The Existence of Zeros of Multivalued Functionals, Coincidence Points, and Fixed Points in $f$-Quasimetric Spaces

T. N. Fomenko

Matematičeskie zametki, Tome 110 (2021) no. 4, pp. 598-609

Résumé

The notion of a $\lambda$-generalized-search multivalued functional on an $f$-quasimetric space is introduced. An existence theorem for zeros of such functionals is proved. As corollaries, theorems on coincidence and fixed points of multivalued mappings of $f$-quasimetric spaces are proved. In particular, Nadler's well-known theorem on fixed points of multivalued contraction mappings is generalized to the case of an $f$-quasimetric space. For a large class of single-valued mappings, including generalized contractions, a theorem on the existence of a (not necessarily unique) fixed point is proved. This theorem extends the existence part of E. S. Zhukovskii's recent fixed-point theorem for generalized contractions, which is a generalization to $f$-quasimetric spaces of Krasnosel'skii's well-known fixed-point theorem and Browder's fixed-point theorem (equivalent to Krasnosel'skii's theorem).

Détail

Mots-clés : $f$-quasimetric space
Keywords: $\lambda$-generalized-search functional, coincidence point, fixed point, generalized contraction.

Bibliographie

[1] A. V. Arutyunov, A. V. Greshnov, L. V. Lokoutsievskii, K. V. Storozhuk, “Topological and geometrical properties of spaces with symmetric and nonsymmetric $f$-quasimetrics”, Topology Appl., 221 (2017), 178–194 | DOI | MR

[2] E. S. Zhukovskii, “Nepodvizhnye tochki szhimayuschikh otobrazhenii $f$-kvazimetricheskikh prostranstv”, Sib. matem. zhurn., 59:6 (2018), 1338–1350 | DOI

[3] A. D. Pitcher, E. W. Chittenden, “On the foundations of the calcul fonctionnel of Fréchet”, Trans. Amer. Math. Soc., 19:1 (1918), 66–78 | MR

[4] I. A. Bakhtin, “Printsip szhatykh otobrazhenii v pochti metricheskikh prostranstvakh”, Funktsionalnyi analiz, 30, Ulyanov. gos. ped. in-t im. I. N. Ulyanova, Ulyanovsk, 1989, 26–37

[5] A. V. Arutyunov, A. V. Greshnov, “$(q_1,q_2)$-kvazimetricheskie prostranstva. Nakryvayuschie otobrazheniya i tochki sovpadeniya”, Izv. RAN. Ser. matem., 82:2 (2018), 3–32 | DOI | MR

[6] M. A. Krasnoselskii, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskii, V. Ya. Stetsenko, Priblizhennoe reshenie operatornykh uravnenii, Nauka, M., 1969 | MR | Zbl

[7] F. E. Browder, “On the convergence of successive approximations for nonlinear functional equations”, Nederl. Akad. Wetensch. Proc. Ser. A, 71 (1968), 27–35 | DOI | MR

[8] J. Jachymski, “Around Browder's fixed point theorem for contractions”, J. Fixed Point Theory Appl., 5:5 (2009), 47–61 | DOI | MR

[9] T. N. Fomenko, “O priblizhenii k tochkam sovpadeniya i obschim nepodvizhnym tochkam nabora otobrazhenii metricheskikh prostranstv”, Matem. zametki, 86:1 (2009), 110–125 | DOI | MR | Zbl

[10] T. N. Fomenko, “K zadache kaskadnogo poiska mnozhestva sovpadenii nabora mnogoznachnykh otobrazhenii”, Matem. zametki, 86:2 (2009), 304–309 | DOI | MR | Zbl

[11] 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 (2010), 760–773 | DOI | MR

[12] T. N. Fomenko, “Kaskadnyi poisk proobrazov i sovpadenii: globalnaya i lokalnaya versii”, Matem. zametki, 93:1 (2013), 127–143 | DOI | MR | Zbl

[13] T. N. Fomenko, “Poisk nulei funktsionalov, nepodvizhnye tochki i sovpadeniya otobrazhenii v kvazimetricheskikh prostranstvakh”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2019, no. 6, 14–22

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

[15] S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales”, Fund. Math., 3 (1922), 133–181 | DOI | MR