Abstract
In Sect. 1, an extension to semigroup couple metric spaces is given for the fixed point result in Matkowski (Diss Math 127:1–68, 1975). In Sect. 2, we show that the simulation-type contractive maps in quasi-metric spaces introduced by Alsulami et al. (Discrete Dyn Nat Soc 2014, Article ID 269286, 2014) are in fact Meir–Keeler maps. Finally, in Sect. 3, the Brezis–Browder ordering principle (Adv Math 21:355–364, 1976) is used to get a proof, in the reduced axiomatic system (ZF-AC+DC), of a fixed point result [in the complete axiomatic system (ZF)] over Cantor complete ultrametric spaces due to Petalas and Vidalis (Proc Am Math Soc 118:819–821, 1993). The methodological approach we chose consisted in treating each section from a self-contained perspective; so, ultimately, these are independent units of the present exposition.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Agarwal, R.P., El-Gebeily, M.A., O’Regan, D.: Generalized contractions in partially ordered metric spaces. Appl. Anal. 87, 109–116 (2008)
Alsulami, H.H., Karapinar, E., Khojasteh, F., Roldan, A.-F.: A proposal to the study of contractions in quasi-metric spaces. Discrete Dyn. Nat. Soc. 2014 (2014). Article ID 269286
An, T.V., Dung, N.V., Hang, V.T.L.: A new approach to fixed point theorems on G -metric spaces. Topol. Appl. 160, 1486–1493 (2013)
Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133–181 (1922)
Bernays, P.: A system of axiomatic set theory: part III. Infinity and enumerability analysis. J. Symb. Log. 7, 65–89 (1942)
Bourbaki, N.: Sur le théorème de Zorn. Arch. Math. 2, 434–437 (1949/1950)
Bourbaki, N.: General Topology (Chapters 1–4). Springer, Berlin (1987)
Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)
Brezis, H., Browder, F.E.: A general principle on ordered sets in nonlinear functional analysis. Adv. Math. 21, 355–364 (1976)
Brøndsted, A.: Fixed points and partial orders. Proc. Am. Math. Soc. 60, 365–366 (1976)
Brunner, N.: Topologische Maximalprinzipien. Z. Math. Logik Grundl. Math. 33, 135–139 (1987)
Cârjă, O., Necula, M., Vrabie, I.I.: Viability, Invariance and Applications. North Holland Mathematics Studies, vol. 207. Elsevier B. V., Amsterdam (2007)
Choudhury, B.S., Metiya, N.: Fixed points of weak contractions in cone metric spaces. Nonlinear Anal. 72, 1589–1593 (2010)
Cirić, L.B.: A new fixed-point theorem for contractive mappings. Publ. Inst. Math. 30 (44), 25–27 (1981)
Cohen, P.J.: Set Theory and the Continuum Hypothesis. Benjamin, New York (1966)
Collaco, P., E Silva, J.C.: A complete comparison of 25 contractive definitions. Nonlinear Anal. 30, 441–476 (1997)
Cristescu, R.: Topological Vector Spaces. Noordhoff International Publishers, Leyden (1977)
Dhage, B.C.: Generalized metric spaces and mappings with fixed point. Bull. Calcutta Math. Soc. 84, 329–336 (1992)
Di Bari, C., Vetro, P.: \(\varphi\)-pairs and common fixed points in cone metric spaces. Rend. Circ. Mat. Palermo 57, 279–285 (2008)
Dodu, J., Morillon, M.: The Hahn-Banach property and the axiom of choice. Math. Log. Q. 45, 299–314 (1999)
Du, W.-S.: A note on cone metric fixed point theory and its equivalence. Nonlinear Anal. 72, 2259–2261 (2010)
Du, W.-S., Khojasteh, F.: New results and generalizations for approximate fixed point property and their applications. Abstr. Appl. Anal. 2014 (2014). Article ID 581267
Dutta, P.N., Choudhury, B.S.: A generalisation of contraction principle in metric spaces. Fixed Point Theory Appl. 2008 (2008). Article ID 406368
