Abstract
The famous Banach contraction principle holds in complete metric spaces, but completeness is not a necessary condition: there are incomplete metric spaces on which every contraction has a fixed point. The aim of this paper is to present various circumstances in which fixed point results imply completeness. For metric spaces, this is the case of Ekeland’s variational principle and of its equivalent, Caristi’s fixed point theorem. Other fixed point results having this property will also be presented in metric spaces, in quasi-metric spaces, and in partial metric spaces. A discussion on topology and order and on fixed points in ordered structures and their completeness properties is included as well.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Abdeljawad, E. Karapınar, and K. Taş, “Existence and uniqueness of a common fixed point on partial metric spaces,” Appl. Math. Lett., 24, No. 11, 1900–1904 (2011).
S. Abramsky and A. Jung, “Domain theory,” in: Handbook of Logic in Computer Science, Vol. 3, Oxford Univ. Press, New York (1994), pp. 1–168.
Ö. Acar, I. Altun, and S. Romaguera, “Caristi’s type mappings on complete partial metric spaces,” Fixed Point Theory, 14, No. 1, 3–9 (2013).
A. Aghajani and A. Razani, “Some completeness theorems in the Menger probabilistic metric space,” Appl. Sci., 10, 1–8 (2008).
C. Alegre, J. Marín, and S. Romaguera, “A fixed point theorem for generalized contractions involving w-distances on complete quasi-metric spaces,” Fixed Point Theory Appl., 40 (2014).
M. A. Alghamdi, N. Shahzad, and O. Valero, “On fixed point theory in partial metric spaces,” Fixed Point Theory Appl., 175 (2012).
M. A. Alghamdi, N. Shahzad, and O. Valero, “New results on the Baire partial quasi-metric space, fixed point theory and asymptotic complexity analysis for recursive programs,” Fixed Point Theory Appl., 14 (2014).
M. Alimohammady, A. Esmaeli, and R. Saadati, “Completeness results in probabilistic metric spaces,” Chaos Solitons Fractals, 39, No. 2, 765–769 (2009).
T. Alsiary and A. Latif, “Generalized Caristi fixed point results in partial metric spaces,” J. Nonlinear Convex Anal., 16, No. 1, 119–125 (2015).
I. Altun and S. Romaguera, “Characterizations of partial metric completeness in terms of weakly contractive mappings having fixed point,” Appl. Anal. Discrete Math., 6, No. 2, 247–256 (2012).
P. Amato, “A method for reducing fixed-point problems to completeness problems and vice versa,” Boll. Un. Mat. Ital. B (6), 3, No. 2, 463–476 (1984).
P. Amato, “The completion classes of a metric space,” Rend. Circ. Mat. Palermo (2) Suppl., No. 12, 157–168 (1986).
P. Amato, “Some properties of completion classes for normed spaces,” Note Mat., 13, No. 1, 123–134 (1993).
A. Amini-Harandi, “Metric-like spaces, partial metric spaces and fixed points,” Fixed Point Theory Appl., 204 (2012).
V. G. Angelov, “Fixed point theorem in uniform spaces and applications,” Czech. Math. J., 37 (112), No. 1, 19–33 (1987).
V. G. Angelov, “A converse to a contraction mapping theorem in uniform spaces,” Nonlinear Anal., 12, No. 10, 989–996 (1988).
V. G. Angelov, “Corrigendum: ‘A converse to a contraction mapping theorem in uniform spaces’ [Nonlinear Anal. 12, No. 10, 989–996 (1988); MR0962764 (89k:54101)],” Nonlinear Anal., 23, No. 11, 1491 (1994).
V. G. Angelov, “An extension of Kirk–Caristi theorem to uniform spaces,” Antarct. J. Math., 1, No. 1, 47–51 (2004).
M.-C. Anisiu and V. Anisiu, “On the characterization of partial metric spaces and quasimetrics,” Fixed Point Theory, 17, No. 1 (2016).
Q. H. Ansari and L.-J. Lin, “Ekeland-type variational principles and equilibrium problems,” Topics in Nonconvex Optimization, Springer Optim. Appl., Vol. 50, Springer, New York (2011), pp. 147–174.
