buen B., García Falset J.: Banach spaces which are somewhat uniformly noncreasy. J. Math. Anal. Appl. 285, 444–455 (2003)
Fetter Nathansky, H.; Llorens-Fuster, E.: A product space with the fixed point property. Fixed Point Theory Appl. 2012, 91 (2012)
García-Falset, J.: Banach spaces satisfying the fixed point property for nonexpansive mappings. Ph. D. Thesis, Univ. of Valencia, Spain (1990)
García-Falset J.: Fixed point property in Banach spaces whose characteristic of convexity is less than 2. J. Austral. Math. Soc. Ser A. 54, 169–173 (1993)
García-Falset J.: Stability and fixed points for nonexpansive mappings. Houston J. Math. 20, 495–505 (1994)
García-Falset J.: The fixed point property in Banach spaces with NUS property, J. Math. Anal. Appl. 215, 532–542 (1997)
García-Falset J., Llorens Fuster E.: A geometric property of banach spaces related to the fixed point property. J. Math. Anal. Appl. 172, 39–52 (1993)
García-Falset, J.; Jiménez-Melado, A.; Llorens-Fuster, E.: Stability of the fixed point property for nonexpansive mappings. In: Kirk, W.A.; Sims, B. (eds.) Handbook of Metric Fixed Point Theory, pp. 231–238. Kluwer Academic Publications (2001)
García-Falset J., Llorens Fuster E., Mazcuñán Navarro E.: Banach spaces which are r-uniformly noncreasy. Nonlinear Anal. 53, 957–975 (2003)
García-Falset J., Llorens Fuster E., Mazcuñán Navarro E.: Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings. J. Funct. Anal. 233, 494–514 (2006)
Giles J.R., Sims B., Swaminathan S.: A geometrically aberrant Banach space with normal structure. Bull. Austal. Math. Soc. 31, 75–81 (1985)
Goebel K.: Convexivity of balls and fixed-point theorems for mappings with nonexpansive square. Compositio Math. 22, 269–274 (1970)
Goebel, K.; Kirk W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press (1990)
Gossez J.-P., Lami Dozo E.: Some geometric properties related to the fixed point theory for nonexpansive mappings. Pac. J. Math. 40, 565–573 (1972)
Jiménez-Melado, A.: Una propiedad geométrica de los espacios de Banach relacionada con la Teorí a del Punto Fijo. Ph.D. dissertation, Univ. de Málaga (Spain) (1988)
Jiménez-Melado A.: Stability of weak normal structure in James quasi reflexive space. Bull. Austral. Math. Soc. 46(3), 367–372 (1992)
Jiménez-Melado A., Llorens-Fuster E.: A sufficient condition for the fixed point property. Nonlinear Anal. 20, 849–853 (1993)
Kato M., Tamura T.: Weak nearly uniform smoothness and worth property of ψ-direct sums of Banach spaces \({X \bigoplus _{\psi} Y}\), Comment. Math. Prace. Mat. 46, 113–129 (2006)
Jiménez-Melado, A.; Llorens-Fuster, E.: A renorming of ℓ2, rare but with the fixed-point property. Int. J. Math. Math. Sci. 2003(65), 4115–4129 (2003)
Jiménez-Melado, A.; Llorens-Fuster, E.: A class of renormings of ℓ2 with the fixed point property. J. Nonlinear Convex Anal. (in press)
Jiménez-Melado A, Llorens-Fuster E., Saejung S.: The von Newman-Jordan constant, weak orthogonality and normal structure in Banach spaces. Proc. Am. Math. Soc. 134, 355–364 (2006)
Karlovitz L.A.: Existence of a fixed point for a nonexpansive map in a space without normal structure, Pac. J. Math. 66, 153–159 (1976)
Khamsi, M.A.: étude de la propriété du point fixe dans les espaces de Banach et les espaces métriques, Thèse de Doctorat de l’Université Paris VI (1987)
Khamsi M.A., Kirk W.A.: An Introduction To Metric Spaces and Fixed Point Theory Pure and Applied Mathematics. Wiley and Sons, N.Y (2001)
Kirk W.A.: A fixed point theorem for mappings which do not increase distances. Am. Math. Monthly 72, 1004–1006 (1965)
Kirk, W. A.: Some questions in metric fixed point theory. Recent Advances on Metric Fixed Point Theory (Seville, 1995), vol. 48, pp. 73–97, Ciencias, Univ. Sevilla, Seville (1996)
