Fixed points and coincidences of families of mappings between
Izvestiya. Mathematics , Tome 83 (2019) no. 1, pp. 151-172.

Voir la notice de l'article provenant de la source Math-Net.Ru

We present theorems on common fixed points and coincidences for families of multi-valued mappings of ordered sets. This generalizes some recent results of the author and Podoprikhin as well as the classical theorems of Knaster–Tarski, Smithson and Zermelo. We also consider connections with the Caristi fixed-point theorem and other metrical results.
Keywords: ordered set, multi-valued mapping, fixed point, coincidence point.
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T. N. Fomenko. Fixed points and coincidences of families of mappings between. Izvestiya. Mathematics , Tome 83 (2019) no. 1, pp. 151-172. http://geodesic.mathdoc.fr/item/IM2_2019_83_1_a6/

[1] T. N. Fomenko, “Approximation of coincidence points and common fixed points of a collection of mappings of metric spaces”, Math. Notes, 86:1 (2009), 107–120 | DOI | DOI | MR | Zbl

[2] T. N. Fomenko, “Cascade search of the coincidence set of collections of multivalued mappings”, Math. Notes, 86:2 (2009), 276–281 | DOI | DOI | MR | Zbl

[3] T. N. Fomenko, “Cascade search principle and its applications to the coincidence problems of $n$ one-valued or multi-valued mappings”, Topology Appl., 157:4 (2010), 760–773 | DOI | MR | Zbl

[4] T. N. Fomenko, “Browder functions and theorems on fixed points and coincidences”, Izv. Math., 79:5, 1087–1095 | DOI | DOI | MR | Zbl

[5] T. N. Fomenko, “Cascade search for preimages and coincidences: global and local versions”, Math. Notes, 93:1 (2013), 172–186 | DOI | DOI | MR | Zbl

[6] T. N. Fomenko, “Functionals strictly subjected to convergent series and search for singularities of mappings”, J. Fixed Point Theory Appl., 14:1 (2013), 21–40 | DOI | MR | Zbl

[7] S. R. Gainullova, T. N. Fomenko, “Functionals subordinate to converging series and some applications”, Math. Notes, 96:2 (2014), 294–297 | DOI | DOI | MR | Zbl

[8] T. N. Fomenko, “Functionals strictly subordinate to series and search for solutions of equations”, Dokl. Math., 88:3 (2013), 748–750 | DOI | DOI | MR | Zbl

[9] T. N. Fomenko, “Approximation theorems in metric spaces and functionals strictly subordinated to convergent series”, Topology Appl., 179 (2015), 81–90 | DOI | MR | Zbl

[10] S. Abian, A. B. Brown, “A theorem on partially ordered sets, with applications to fixed point theorems”, Canad. J. Math., 13 (1961), 78–82 | DOI | MR | Zbl

[11] Handbook of metric fixed point theory, eds. W. A. Kirk, B. Sims, Kluwer Acad. Publ., Dordrecht, 2001, xiv+703 pp. | DOI | MR | Zbl

[12] R. E. Smithson, “Fixed points of order preserving multifunctions”, Proc. Amer. Math. Soc., 28 (1971), 304–310 | DOI | MR | Zbl

[13] E. Zermelo, “Beweis, daß jede Menge wohlgeordnet werden kann”, Math. Ann., 59:4 (1904), 514–516 | DOI | MR | Zbl

[14] E. Zermelo, “Neuer Beweis für die Möglichkeit einer Wohlordnung”, Math. Ann., 65:1 (1907), 107–128 | DOI | MR | Zbl

[15] A. V. Arutyunov, E. S. Zhukovskiy, S. E. Zhukovskiy, “Coincidence points of set-valued mappings in partially ordered spaces”, Dokl. Math., 88:3 (2013), 727–729 | DOI | DOI | MR | Zbl

[16] A. V. Arutyunov, E. S. Zhukovskiy, S. E. Zhukovskiy, “Coincidence points principle for set-valued mappings in partially ordered spaces”, Topology Appl., 201 (2016), 330–343 | DOI | MR | Zbl

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

[18] D. A. Podoprikhin, T. N. Fomenko, “On coincidences of families of mappings on ordered sets”, Dokl. Math., 94:3 (2016), 620–622 | DOI | DOI | MR | Zbl

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

[20] A. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia Math. Appl., 54, Cambridge Univ. Press, Cambridge, 1995, xviii+802 pp. | DOI | MR | Zbl

