Abstract
In previous papers of the author, the cascade search principle was proposed, which makes it possible to construct a set-valued self-map of a metric spaceX from a set-valued functional or a collection of set-valued maps of X so that the new map generates a multicascade, i.e., a set-valued discrete dynamical system whose limit set coincides with the zero set of the given functional, with the coincidence set of the given collection, or with the common preimage of a closed subspace under the maps from this collection. Stability issues of cascade search were studied. This paper is devoted to a generalization and local modifications of the cascade search principle and their applications to problems concerning local search and approximation of common preimages of subspaces and coincidence sets for finite collections of set-valued maps of metric spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T. N. Fomenko, “Approximation of coincidence points and common fixed points of a collection of mappings of metric spaces,” Mat. Zametki 86(1), 110–125 (2009) [Math. Notes 86 (1), 107–120 (2009)].
T. N. Fomenko, “Cascade search of the coincidence set of collections of multivalued mappings,” Mat. Zametki 86(2), 304–309 (2009) [Math. Notes 86 (2), 276–281 (2009)].
T. N. Fomenko, “Cascade search principle and its applications to the coincidence problem of n one-valued or multi-valued mappings,” Topology Appl. 157(4), 760–773 (2010).
T. N. Fomenko, “On approximation of coincidence points for a finite set of maps of metric spaces,” in Abstracts of the Fifth International Conference of Differential and Functional Differential Equations (DFDE-2008), Moscow, Russia, August 17–24, 2008 (Moscow, 2008), p. 119 [in Russian].
T. N. Fomenko, “The cascade search principle and coincidence of N mappings,” in Proceedings of the International Conference “Modern Problems of Mathematics, Mechanics, and Their Applications” Dedicated to the 70th Birthday of V. A. Sadovnichii, Moscow, Russian, March 30–April 2, 2009 (Moskov. Univ., Moscow, 2009), p. 99 [in Russian].
A. V. Arutyunov, “Covering maps of metric spaces and fixed points,” Dokl. Ross. Akad. Nauk 416(2), 151–155 (2007) [Russian Acad. Sci. Dokl. Math. 76 (2), 665–668 (2007)].
A. V. Arutyunov, “Stability of coincidence points and properties of covering mappings,” Mat. Zametki 86(2), 163–169 (2009) [Math. Notes 86 (2), 153–158 (2009)].
T. N. Fomenko, “The stability of Cascade Search Principle,” in Abstracts of the 2010 International Conference on Topology and its Applications, Nafpaktos, Greece, June 26–30, 2010 (Nafpaktos, 2010), p. 99.
T. N. Fomenko, “Stability of cascade search,” Izv. Ross. Akad. Nauk Ser. Mat. 74(5), 171–190 (2010) [Russian Acad. Sci. Izv. Math. 74 (5), 1051–1068 (2010)].
T. N. Fomenko, “Cascade search: Stability of attainable limit points,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 5, 3–9 (2010) [Moscow Univ. Math. Bull. 65 (5), 179–185 (2010)].
T. N. Fomenko, “New developments in the cascade search theory,” in Abstracts of the 8-th International ISAAC Congress, Moscow, Russia, August 22–27, 2010 (Moscow, 2011), p. 369.
A. Arutyunov, E. Avakov, B. Gel’man, A. Dmitruk, and V. Obukhovskii, “Locally covering maps in metric spaces and coincidence points,” J. Fixed Point Theory Appl., No. 1, 105–127 (2009).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © T. N. Fomenko, 2013, published in Matematicheskie Zametki, 2013, Vol. 93, No. 1, pp. 127–143.
Rights and permissions
About this article
Cite this article
Fomenko, T.N. Cascade search for preimages and coincidences: Global and local versions. Math Notes 93, 172–186 (2013). https://doi.org/10.1134/S0001434613010173
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434613010173