H. Aydi, E. Karapınar, and P. Kumam, “A note on modified proof of Caristi’s fixed point theorem on partial metric spaces,” J. Inequal. Appl., 355 (2013).
H. Aydi, E. Karapınar, and C. Vetro, “On Ekeland’s variational principle in partial metric spaces,” Appl. Math. Inform. Sci., 9, No. 1, 257–262 (2015).
A. C. Babu, “A converse to a generalized Banach contraction principle,” Publ. Inst. Math. (Beograd) (N.S.), 32 (46), 5–6 (1982).
S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,” Fund. Math., 3, 133–181 (1922).
T. Q. Bao, S. Cobzaş, and A. Soubeyran, Variational Principles and Completeness in Pseudo-Quasimetric Spaces, Preprint (2016).
T. Q. Bao, B. S. Mordukhovich, and A. Soubeyran, “Fixed points and variational principles with applications to capability theory of wellbeing via variational rationality,” Set-Valued Var. Anal., 23, No. 2, 375–398 (2015).
T. Q. Bao, B. S. Mordukhovich, and A. Soubeyran, “Variational analysis in psychological modeling,” J. Optim. Theory Appl., 164, No. 1, 290–315 (2015).
T. Q. Bao and A. Soubeyran, Variational Analysis and Applications to Group Dynamics, Preprint (2015), http://www.optimization-online.org.
T. Q. Bao and M. A. Théra, “On extended versions of Dancs–Hegedüs–Medvegyev’s fixed-point theorem,” Optimization (2015).
V. Berinde and M. Choban, “Remarks on some completeness conditions involved in several common fixed point theorems,” Creat. Math. Inform., 19, No. 1, 1–10 (2010).
V. Berinde and M. Choban, “Generalized distances and their associate metrics. Impact on fixed point theory,” Creat. Math. Inform., 22, No. 1, 23–32 (2013).
C. Bessaga, “On the converse of the Banach ‘fixed-point principle’,” Colloq. Math., 7, 41–43 (1959).
F. Blanqui, “A point on fixpoints in posets,” arXiv:1502.06021 [math.LO] (2014).
J. M. Borwein, “Completeness and the contraction principle,” Proc. Am. Math. Soc., 87, No. 2, 246–250 (1983).
N. Bourbaki, “Sur le théorème de Zorn,” Arch. Math. (Basel), 2, 434–437 (1951).
A. Branciari, “A fixed point theorem of Banach–Caccioppoli type on a class of generalized metric spaces,” Publ. Math. Debrecen, 57, No. 1-2, 31–37 (2000).
M. Bukatin, R. Kopperman, S. Matthews, and H. Pajoohesh, “Partial metric spaces,” Am. Math. Mon., 116, No. 8, 708–718 (2009).
C.-S. Chuang, L.-J. Lin, and W. Takahashi, “Fixed point theorems for single-valued and set-valued mappings on complete metric spaces,” J. Nonlinear Convex Anal., 13, No. 3, 515–527 (2012).
S. Cobzaş, “Completeness in quasi-metric spaces and Ekeland Variational Principle,” Topol. Appl., 158, No. 8, 1073–1084 (2011).
S. Cobzaş, “Ekeland variational principle in asymmetric locally convex spaces,” Topol. Appl., 159, No. 10-11, 2558–2569 (2012).
S. Cobzaş, “Functional analysis in asymmetric normed spaces,” in: Frontiers in Mathematics, Birkhäuser; Springer, Basel (2013).
E. H. Connell, “Properties of fixed point spaces,” Proc. Am. Math. Soc., 10, 974–979 (1959).
S. Dancs, M. Hegedűs, and P. Medvegyev, “A general ordering and fixed-point principle in complete metric space,” Acta Sci. Math. (Szeged), 46, No. 1-4, 381–388 (1983).
A. C. Davis, “A characterization of complete lattices,” Pacific J. Math., 5, 311–319 (1955).
K. Deimling, Nonlinear Functional Analysis, Springer, Berlin (1985).
M. M. Deza and E. Deza, Encyclopedia of Distances, Springer, Heidelberg (2014).
S. Dhompongsa, W. Inthakon, and A. Kaewkhao, “Edelstein’s method and fixed point theorems for some generalized nonexpansive mappings,” J. Math. Anal. Appl., 350, No. 1, 12–17 (2009).