Kutzarova D., Prus S., Sims B.: Remarks on orthogonal convexity of Banach spaces. Houston J. Math. 19, 603–614 (1993)
Kirk. W.A.; Sims, B. (eds.) Handbook of Metric Fixed Point Theory. Kluwer Academis Publishers, Dordrecht (2001)
Lifshitz, E. A.: Fixed point theorems for operators in strongly convex spaces. Voronez Gos. Univ. Trudy Mat. Fak. 16, 23–28 (1975) (in Russian)
Lim T.C.: Asymptotic centers and nonexpansive mappings in conjugate Banach spaces. Pac. J. Math. 90, 135–143 (1980)
Lin P.K.: Unconditional bases and fixed points of nonexpansive mappings. Pac. J. Math. 116, 69–76 (1985)
Llorens-Fuster E.: Semigroups of mappings with rigid Lipschitz constant. Proc. Am. Math. Soc. 130, 1407–1412 (2002)
Llorens-Fuster, E.; Muñiz-Perez, O.: Some relationships between sufficient conditions for the fixed point property (Submitted)
Naidu S.V.R., Sastry K.P.R.: Convexity conditions in normed linear spaces. J. Reine Angew. Math. 297, 35–53 (1976)
Nelson, J.L.; Singh, K. L.; Whitfield, J.H.M.: Normal structures and nonexpansive mappings in Banach spaces. Nonlinear analysis, pp. 433–492. World Science Publishing, Singapore (1987)
Mazcuñ án Navarro, E.: Geometry of Banach Spaces in Metric Fixed Point Theory. Ph. D. Thesis, Univ. of Valencia, Spain (2003)
Mazcuñán Navarro E.: Stability of the fixed point property in Hilbert spaces. Proc. Am. Math. Soc. 134, 129–138 (2006)
Prus S.: Banach spaces which are uniformly noncreasy. Nonlinear Anal. 30, 2317–2324 (1997)
Prus S., Szczepanik M.: Nearly uniformly noncreasy Banach spaces. J. Math. Anal. Appl. 307, 255–273 (2005)
Saejung S.: Convexity conditions and normal structure of Banach spaces. J. Math. Anal. Appl. 344, 851–856 (2008)
Sims B.: Orthogonality and fixed points of nonexpansive maps. Proc. Center Austral. Math. Nat. Univ. 20, 179–186 (1988)
Sims B.: A class of spaces with weak normal structure. Bull. Austral. Math. Soc. 49, 523–528 (1994)
Sims, B.; Smyth, M.: On non-uniform conditions giving weak normal structure. First International Conference in Abstract Algebra (Kruger Park, 1993). Quaestiones Math. 18(1-3), 9–19 (1995)
Sims B., Smyth M.A.: On some Banach space properties sufficient for weak normal structure and their permanence properties. Trans. Am. Math. Soc 351, 497–513 (1999)
Smith M.A., Turret B.: A reflexive LUR Banach space which laks normal structure. Can. Math. Bull. 28, 492–494 (1985)
Sullivan F.: A generalization of uniformly rotund Banach spaces. Can. J. Math. 31, 628–636 (1979)
Tan K.-K., Xu H.K.: On fixed point theorems of nonexpansive mappings in product spaces. Proc. Am. Math. Soc. 113(4), 983–989 (1991)
Smyth, M.A.: Aspects of the Fixed Point Theory for some Metrically Defined Maps. Ph. D. dissertation, University of Newcastle, Australia (1994)
Turett, B.: A dual view of a theorem of Baillon. Nonlinear Analysis and Applications (St. Johns, Nfld., 1981). Lecture Notes Pure and Applied Mathematics, vol. 80, pp. 279–286. Dekker, New York (1982)
van Dulst, D.: Some geometric properties related to normal structure. Nonlinear Analysis and Applications (St Johns, Nfld., 1981). Lecture Notes in Pure and Applied Mathematics, vol. 80, pp. 155–162. Dekker, New York (1982)
van Dulst, D.; Sims, B.: Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK). Banach Space Theory and its Applications (Bucharest, 1981). Lecture Notes in Mathematics, vol. 991. Springer, Berlin, pp. 35–43 (1983)
Wiśnicki A.: Towards the fixed point property for superreflexive spaces, Bull. Austral. Math. Soc. 64, 435–444 (2001)
Zidler V.: On some rotundity and smoothness properties of Banach spaces. Dissertationes Math. Rozprawy Mat. 87, 33 (1971)
Acknowledgments
Research partially supported by a Grant from Ministerio de Educación y Cultura (Spain) MTM2009-10696-C02-02.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.