Geodesic
Preservation of the existence of coincidence points under some discrete transformations of a pair of mappings of metric spaces
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 292-300 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In topology there are known results on the preservation under homotopy of the fixed point property of self-mappings in some spaces if the Lefschetz number of the initial mapping is nonzero. For the class of contracting mappings of metric spaces and for some of their generalizations, there are M. Frigon's known results on the preservation of the contraction property and hence of the fixed point property under homotopies of some special type. In 1984 J. W. Walker introduced a discrete counterpart of homotopy for mappings in an ordered set, which he called an order isotone homotopy. R. E. Stong showed the naturalness of this notion and its relation to the usual continuous homotopy. Recently, the author and D. A. Podoprikhin have generalized Walker's notion of order isotone homotopy and suggested sufficient conditions for the preservation under such discrete homotopy (a pair of homotopies) of the property of a mapping (a pair of mappings) of ordered sets to have a fixed point (a coincidence point). This paper contains metric counterparts of the obtained results and some corollaries. The method of ordering a metric space proposed by A. Brøndsted in 1974 is used.
Keywords: fixed point, coincidence point, Brøndsted's order, order homotopy, discrete counterpart of homotopy. 
@article{TIMM_2017_23_4_a27,
     author = {T. N. Fomenko},
     title = {Preservation of the existence of coincidence points under some discrete transformations of a pair of mappings of metric spaces},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {292--300},
     year = {2017},
     volume = {23},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2017_23_4_a27/}
}
TY  - JOUR
AU  - T. N. Fomenko
TI  - Preservation of the existence of coincidence points under some discrete transformations of a pair of mappings of metric spaces
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2017
SP  - 292
EP  - 300
VL  - 23
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TIMM_2017_23_4_a27/
LA  - ru
ID  - TIMM_2017_23_4_a27
ER  - 
%0 Journal Article
%A T. N. Fomenko
%T Preservation of the existence of coincidence points under some discrete transformations of a pair of mappings of metric spaces
%J Trudy Instituta matematiki i mehaniki
%D 2017
%P 292-300
%V 23
%N 4
%U http://geodesic.mathdoc.fr/item/TIMM_2017_23_4_a27/
%G ru
%F TIMM_2017_23_4_a27
T. N. Fomenko. Preservation of the existence of coincidence points under some discrete transformations of a pair of mappings of metric spaces. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 292-300. http://geodesic.mathdoc.fr/item/TIMM_2017_23_4_a27/

[1] “On continuation methods for contractive and nonexpansive mappings”, Recent Advances on Metric Fixed Point Theory, Universidad de Sevilla, Sevilla, 1996, 19–29

[2] Podoprikhin D.A., Fomenko T.N., “Sokhranenie svoistva nepodvizhnoi tochki i svoistva sovpadeniya pri gomotopii otobrazhenii uporyadochennykh mnozhestv”, Dokl. AN, 477:4 (2017), 402–405 | DOI

[3] Brondsted A., “On a lemma of Bishop and Phelps”, Pacific J. Math., 55:2 (1974), 335–341 | DOI | MR | Zbl

[4] Walker J.W., “Isotone relations and the fixed point property for posets”, Discr. Math., 48:2–3 (1984), 275–288 | DOI | MR | Zbl

[5] Stong R.E., “Finite topological spaces”, Trans. Amer. Math. Soc., 123:2 (1966), 325–340 | DOI | MR | Zbl

[6] Handbook of metric fixed point theory, eds. eds. W.A. Kirk, B. Sims, Springer Science Business Media, NY, 2001, 704 pp. | DOI | MR

[7] Podoprikhin D.A., Fomenko T.N., “O sovpadeniyakh semeistv otobrazhenii uporyadochennykh mnozhestv”, Dokl. RAN. Matematika, 471:1 (2016), 16–18 | DOI | MR | Zbl

[8] Fomenko T.N., Podoprikhin D., “Common fixed points and coincidences of mapping families on partially ordered set”, Topology Appl., 221 (2017), 275–285 | DOI | MR | Zbl

[9] Arutyunov A.V., Zhukovskii E.S., Zhukovskii S.E., “O tochkakh sovpadeniya otobrazhenii v chastichno uporyadochennykh prostranstvakh”, Dokl. RAN, 88:3 (2013), 710–713 | MR | Zbl

[10] Arutyunov A.V., Zhukovskiy E.S., and Zhukovskiy S.E., “Coincidence points principle for mappings in partially ordered spaces”, Topology Appl., 179 (2015), 13–33 | DOI | MR | Zbl

[11] Fomenko T.N., Podoprikhin D.A., “Fixed points and coincidences of mappings of partially ordered sets”, J. Fixed Point Theory Appl., 18:4 (2016), 823–842 | DOI | MR | Zbl

[12] Podoprikhin D.A., “Fixed points of mappings on ordered sets”, Lobachevskii J. Math., 38:6 (2017), 1069–1074 | DOI