S. Dhompongsa and A. Kaewcharoen, “Fixed point theorems for nonexpansive mappings and Suzuki-generalized nonexpansive mappings on a Banach lattice,” Nonlinear Anal., 71, No. 11, 5344–5353 (2009).
S. Dhompongsa and H. Yingtaweesittikul, “Fixed points for multivalued mappings and the metric completeness,” Fixed Point Theory Appl., 972395 (2009).
M. Edelstein, “An extension of Banach’s contraction principle,” Proc. Am. Math. Soc., 12, 7–10 (1961).
M. Edelstein, “On fixed and periodic points under contractive mappings,” J. London Math. Soc., 37, 74–79 (1962).
M. Edelstein, “A theorem on fixed points under isometries,” Am. Math. Mon., 70, 298–300 (1963).
M. Edelstein, “A short proof of a theorem of L. Janos,” Proc. Am. Math. Soc., 20, 509–510 (1969).
M. Elekes, “On a converse to Banach’s fixed point theorem,” Proc. Am. Math. Soc., 137, No. 9, 3139–3146 (2009).
O. Frink, “Topology in lattices,” Trans. Am. Math. Soc., 51, 569–582 (1942).
J. García-Falset, E. Llorens-Fuster, and T. Suzuki, “Fixed point theory for a class of generalized nonexpansive mappings,” J. Math. Anal. Appl., 375, No. 1, 185–195 (2011).
P. G. Georgiev, “The strong Ekeland variational principle, the strong drop theorem and applications,” J. Math. Anal. Appl., 131, No. 1, 1–21 (1988).
P. Ghosh and A. Deb Ray, “A characterization of completeness of generalized metric spaces using generalized Banach contraction principle,” Demonstratio Math., 45, No. 3, 717–724 (2012).
G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, Continuous Lattices and Domains, Encycl. Math. Its Appl., Vol. 93, Cambridge Univ. Press, Cambridge (2003).
K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math., Vol. 28, Cambridge Univ. Press, Cambridge (1990).
J. Goubault-Larrecq, Non-Hausdorff Topology and Domain Theory: Selected Topics in Point-Set Topology, New Math. Monogr., Vol. 22, Cambridge University Press, Cambridge (2013).
M. Grabiec, Y. J. Cho, and R. Saadati, “Completeness and fixed points in probabilistic quasi-pseudo-metric spaces,” Bull. Stat. Econ., 2, No. A08, 39–47 (2008).
A. Granas and J. Dugundji, Fixed point theory, Springer Monogr. Math., Springer, New York (2003).
R. H. Haghi, Sh. Rezapour, and N. Shahzad, “Be careful on partial metric fixed point results,” Topol. Appl., 160, No. 3, 450–454 (2013).
P. Hitzler and A. K. Seda, “Dislocated topologies,” J. Electr. Eng., 51, No. 12/s, 3–7 (2000).
P. Hitzler and A. K. Seda, “A ‘converse’ of the Banach contraction mapping theorem,” J. Electr. Eng., 52, No. 10/s, 3–6 (2001).
P. Hitzler and A. K. Seda, “The fixed-point theorems of Priess-Crampe and Ribenboim in logic programming,” in: Valuation Theory and Its Applications, Vol. I (Saskatoon, SK, 1999 ), Fields Inst. Commun., Vol. 32, Amer. Math. Soc., Providence (2002), pp. 219–235.
S. B. Hosseini and R. Saadati, “Completeness results in probabilistic metric spaces, I,” Commun. Appl. Anal., 9, No. 3-4, 549–553 (2005).
P. Howard and J. E. Rubin, Consequences of the Axiom of Choice, Math. Surveys Monogr., Vol. 59, Amer. Math. Soc., Providence (1998).
T. K. Hu, “On a fixed-point theorem for metric spaces,” Am. Math. Mon., 74, 436–437 (1967).
H. Huang, “Global weak sharp minima and completeness of metric space,” Acta Math. Sci. Ser. B Engl. Ed., 25, No. 2, 359–366 (2005).
