Fixed point free maps of a closed ball with small measures of noncompactness

• Autores: Martin Väth
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 52, Fasc. 2, 2001, págs. 101-116
• Idioma: inglés
• Títulos paralelos:
  • Mapas libres de puntos fijos de una bola cerrada con pequeñas medidas de no compacidad
• Enlaces
  • Texto completo
• Resumen

  • We show that in all infinite-dimensional normed spaces it is possible to construct a fixed point free continuous map of the unit ball whose measure of noncompactness is bounded by 2. Moreover, for a large class of spaces (containing separable spaces, Hilbert spaces and l-infinity (S)) even the best possible bound 1 is attained for certain measures of noncompactness.

• Referencias bibliográficas
  • R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskiĭ, Measures of noncompactness and condensing operators, Birkhäuser...
  • D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Ann. of Math. (2) 88 (1968), 35–46.
  • J. Appell, E. Giorgieri, and M. Väth, On a class of maps related to the Furi-Martelli-Vignoli spectrum, Ann. Mat. Pura Appl., to appear.
  • J. M. Ayerbe Toledano, T. Domínguez Benavides, and G. López Acedo, Measures of noncompactness in metric fixed point theory, Birkhäuser Verlag,...
  • Y. Benyamini and Y. Sternfeld, Spheres in infinite-dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. Soc. 83(3) (1983),...
  • G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84–92.
  • M. M. Day, Normed linear spaces, 3rd ed., Springer-Verlag, Berlin, Heidelberg, New York, 1973.
  • K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, Heidelberg, 1985.
  • J. Diestel, Geometry of Banach spaces-selected topics, Lecture Notes in Mathematics, 485, Springer-Verlag, Berlin, Heidelberg, New York, 1975.
  • J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353–367.
  • W. Feng, A new spectral theory for nonlinear operators and its applications, Abstr. Appl. Anal. 2 (1997), 163–183.
  • C. Franchetti, Lipschitz maps on the unit ball of normed spaces, Confer. Sem. Mat. Univ. Bari 202 (1985), 1–11.
  • C. Franchetti, Lipschitz maps and the geometry of the unit ball in normed spaces, Arch. Math. (Basel) 46(1) (1986), 76–84.
  • M. Furi and A. Martelli, On the minimal displacement of points under $\alpha$-Lipschitz maps in normed spaces, Boll. Un. Mat. Ital. (4) 9...
  • K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990.
  • P. Habala, P. Hájek, and V. Zizler, Introduction to Banach spaces II, Matfyzpress, Karlovy, 1996.
  • T. J. Jech, The Axiom of Choice, North-Holland Publ. Company, Amsterdam, 1973.
  • S. Kakutani, Topological properties of the unit sphere of a Hilbert space, Proc. Imp. Acad. Tokyo 19 (1943), 269–271.
  • V. L. Klee, Jr., Some topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955), 30–45.
  • P. K. Lin and Y. Sternfeld, Convex sets with the Lipschitz fixed point property are compact, Proc. Amer. Math. Soc. 93(4) (1985), 633–639.
  • S. Reich, The fixed point property for nonexpansive mappings I, Amer. Math. Monthly 83 (1976), 266–268.
  • S. Reich, A minimal displacement problem, Comment. Math. Univ. St. Pauli 26(2) (1977/78), 131–135.
  • S. Reich, The fixed point property for nonexpansive mappings II, Amer. Math. Monthly 87 (1980), 292–294.
  • B. N. Sadovskiĭ, Limit-compact and condensing operators (Russian), Uspehi Mat. Nauk 27(1) (1972), 81–146.
  • E. U. Tarafdar and H. B. Thompson, On the solvability of nonlinear, noncompact operator equations, J. Austral. Math. Soc. Ser. A 43(1) (1987),...