The ε-approximated complete invariance property

• García, Gonzalo ^[1]
  1. [1] Universidad Nacional de Educación a Distancia

     Universidad Nacional de Educación a Distancia

     Madrid, España

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 23, Nº. 2, 2022, págs. 453-462
• Idioma: inglés
• DOI: 10.4995/agt.2022.16641
• Enlaces
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• Resumen

  • In the present paper we introduce a generalization of the complete invariance property (CIP) for metric spaces, which we will call the ε-approximated complete invariance property (ε-ACIP). For our goals, we will use the so called degree of nondensifiability (DND) which, roughly speaking, measures (in the specified sense) the distance from a bounded metric space to its class of Peano continua. Our main result relates the ε-ACIP with the DND and, in particular, proves that a densifiable metric space has the ε-ACIP for each ε>0. Also, some essentials differences between the CIP and the ε-ACIP are shown.

• Referencias bibliográficas
  • Y. Cherruault and G. Mora, Optimisation Globale. Théorie des Courbes α-denses, Económica, Paris, 2005.
  • R. Dubey and A. Vyas, Wavelets and the complete invariance property, Mat. Vesnik, 62 (2010), 183-188.
  • G. García and G. Mora, A fixed point result in Banach algebras based on the degree of nondensifiability and applications to quadratic integral...
  • G. García and G. Mora, The degree of convex nondensifiability in Banach spaces, J. Convex Anal. 22 (2015), 871-888.
  • K. H. Heinrich and J. R. Martin, G-spaces and fixed point sets, Geom. Dedicata 83 (2000), 39-61. https://doi.org/10.1023/A:1005246831488
  • J. R. Martin, Fixed point sets of metric and nonmetric spaces, Trans. Amer. Math. Soc. 284 (1984), 337-353. https://doi.org/10.1090/S0002-9947-1984-0742428-1
  • J. R. Martin, Fixed point sets of LC∞,C∞ continua, Proc. Amer. Math. Soc. 81 (1981), 325-328. https://doi.org/10.1090/S0002-9939-1981-0593482-4
  • J. R. Martin, Fixed point sets of Peano continua, Pacific J. Math. 74 (1978), 163-166. https://doi.org/10.2140/pjm.1978.74.163
  • J. R. Martin and S. B. Nadler, Examples and questions in the theory of fixed point sets, Canad. J. Math. 31 (1979), 1017-1032. https://doi.org/10.4153/CJM-1979-094-5
  • J. R. Martin and E. D. Tymchatyn, Fixed point sets of 1-dimensional Peano Continua, Pacific J. Math. 89 (1980), 147-149. https://doi.org/10.2140/pjm.1980.89.147
  • D. Masood and P. Singh, Complete invariance property on hyperspaces, JP J. Geom. Topol. 17 (2015), 83-94. https://doi.org/10.17654/JPGTMay2015_083_094
  • D. Masood and P. Singh, On equivariant complete invariance property, Sci. Math. Jpn. 77 (2013), 1-6.
  • S. C. Maury, Hyperspaces and the S-equivariant complete invariance property, Kyungpook Math. J. 55 (2015), 219-224. https://doi.org/10.5666/KMJ.2015.55.1.219
  • G. Mora, The Peano curves as limit of α-dense curves, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 9 (2005), 23-28.
  • G. Mora and Y. Cherruault, Characterization and generation of α-dense curves, Computers Math. Applic. 33 (1997), 83-91. https://doi.org/10.1016/S0898-1221(97)00067-9
  • G. Mora and J. A. Mira, Alpha-dense curves in infinite dimensional spaces, Int. J. Pure Appl. Math. 5 (2003), 437-449.
  • G. Mora and D. A. Redtwitz, Densifiable metric spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 105 (2011), 71-83. https://doi.org/10.1007/s13398-011-0005-y
  • M. Rahal, R. Ziadi and A. Ellaia, Generating α-dense curves in non-convex sets to solve a class of non-smooth constrained global optimization,...
  • H. Sagan, Space-Filling Curves, Springer-Verlag, New York 1994. https://doi.org/10.1007/978-1-4612-0871-6
  • L. E. Ward, Fixed point sets, Pacific J. Math. 47 (1973), 553-565. https://doi.org/10.2140/pjm.1973.47.553
  • S. Willard, General Topology, Dover Pub. Inc., New York 1970.
  • D. X. Zhou, Complete invariance property with respect to homeomorphism over frame multiwavelet and super-wavelet spaces, Journal of Mathematics...