S. Iemoto, W. Takahashi, and H. Yingtaweesittikul, “Nonlinear operators, fixed points and completeness of metric spaces,” in: Fixed Point Theory and Its Applications, Yokohama Publ., Yokohama (2010), pp. 93–101.
A. A. Ivanov, “Fixed points of mappings of metric spaces,” in: Studies in Topology, II, Zap. Naučn. Sem. Leningr. Otdel. Mat. Inst. Steklov. (LOMI), 66, 5–102 (1976).
J. Jachymski, “An iff fixed point criterion for continuous self-mappings on a complete metric space,” Aequationes Math., 48, No. 2-3, 163–170 (1994).
J. Jachymski, “Some consequences of fundamental ordering principles in metric fixed point theory,” Proceedings of Workshop on Fixed Point Theory (Kazimierz Dolny), Ann. Univ. Mariae Curie-Skłodowska Sect. A, 51, No. 2, 123–134 (1997),
J. Jachymski, “Fixed point theorems in metric and uniform spaces via the Knaster–Tarski principle,” Nonlinear Anal., 32, No. 2, 225–233 (1998).
J. Jachymski, “Some consequences of the Tarski–Kantorovitch ordering theorem in metric fixed point theory,” Quaestiones Math., 21, No. 1-2, 89–99 (1998).
J. Jachymski, “A short proof of the converse to the contraction principle and some related results,” Topol. Methods Nonlinear Anal., 15, No. 1, 179–186 (2000).
J. Jachymski, “Order-theoretic aspects of metric fixed point theory,” in: Handbook of Metric Fixed Point Theory, Kluwer Academic, Dordrecht (2001), pp. 613–641.
J. Jachymski, “Converses to fixed point theorems of Zermelo and Caristi,” Nonlinear Anal., 52, No. 5, 1455–1463 (2003).
J. Jachymski, “Equivalent conditions for generalized contractions on (ordered) metric spaces,” Nonlinear Anal., 74, No. 3, 768–774 (2011).
J. Jachymski, “A stationary point theorem characterizing metric completeness,” Appl. Math. Lett., 24, No. 2, 169–171 (2011).
S. Janković, Z. Kadelburg, and S. Radenović, “On cone metric spaces: a survey,” Nonlinear Anal., 74, No. 7, 2591–2601 (2011).
L. Janoš, “A converse of Banach’s contraction theorem,” Proc. Am. Math. Soc., 18, 287–289 (1967).
L. Janoš, “A converse of the generalized Banach’s contraction theorem,” Arch. Math. (Basel), 21, 69–71 (1970).
G.-J. Jiang, “On characterization of metric completeness,” Turk. J. Math., 24, No. 3, 267–272 (2000).
O. Kada, T. Suzuki, and W. Takahashi, “Nonconvex minimization theorems and fixed point theorems in complete metric spaces,” Math. Japon., 44, No. 2, 381–391 (1996).
E. Karapinar, “Generalizations of Caristi Kirk’s theorem on partial metric spaces,” Fixed Point Theory Appl., 4 (2011).
E. Karapınar and S. Romaguera, “On the weak form of Ekeland’s variational principle in quasimetric spaces,” Topol. Appl., 184, 54–60 (2015).
E. Karapınar and P. Salimi, “Dislocated metric space to metric spaces with some fixed point theorems,” Fixed Point Theory Appl., 222 (2013).
S. Kasahara, “Classroom notes: A remark on the converse of Banach’s Contraction Theorem,” Am. Math. Mon., 75, No. 7, 775–776 (1968).
K. Keimel, “Topological cones: functional analysis in a T0-setting,” Semigroup Forum, 77, No. 1, 109–142 (2008).
K. Keimel, “Weak topologies and compactness in asymmetric functional analysis,” Topol. Appl., 185/186, 1–22 (2015).
J. L. Kelley, General topology, Springer, Berlin (1975).
J. C. Kelly, “Bitopological spaces,” Proc. London Math. Soc. (3), 13, 71–89 (1963).
M. A. Khamsi, “Generalized metric spaces: A survey,” J. Fixed Point Theory Appl., 17, No. 3, 455–475 (2015).
M. Kikkawa and T. Suzuki, “Some similarity between contractions and Kannan mappings,” Fixed Point Theory Appl., 649749 (2008).