Edelstein, M.: An extension of Banach’s contraction principle. Proc. Am. Math. Soc. 12, 7–10 (1961)
Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc. (N. Ser.) 1, 443–474 (1979)
Gajić, L.: On ultrametric spaces. Novi Sad J. Math. 31, 69–71 (2001)
Găvruţa, L., Găvruţa, P., Khojasteh, F.: Two classes of Meir-Keeler contractions. Arxiv 1405-5034-v1, 20 May 2014
Goepfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Canadian Mathematical Society Books in Mathematics, vol. 17. Springer, New York (2003)
Hicks, T.L., Rhoades, B.E.: Fixed point theory in symmetric spaces with applications to probabilistic spaces. Nonlinear Anal. (A) 36, 331–344 (1999)
Hitzler, P.: Generalized metrics and topology in logic programming semantics. PhD Thesis, Natl. Univ. Ireland, Univ. College Cork (2001)
Huang, L.-G., Zhang, X.: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332, 1468–1476 (2007)
Hyers, D.H., Isac, G., Rassias, T.M.: Topics in Nonlinear Analysis and Applications. World Scientific Publishing, Singapore (1997)
Jachymski, J.: Common fixed point theorems for some families of mappings. Indian J. Pure Appl. Math. 25, 925–937 (1994)
Jachymski, J.: Equivalent conditions and the Meir-Keeler type theorems. J. Math. Anal. Appl. 194, 293–303 (1995)
Janković, S., Kadelburg, Z., Radenović, S.: On cone metric spaces: a survey. Nonlinear Anal. 74, 2591–2601 (2011)
Jleli, M., Samet, B.: Remarks on G-metric spaces and fixed point theorems. Fixed Point Theory Appl. 2012, 210 (2012)
Kang, B.G., Park, S.: On generalized ordering principles in nonlinear analysis. Nonlinear Anal. 14, 159–165 (1990)
Kasahara, S.: On some generalizations of the Banach contraction theorem. Publ. Res. Inst. Math. Sci. Kyoto Univ. 12, 427–437 (1976)
Khamsi, M.A.: Remarks on cone metric spaces and fixed point theorems of contractive mappings. Fixed Point Theory Appl. 2010 (2010). Article ID 315398
Khamsi, M.A., Kirk, W.A.: An Introduction to Metric Spaces and Fixed Point Theory. Wiley, New York (2001)
Khojasteh, F., Shukla, S., Radenović, S.: Formalization of many contractions via the simulation functions. Arxiv 1109-3021-v2, 13 August 2013
Kincses, J., Totik, V.: Theorems and counterexamples on contractive mappings. Math. Balkanica 4, 69–99 (1999)
Leader, S.: Fixed points for general contractions in metric spaces. Math. Jpn. 24, 17–24 (1979)
Matkowski, J.: Integrable solutions of functional equations. Diss. Math. 127 (1975). Polish Scientific Publishers, Warsaw
Matkowski, J.: Fixed point theorems for contractive mappings in metric spaces. Časopis Pest. Mat. 105, 341–344 (1980)
Matthews, S.G.: Partial metric topology (Proc. 8th Summer Conf. Gen. Top. Appl.). Ann. New York Acad. Sci. 728, 183–197 (1994)
Meir, A., Keeler, E.: A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326–329 (1969)
Mishra, S.N., Pant, R.: Generalization of some fixed point theorems in ultrametric spaces. Adv. Fixed Point Theory 4, 41–47 (2014)
Moore, G.H.: Zermelo’s Axiom of Choice: Its Origin, Development and Influence. Springer, New York (1982)
Mustafa, Z., Sims, B.: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 7, 289–297 (2006)
Nieto, J.J., Rodriguez-Lopez, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223–239 (2005)
Pant, R.: Some new fixed point theorems for contractive and nonexpansive mappings. Filomat 28, 313–317 (2014)