[21] V. I. Arnol'd, Mathematical methods of classical mechanics, Grad. Texts in Math., 60, 2nd ed., Springer-Verlag, New York, 1989, xvi+508 pp. | DOI | MR | MR | Zbl | Zbl

[22] J. Palis, Jr., W. de Melo, Geometric theory of dynamical systems. An introduction, Transl. from the Portuguese, Springer-Verlag, New York–Berlin, 1982, xii+198 pp. | MR | MR | Zbl

[23] K. Schmidt, Dynamical systems of algebraic origin, Progr. Math., 128, Birkhäuser Verlag, Basel, 1995, xviii+310 pp. | MR | Zbl

[24] T. N. Fomenko, K. S. Yastrebov, “Convergence of Noor-type iteration scheme with errors in a convex cone metric space”, Moscow Univ. Math. Bull., 71:1 (2016), 39–42 | DOI | MR | Zbl

[25] S. Itoh, W. Takahashi, “The common fixed point theory of singlevalued mappings and multivalued mappings”, Pacific J. Math., 79:2 (1978), 493–508 | DOI | MR | Zbl

[26] M. Berzig, “Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications”, J. Fixed Point Theory Appl., 12:1-2 (2012), 221–238 | DOI | MR | Zbl

[27] P. D. Proinov, I. A. Nikolova, “Approximation of point of coincidence and common fixed points of quasi-contraction mappings using the Jungck iteration scheme”, Appl. Math. Comput., 264 (2015), 359–365 | DOI | MR | Zbl

[28] V. Berinde, “Common fixed points of noncommuting almost contractions in cone metric spaces”, Math. Commun., 15:1 (2010), 229–241 | MR | Zbl

[29] M. Abbas, V. Rakočević, A. Iqbal, “Coincidence and common fixed points of Perov type generalized Ćirić-contraction mappings”, Mediterr. J. Math., 13:5 (2016), 3537–3555 | DOI | MR | Zbl

[30] R. E. Smithson, “A common fixed point theorem for nested spaces”, Pacific J. Math., 82:2 (1979), 533–537 | DOI | MR | Zbl

[31] D. Turkoglu, “Some common fixed point theorems for weakly compatible mappings in uniform spaces”, Acta Math. Hungar., 128:1-2 (2010), 165–174 | DOI | MR | Zbl

[32] M. L. Diviccaro, “Commutative multifunctions in posets with minimum common fixed point”, Int. Math. Forum, 5:13-16 (2010), 755–759 | DOI | MR | Zbl

[33] T. Nazir, S. Silvestrov, “Common fixed point results for family of generalized multivalued $F$-contraction mappings in ordered metric spaces”, Engineering mathematics II, Springer Proc. Math. Stat., 179, Springer, Cham, 2016, 419–432 ; arXiv: 1606.05299 | DOI | MR | Zbl

[34] P. Cousot, R. Cousot, “Constructive versions of Tarski's fixed point theorems”, Pacific J. Math., 82:1 (1979), 43–57 | DOI | MR | Zbl

[35] J. S. W. Wong, “Common fixed points of commuting monotone mappings”, Canad. J. Math., 19 (1967), 617–620 | DOI | MR | Zbl

[36] R. DeMarr, “Common fixed points for commuting contraction mappings”, Pacific J. Math., 13:4 (1963), 1139–1141 | DOI | MR | Zbl

[37] E. Karapinar, U. Yüksel, “Some common fixed point theorems in partial metric spaces”, J. Appl. Math., 2011 (2011), 263621, 16 pp. | DOI | MR | Zbl

[38] G. F. Jungck, “Common fixed point theorems for compatible self-maps of Hausdorff topological spaces”, Fixed Point Theory Appl., 2005:3 (2005), 355–363 | DOI | MR | Zbl

[39] L. Gajić, V. Rakočević, “Quasicontraction nonself-mappings on convex metric spaces and common fixed point theorems”, Fixed Point Theory Appl., 2005:3 (2005), 365–375 | DOI | MR | Zbl

[40] V. V. Popova, Obschie nepodvizhnye tochki semeistva monotonnykh operatorov, Avtoreferat diss. ... kand. fiz.-matem. nauk, Gos. ped. un-t im. A. I. Gertsena, SPb., 1995, 12 pp.

[41] V. V. Popova, “Obschie nepodvizhnye tochki monotonnykh operatorov, imeyuschikh minorantu”, Dep. v VINITI, No 668-84, RZhMat, 1984, 5B906, 6 pp.