W. Kirk and N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, Cham (2014).
W. A. Kirk, “Contraction mappings and extensions,” in: Handbook of Metric Fixed Point Theory, Kluwer Academic, Dordrecht, 1–34 (2001).
W. A. Kirk and L. M. Saliga, “The Brézis–Browder order principle and extensions of Caristi’s theorem,” Nonlinear Anal., 47, No. 4, 2765–2778 (2001).
W. A. Kirk and B. Sims, eds., Handbook of Metric Fixed Point Theory, Kluwer Academic, Dordrecht (2001).
V. L. Klee, Jr., “Some topological properties of convex sets,” Trans. Am. Math. Soc., 78, 30–45 (1955).
J. Klimeš, “Characterizations of completeness for semilattices by using of fixed points,” Scripta Fac. Sci. Natur. Univ. Purk. Brun., 12, No. 10, 507–513 (1982).
J. Klimeš, “A characterization of a semilattice completeness,” Scripta Fac. Sci. Natur. Univ. Purk. Brun., 14, No. 8, 399–407 (1984).
J. Klimeš, “Fixed point characterization of completeness on lattices for relatively isotone mappings,” Arch. Math. (Brno), 20, No. 3, 125–132 (1984).
J. Klimeš, “A characterization of inductive posets,” Arch. Math. (Brno), 21, No. 1, 39–42 (1985).
R. D. Kopperman, “Which topologies are quasimetrizable?” Topol. Appl., 52, No. 2, 99–107 (1993).
B. K. Lahiri, M. K. Chakrabarty, and A. Sen, “Converse of Banach’s contraction principle and star operation,” Proc. Natl. Acad. Sci. India Sect. A, 79, No. 4, 367–374 (2009).
S. Leader, “A topological characterization of Banach contractions,” Pacific J. Math., 69, No. 2, 461–466 (1977).
S. Leader, “Uniformly contractive fixed points in compact metric spaces,” Proc. Am. Math. Soc., 86, No. 1, 153–158 (1982).
S. Leader, “Equivalent Cauchy sequences and contractive fixed points in metric spaces,” Studia Math., 76, No. 1, 63–67 (1983).
W. Lee and Y. Choi, “A survey on characterizations of metric completeness,” Nonlinear Anal. Forum, 19, 265–276 (2014).
L.-J. Lin and W.-S. Du, “Ekeland’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces,” J. Math. Anal. Appl., 323, No. 1, 360–370 (2006).
L.-J. Lin and W.-S. Du, “Some equivalent formulations of the generalized Ekeland’s variational principle and their applications,” Nonlinear Anal., 67, No. 1, 187–199 (2007).
L.-J. Lin and W.-S. Du, “On maximal element theorems, variants of Ekeland’s variational principle and their applications,” Nonlinear Anal., 68, No. 5, 1246–1262 (2008).
Z. Liu, “Fixed points and completeness,” Turk. J. Math., 20, No. 4, 467–472 (1996).
Z. Liu and S. M. Kang, “On characterizations of metric completeness,” Indian J. Math., 44, No. 2, 183–187 (2002).
Z. Liu and S. M. Kang, “On characterizations of ≤-completeness and metric completeness,” Southeast Asian Bull. Math., 27, No. 2, 325–331 (2003).
R. Mańka, “Connection between set theory and the fixed point property,” Colloq. Math., 53, No. 2, 177–184 (1987).
R. Mańka, “Some forms of the axiom of choice,” Jbuch. Kurt-Gödel-Ges., 24–34 (1988).
R. Mańka, “Turinici’s fixed point theorem and the axiom of choice,” Rep. Math. Logic, 22, 15–19 (1988).
J. Marín, S. Romaguera, and P. Tirado, “Weakly contractive multivalued maps and w-distances on complete quasi-metric spaces,” Fixed Point Theory Appl., 2 (2011).
G. Markowsky, “Chain-complete posets and directed sets with applications,” Algebra Universalis, 6, No. 1, 53–68 (1976).
S. G. Matthews, Partial Metric Spaces, Research Report, No. 212, Univ. of Warwick, UK (1992).
S. G. Matthews, The Cycle Contraction Mapping Theorem, Research Report, No. 228, Univ. of Warwick, UK (1992).
S. G. Matthews, The Topology of Partial Metric Spaces, Research Report, No. 222, Univ. of Warwick, UK (1992).
S. G. Matthews, “Partial metric topology,” in: Papers on General Topology and Applications (Flushing, NY, 1992), Ann. New York Acad. Sci., Vol. 728, New York Acad. Sci., New York (1994), pp. 183–197.
