On the topology and \(wt\)-distance on metric type spaces.

*(English)*Zbl 1333.54035Summary: Recently, M. A. Khamsi and the first author [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 9, 3123–3129 (2010; Zbl 1321.54085)] discussed a natural topology defined on any metric type space and noted that this topology enjoys most of the metric topology like properties. In this paper, we define topologically complete type metrizable space and prove that being of metrizability type is preserved under a countable Cartesian product and establish the fact that any set in a complete metric type space is a topologically metrizable type space. Next, we introduce the concept of \(wt\)-distance on a metric type space and prove some fixed point theorems in a partially ordered metric type space with some weak contractions induced by the \(wt\)-distance.

##### MSC:

54E35 | Metric spaces, metrizability |

54H25 | Fixed-point and coincidence theorems (topological aspects) |

54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |

##### Keywords:

metric type space; topologically complet; Alexandroff theorem; \(wt\)-distance; fixed point theorem
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\textit{N. Hussain} et al., Fixed Point Theory Appl. 2014, Paper No. 88, 14 p. (2014; Zbl 1333.54035)

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##### References:

[1] | Bakhtin, IA, The contraction mapping principle in quasimetric spaces, 26-37, (1989), Unianowsk |

[2] | Czerwik, S, Contraction mappings in \(b\)-metric spaces, Acta Math. Inform. Univ. Ostrav, 1, 5-11, (1993) · Zbl 0849.54036 |

[3] | Hussain, N; Ðorić, D; Kadelburg, Z; Radenović, S, Suzuki-type fixed point results in metric type spaces, No. 2012, (2012) · Zbl 1274.54128 |

[4] | Hussain, N; Parvaneh, V; Roshan, JR; Kadelburg, Z, Fixed points of cyclic weakly [inlineequation not available: see fulltext.]-contractive mappings in ordered \(b\)-metric spaces with applications, No. 2013, (2013) |

[5] | Shah, MH; Simic, S; Hussain, N; Sretenovic, A; Radenović, S, Common fixed points theorems for occasionally weakly compatible pairs on cone metric type spaces, J. Comput. Anal. Appl, 14, 290-297, (2012) · Zbl 1256.54081 |

[6] | Alghamdi, MA; Hussain, N; Salimi, P, Fixed point and coupled fixed point theorems on \(b\)-metric-like spaces, No. 2013, (2013) · Zbl 06275443 |

[7] | Boriceanu, M; Petrusel, A; Rus, IA, Fixed point theorems for some multivalued generalized contractions in \(b\)-metric spaces, Int. J. Math. Stat, 6, 65-76, (2010) |

[8] | Bota, M; Molnár, A; Varga, C, On ekeland’s variational principle in \(b\)-metric spaces, Fixed Point Theory, 12, 21-28, (2011) · Zbl 1278.54022 |

[9] | Agarwal, RP; Hussain, N; Taoudi, M-A, Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations, No. 2012, (2012) |

[10] | Cho, YJ; Saadati, R; Wang, S, Common fixed point theorems on generalized distance in order cone metric spaces, Comput. Math. Appl, 61, 1254-1260, (2011) · Zbl 1217.54041 |

[11] | Graily, E; Vaezpour, SM; Saadati, R; Cho, YJ, Generalization of fixed point theorems in ordered metric spaces concerning generalized distance, No. 2011, (2011) |

[12] | Sintunavarat, W; Cho, YJ; Kumam, P, Common fixed point theorems for \(c\)-distance in ordered cone metric spaces, Comput. Math. Appl, 62, 1969-1978, (2011) · Zbl 1231.54028 |

[13] | Khamsi, MA; Hussain, N, KKM mappings in metric type spaces, Nonlinear Anal, 73, 3123-3129, (2010) · Zbl 1321.54085 |

[14] | Hussain, N; Shah, MH, KKM mappings in cone \(b\)-metric spaces, Comput. Math. Appl, 62, 1677-1684, (2011) · Zbl 1231.54022 |

[15] | Aghajani, A, Abbas, M, Roshan, JR: Common fixed point of generalized weak contractive mappings in partially ordered \(b\)-metric spaces. Math. Slovaca (2012, in press) · Zbl 1349.54078 |

[16] | Kada, O; Suzuki, T; Takahashi, W, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Jpn, 44, 381-391, (1996) · Zbl 0897.54029 |

[17] | Ilić, D; Rakočević, V, Common fixed points for maps on metric space with \(w\)-distance, Appl. Math. Comput, 199, 599-610, (2008) · Zbl 1143.54018 |

[18] | Jungck, G, Commuting mappings and fixed points, Am. Math. Mon, 83, 261-263, (1976) · Zbl 0321.54025 |

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