[42] V. V. Popova, “Obschie nepodvizhnye tochki operatorov vpolne medlennogo rosta”, Issledovaniya po teorii priblizhenii, Sb. nauch. tr., UrGU, Sverdlovsk, 1988, 86–55

[43] V. V. Popova, “O suschestvovanii obschikh nepodvizhnykh tochek monotonnykh operatorov”, Dep. v VINITI, No I6600-83, RZhMat, 1983, 12B1216, 14 pp.

[44] I.A. Bakhtin, V. V. Popova, “Suschestvovanie obschikh nepodvizhnykh tochek monotonnykh poryadkovo nerastyagivayuschikh ne kommutiruyuschikh operatorov”, Dep. v VINITI, No 2297-V 92, RZhMat, 1992, IBI69, 16 pp.

[45] I. A. Bakhtin, V. V. Popova, “Suschestvovanie obschikh nepodvizhnykh tochek monotonnykh ne kommutiruyuschikh operatorov”, Dep. v VINITI, No 2299-V 92, RZhMat, 1992, IBI79, 27 pp.

[46] M. G. Kreĭn, M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Translation, 26, Amer. Math. Soc., New York, 1950, 128 pp. | MR | MR | Zbl

[47] M. A. Krasnosel'skiĭ, Positive solutions of operator equations, P. Noordhoff Ltd., Groningen, 1964, 381 pp. | MR | MR | Zbl | Zbl

[48] A. Yu. Volovikov, “Coincidence points of maps of $\mathbb Z_p^n$-spaces”, Izv. Math., 69:5 (2005), 913–962 | DOI | DOI | MR | Zbl

[49] R. N. Karasev, A. Yu. Volovikov, “Knaster's problem for almost $(Z_{p})^{k}$-orbits”, Topology Appl., 157:5 (2010), 941–945 | DOI | MR | Zbl

[50] E. Bishop, R. R. Phelps, “The support functionals of a convex set”, Convexity, Proc. Sympos. Pure Math., VII, Amer. Math. Soc., Providence, R.I., 1963, 27–35 | DOI | MR | Zbl

[51] R. DeMarr, “Partially ordered spaces and metric spaces”, Amer. Math. Monthly, 72:6 (1965), 628–631 | DOI | MR | Zbl

[52] I. Ekeland, “On the variational principle”, J. Math. Anal. Appl., 47:2 (1974), 324–353 | DOI | MR | Zbl

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

[54] J. Caristi, “Fixed point theorems for mappings satisfying inwardness conditions”, Trans. Amer. Math. Soc., 215 (1976), 241–251 | DOI | MR | Zbl

[55] T. N. Fomenko, “Fixed-point and coincidence theorems in ordered sets”, Dokl. Math., 95:3 (2017), 264–266 | DOI | DOI | MR | Zbl

[56] T. N. Fomenko, “Brondsted order in a metric space and generalizations of Caristi theorem”, Moscow Univ. Math. Bull., 72:5 (2017), 199–202 | DOI | MR | Zbl

[57] S. B. Nadler, Jr., “Multi-valued contraction mappings”, Pacific J. Math., 30:2 (1969), 475–488 | DOI | MR | Zbl

[58] T. N. Fomenko, “Sokhranenie suschestvovaniya tochki sovpadeniya pri nekotorykh diskretnykh preobrazovaniyakh pary otobrazhenii metricheskikh prostranstv”, Tr. IMM UrO RAN, 23, no. 4, 2017, 292–300 | DOI | MR

[59] J. R. Jachymski, “Some consequences of fundamental ordering principles in metric fixed point theory”, Ann. Univ. Mariae Curie-Skłodowska Sect. A, 51:2 (1997), 123–134 | MR | Zbl

[60] J. R. Jachymski, “Caristi's fixed point theorem and selections of set-valued contractions”, J. Math. Anal. Appl., 227:1 (1998), 55–67 | DOI | MR | Zbl

[61] W. A. Kirk, “Caristi's fixed point theorem and metric convexity”, Colloq. Math., 36:1 (1976), 81–86 | DOI | MR | Zbl

[62] A. V. Arutyunov, “Covering mappings in metric spaces and fixed points”, Dokl. Math., 76:2 (2007), 665–668 | DOI | MR | Zbl

[63] B. D. Gelman, V. K. Musienko, “O teoreme A. V. Arutyunova”, Aktualnye problemy matematiki i informatiki. Trudy matematicheskogo fakulteta, 2010, no. 2, 81–91