P. R. Meyers, “Some extensions of Banach’s contraction theorem,” J. Res. Natl. Bur. Stand. Sect. B, 69B, 179–184 (1965).
P. R. Meyers, “A converse to Banach’s contraction theorem,” J. Res. Nat. Bur. Stand. Sect. B, 71B, 73–76 (1967).
R. N. Mukherjee and T. Som, “An application of Meyer’s theorem on converse of Banach’s contraction principle,” Bull. Inst. Math. Acad. Sinica, 12, No. 3, 253–255 (1984).
V. V. Nemytskiĭ, “The fixed point method in analysis,” Usp. Mat. Nauk, 1, 141–174 (1936).
A.-M. Nicolae, On Completeness and Fixed Points, Master Thesis, Babeş–Bolyai University, Fac. of Math. and Comput. Sci., Cluj-Napoca (2008).
J. J. Nieto, R. L. Pouso, and R. Rodríguez-López, “Fixed point theorems in ordered abstract spaces,” Proc. Am. Math. Soc., 135, No. 8, 2505–2517 (2007).
J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, 22, No. 3, 223–239 (2005).
J. J. Nieto and R. Rodríguez-López, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,” Acta Math. Sin. (Engl. Ser.), 23, No. 12, 2205–2212 (2007).
S. J. O’Neill, “Partial metrics, valuations, and domain theory,” Ann. N.Y. Acad. Sci., 806, 304–315 (1996).
S. Oltra and O. Valero, “Banach’s fixed point theorem for partial metric spaces,” Rend. Istit. Mat. Univ. Trieste, 36, No. 1-2, 17–26 (2004).
V. I. Opoitsev, “A converse of the contraction mapping principle,” Usp. Mat. Nauk, 31, No. 4 (190), 169–198 (1976).
D. Paesano and P. Vetro, “Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces,” Topol. Appl., 159, No. 3, 911–920 (2012).
B. Palczewski and A. Miczko, “On some converses of generalized Banach contraction principles,” Nonlinear Functional Analysis and Its Applications (Maratea, 1985 ), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 173, Reidel, Dordrecht (1986), pp. 335–351.
B. Palczewski and A. Miczko, “Converses of generalized Banach contraction principles and remarks on mappings with a contractive iterate at the point,” Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat., 17, No. 1, 71–91 (1987).
S. Park and B. E. Rhoades, “Comments on characterizations for metric completeness,” Math. Japon., 31, No. 1, 95–97 (1986).
L. Pasicki, “Dislocated metric and fixed point theorems,” Fixed Point Theory Appl., 82 (2015).
J.-P. Penot, “The drop theorem, the petal theorem and Ekeland’s variational principle,” Nonlinear Anal., 10, No. 9, 813–822 (1986).
C. Petalas and T. Vidalis, “A fixed point theorem in non-Archimedean vector spaces,” Proc. Am. Math. Soc., 118, No. 3, 819–821 (1993).
A. Petruşel, “Multivalued weakly Picard operators and applications,” Sci. Math. Jpn., 59, No. 1, 169–202 (2004).
A. Petruşel and G. Petruşel, “Multivalued Picard operators,” J. Nonlinear Convex Anal., 13, No. 1, 157–171 (2012).
A. Petruşel and I. A. Rus, “Fixed point theorems in ordered L-spaces,” Proc. Am. Math. Soc., 134, No. 2, 411–418 (2006).
S. Prieß-Crampe, “Der Banachsche Fixpunktsatz für ultrametrische Räume,” Results Math., 18, No. 1-2, 178–186 (1990).
A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proc. Am. Math. Soc., 132, No. 5, 1435–1443 (2004).
S. Romaguera, “A Kirk type characterization of completeness for partial metric spaces,” Fixed Point Theory Appl., 493298 (2010).