Pathak, H.K., Shahzad, N.: Fixed point results for generalized quasi-contraction mappings in abstract metric spaces. Nonlinear Anal. 71, 6068–6076 (2009)
Petalas, C., Vidalis, T.: A fixed point theorem in non-Archimedean vector spaces. Proc. Am. Math. Soc. 118, 819–821 (1993)
Popa, E.: Espaces pseudométriques et pseudonormés. An. Şt. Univ. “A. I. Cuza” Iaşi (S. I-a, Mat.) 14, 383–390 (1968)
Ran, A.C.M., Reurings, M.C.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 1435–1443 (2004)
Rhoades, B.E.: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 226, 257–290 (1977)
Roldan, A.-F., Karapinar, E., Roldan, C., Martinez-Moreno, J.: Coincidence point theorems on metric spaces via simulation functions. J. Comput. Appl. Math. 275, 345–355 (2015)
Rus, I.A.: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca (2001)
Samet, B., Vetro, C., Yazidi, H.: A fixed point theorem for a Meir-Keeler type contraction through rational expression. J. Nonlinear Sci. Appl. 6, 162–169 (2013)
Sayed, A.F.: Common fixed point theorems of multivalued maps in fuzzy ultrametric spaces. J. Math. 2013 (2013). Article ID 617532
Schechter, E.: Handbook of Analysis and its Foundation. Academic Press, New York (1997)
Tarski, A.: Axiomatic and algebraic aspects of two theorems on sums of cardinals. Fund. Math. 35, 79–104 (1948)
Tataru, D.: Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms. J. Math. Anal. Appl. 163, 345–392 (1992)
Turinici, M.: Fixed points of implicit contraction mappings. An. Şt. Univ. “Al. I. Cuza” Iaşi (S I-a, Mat) 22, 177–180 (1976)
Turinici, M.: Fixed points for monotone iteratively local contractions. Dem. Math. 19, 171–180 (1986)
Turinici, M.: Abstract comparison principles and multivariable Gronwall-Bellman inequalities. J. Math. Anal. Appl. 117, 100–127 (1986)
Turinici, M.: Minimal points in product spaces. An. Şt. Univ. “Ovidius” Constanţa (Ser. Mat.) 10, 109–122 (2002)
Turinici, M.: Pseudometric versions of the Caristi-Kirk fixed point theorem. Fixed Point Theory 5, 147–161 (2004)
Turinici, M.: Brezis-Browder principle and dependent choice. An. Şt. Univ. “Al. I. Cuza” Iaşi (S. N.) Mat. 57, 263–277 (2011)
Turinici, M.: Ran-Reurings theorems in ordered metric spaces. J. Indian Math. Soc. 78, 207–214 (2011)
Turinici, M.: Wardowski implicit contractions in metric spaces. Arxiv 1211-3164-v2, 15 Sept 2013
Turinici, M.: Contractive maps in Mustafa-Sims metric spaces. Int. J. Nonlinear Anal. Appl. 5, 36–53 (2014)
Turinici, M.: Contractive maps in locally transitive relational metric spaces. Sci. World J. 2014 (2014). Article ID 169358
van Rooij, A.C.M.: Non-Archimedean Functional Analysis. Marcel Dekker, New York (1978)
Wang, Q., Song, M.: Some coupled fixed point theorems in ultra metric spaces. Sci. J. Math. Res. 3, 114–118 (2013)
Wardowski, D.: Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012, 94 (2012)
Wolk, E.S.: On the principle of dependent choices and some forms of Zorn’s lemma. Canad. Math. Bull. 26, 365–367 (1983)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing Co., Singapore (2002)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Turinici, M. (2016). Contraction Maps in Pseudometric Structures. In: Rassias, T., Pardalos, P. (eds) Essays in Mathematics and its Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-31338-2_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-31338-2_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-31336-8
Online ISBN: 978-3-319-31338-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)