S. Romaguera and P. Tirado, “A characterization of Smyth complete quasi-metric spaces via Caristi’s fixed point theorem,” Fixed Point Theory Appl., 183 (2015).
S. Romaguera and O. Valero, “Domain theoretic characterisations of quasi-metric completeness in terms of formal balls,” Math. Struct. Comput. Sci., 20, No. 3, 453–472 (2010).
H. Rubin and J. E. Rubin, Equivalents of the Axiom of Choice. II, Stud. Logic Foundat. Math., Vol. 116, North-Holland, Amsterdam (1985).
I. A. Rus, Metrical Fixed Point Theorems, Univ. “Babeş–Bolyai,” Fac. Mat., Cluj-Napoca (1979).
I. A. Rus, “Weakly Picard mappings,” Comment. Math. Univ. Carolin., 34, No. 4, 769–773 (1993).
I. A. Rus, Generalized Contractions and Applications, Cluj Univ. Press, Cluj-Napoca (2001).
I. A. Rus, “Picard operators and applications,” Sci. Math. Jpn., 58, No. 1, 191–219 (2003).
I. A. Rus, “Fixed point theory in partial metric spaces,” An. Univ. Vest Timiş. Ser. Mat.-Inform., 46, No. 2, 149–160 (2008).
I. A. Rus, A. Petruşel, and G. Petruşel, Fixed Point Theory, Cluj Univ. Press, Cluj-Napoca (2008).
B. Samet, “Discussion on ‘A fixed point theorem of Banach–Caccioppoli type on a class of generalized metric spaces’ by A. Branciari,” Publ. Math. Debrecen 76, No. 3-4, 493–494 (2010).
D. Scott, “Continuous lattices,” Toposes, Algebraic Geometry and Logic (Conf., Dalhousie Univ., Halifax, N. S., 1971), Springer, Berlin (1972), pp. 97–136.
N. Shahzad and O. Valero, “On 0-complete partial metric spaces and quantitative fixed point techniques in denotational semantics,” Abstr. Appl. Anal., 985095 (2013).
N. Shahzad and O. Valero, “A Nemytskii–Edelstein type fixed point theorem for partial metric spaces,” Fixed Point Theory Appl., 26 (2015).
N. Shioji, T. Suzuki, and W. Takahashi, “Contractive mappings, Kannan mappings and metric completeness,” Proc. Am. Math. Soc., 126, No. 10, 3117–3124 (1998).
R. E. Smithson, “Fixed points of order preserving multifunctions,” Proc. Am. Math. Soc., 28, 304–310 (1971).
R. E. Smithson, “Fixed points in partially ordered sets,” Pacific J. Math., 45, 363–367 (1973).
V. Stoltenberg-Hansen, I. Lindström, and E. R. Griffor, Mathematical Theory of Domains, Cambridge Tracts in Theor. Comput. Sci., Vol. 22, Cambridge Univ. Press, Cambridge (1994).
P. V. Subrahmanyam, “Completeness and fixed-points,” Monatsh. Math., 80, No. 4, 325–330 (1975).
F. Sullivan, “A characterization of complete metric spaces,” Proc. Am. Math. Soc., 83, No. 2, 345–346 (1981).
F. Sullivan, “Ordering and completeness of metric spaces,” Nieuw Arch. Wisk. (3), 29, No. 2, 178–193 (1981).
T. Suzuki, “Generalized distance and existence theorems in complete metric spaces,” J. Math. Anal. Appl., 253, No. 2, 440–458 (2001).
T. Suzuki, “Several fixed point theorems concerning τ -distance,” Fixed Point Theory Appl., No. 3, 195–209 (2004).
T. Suzuki, “Counterexamples on τ-distance versions of generalized Caristi’s fixed point theorems,” Bull. Kyushu Inst. Technol. Pure Appl. Math., No. 52, 15–20 (2005).
T. Suzuki, “The strong Ekeland variational principle,” J. Math. Anal. Appl., 320, No. 2, 787–794 (2006).
T. Suzuki, “Fixed point theorems and convergence theorems for some generalized nonexpansive mappings,” J. Math. Anal. Appl., 340, No. 2, 1088–1095 (2008).
T. Suzuki, “A generalized Banach contraction principle that characterizes metric completeness,” Proc. Am. Math. Soc., 136, No. 5, 1861–1869 (2008).
T. Suzuki, “w-distances and τ-distances,” Nonlinear Funct. Anal. Appl., 13(1), 15–27 (2008).
T. Suzuki, “Some notes on τ-distance versions of Ekeland’s variational principle,” Bull. Kyushu Inst. Technol. Pure Appl. Math., No. 56, 19–28 (2009).
T. Suzuki, “Some notes on τ-distance versions of Ekeland’s variational principle,” Bull. Kyushu Inst. Technol. Pure Appl. Math., No. 56, 19–28 (2009).
T. Suzuki, “Characterizations of reflexivity and compactness via the strong Ekeland variational principle,” Nonlinear Anal., 72 , No. 5, 2204–2209 (2010).
T. Suzuki, “Some notes on the class of contractions with respect to τ-distance,” Bull. Kyushu Inst. Technol. Pure Appl. Math., No. 57, 9–18 (2010).
T. Suzuki and W. Takahashi, “Fixed point theorems and characterizations of metric completeness,” Topol. Methods Nonlinear Anal., 8, No. 2, 371–382 (1996).
W. Takahashi, “Existence theorems generalizing fixed point theorems for multivalued mappings,” Fixed Point Theory and Applications (Marseille, 1989), Pitman Res. Notes Math. Ser., Vol. 252, Longman Sci. Tech., Harlow (1991), pp. 397–406.
W. Takahashi, “Existence theorems in metric spaces and characterizations of metric completeness,” NLA98: Convex Analysis and Chaos (Sakado, 1998 ), Josai Math. Monogr., Vol. 1, Josai Univ., Sakado (1999), pp. 67–85.
W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and Its Applications, Yokohama Publ., Yokohama (2000).
W. Takahashi, N.-C. Wong, and J.-C. Yao, “Fixed point theorems for general contractive mappings with W-distances in metric spaces,” J. Nonlinear Convex Anal., 14, No. 3, 637–648 (2013).
M. R. Tasković, “The axiom of choice, fixed point theorems, and inductive ordered sets,” Proc. Am. Math. Soc., 116, No. 4, 897–904 (1992).
M. R. Tasković, “Axiom of choice—100th next,” Math. Morav., 8, No. 1, 39–62 (2004).
M. R. Tasković, “The axiom of infinite choice,” Math. Morav., 16, No. 1, 1–32 (2012).
D. Tataru, “Viscosity solutions of Hamilton–Jacobi equations with unbounded nonlinear terms,” J. Math. Anal. Appl., 163, No. 2, 345–392 (1992).
O. Valero, “On Banach fixed point theorems for partial metric spaces,” Appl. Gen. Topol., 6, No. 2, 229–240 (2005).
O. Valero, “On Banach’s fixed point theorem and formal balls,” Appl. Sci., 10, 256–258 (2008).
G. Wang, B. L. Chen, and L. S. Wang, “A new converse to Banach’s contraction mapping theorem: a nonlinear convergence principle,” Gongcheng Shuxue Xuebao, 16, No. 1, 135–138 (1999).
L. E. Ward, Jr., “Completeness in semi-lattices,” Can. J. Math., 9, 578–582 (1957).
J. D. Weston, “A characterization of metric completeness,” Proc. Am. Math. Soc., 64, No. 1, 186–188 (1977).
E. Witt, “On Zorn’s theorem,” Rev. Math. Hisp.-Am., IV. Ser., 10, 82–85 (1950).
E. S. Wolk, “Dedekind completeness and a fixed-point theorem,” Can. J. Math., 9, 400–405 (1957).
J. S. W. Wong, “Generalizations of the converse of the contraction mapping principle,” Can. J. Math., 18, 1095–1104 (1966).
S.-W. Xiang, “Equivalence of completeness and contraction property,” Proc. Am. Math. Soc., S135, No. 4, 1051–1058 (2007).
E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems, Springer, New York (1986).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 22, No. 1, pp. 127–215, 2018.
Rights and permissions
About this article
Cite this article
Cobzaş, S. Fixed Points and Completeness in Metric and Generalized Metric Spaces. J Math Sci 250, 475–535 (2020). https://doi.org/10.1007/s10958-020-05027-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-05027-1