Characterizing various types of complete generalized metric spaces via fixed point theorems - A survey
Subject Areas : Operator theory
1 - Instituto Universitario de Matem\'{a}tica Pura y Aplicada, Universitat Polit\`{e}cnica de Val\`{e}ncia, 46022, Valencia, Spain
Keywords: Quasi-metric space, G-metric space, partial metric space, fixed point, completeness,
Abstract :
We restore several known extensions of various classical characterizations of metric completeness, established via fixed point results, to the framework of quasi-metric spaces, $G$-metric spaces and partial metric spaces, respectively.
References:
[1] O. Acar, I. Altun, S. Romaguera, Caristi’s type mappings on complete partial metric spaces, Fixed Point Theory. 14 (2013), 3-10.
[2] R. P. Agarwal, E. Karapinar, Remarks on some coupled fixed point theorems in G-metric spaces, Fixed Point Theory Appl. (2013), 2013:2.
[3] R. P. Agarwal, E. Karapınar, D. O’Regan, A. F. Roldán-López-de-Hierro, Fixed Point Theory in Metric Type Spaces, Springer, 2015.
[4] R. P. Agarwal, E. Karapınar, A. F. Roldán-López-de-Hierro, Fixed point theorems in quasimetric spaces and applications to multidimensional fixed points on G-metric spaces, J. Nonlinear Convex Anal. 16 (2015), 1787-1816.
[5] E. S. Ahmed, A. Fulga, The Górnicki-Proinov type contraction on quasi-metric spaces, AIMS Math. 6 (2021), 8815-8834.
[6] C. Alegre, H. Dag, S. Romaguera, P. Tirado, Characterizations of quasi-metric completeness in terms of Kannan-type fixed point theorems, Hacet. J. Math. Stat. 46 (2017), 67-76.
[7] S. Al-Homidan, Q. H. Ansari, J. C. Yao, Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory, Nonlinear Anal. 69 (2008) 126-139.
[8] B. Ali, H. Ali, T. Nazir, Z. Ali, Existence of fixed points of Suzuki-type contractions of quasi-metric spaces, Mathematics. (2023), 2023:4445.
[9] I. Altun, S. Romaguera, Characterizations of partial metric completeness in terms of weakly contractive mappings having fixed point, Appl. Anal. Discr. Math. 6 (2012), 247-256.
[10] T. V. An, N. V. Dung, V. T. L. Hang, A new approach to fixed point theorems on G-metric spaces, Topol. Appl. 160 (2013), 1486-1493.
[11] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241-251.
[12] S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Birkhaüser, 2013.
[13] E. H. Connell, Properties of fixed point spaces, Proc. Amer. Math. Soc. 10 (1959), 974-979.
[14] H. Dag, S. Romaguera, P. Tirado, A characterization of quasi-metric completeness, Proc. WATS’17. (2017), 27-32.
[15] P. Fletcher, W. F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, 1982.
[16] A. Fulga, E. Karapinar, G. Petrusel, On hybrid contractions in the context of quasi-metric spaces, Mathematics. (2020), 2020:675.
[17] T. K. Hu, On a fixed point theorem for metric spaces, Amer. Math. Monthly. 74 (1967), 436-437.
[18] O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Japon. 44 (1996), 381-391.
[19] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71-76.
[20] E. Karapinar, R. P. Agarwal, Fixed Point Theory in Generalized Metric Spaces, Springer, 2022.
[21] E. Karapinar, S. Romaguera, P. Tirado, Characterizations of quasi-metric and G-metric completeness involving w-distances and fixed points, Demonstr. Math. 55 (2022), 939-951.
[22] S. S. Karimizad, G. Soleimani Rad, Pre-symmetric w-cone distances and characterization of TVS-cone metric completeness, Mathematics. (2024), 2024: 1833.
[23] S. S. Karimizad, G. Soleimani Rad, Pre-symmetric c-distances and characterization of complete cone metric spaces, Appl. Gen. Topol. 26 (2025), 529-542.
[24] W. A. Kirk, Caristi’s fixed point theorem and metric convexity, Colloq. Math. 36 (1976), 81-86.
[25] A. Latif, S. A. Al-Mezel, Fixed point results in quasimetric spaces, Fixed Point Theory Appl. (2011), 2011:178306.
[26] S. G. Matthews, Partial metric topology, Proc. 8th Summer Conference on General Topology and Appl., Ann. New York Acad. Sci. 728 (1994), 183-197.
[27] Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal. 7 (2006), 289-297.
[28] Z. Mustafa, B. Sims, Fixed point theorems for contractive mappings in complete G-metric spaces, Fixed Point Theory Appl. (2009), 2009:917175.
[29] D. Paesano, P. Vetro, Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces, Topol. Appl. 159 (2012), 911-920.
[30] S. Park, On generalizations of the Ekeland-type variational principles, Nonlinear Anal. 39 (2000), 881-889.
[31] V. Rakocevic, Fixed Point Results in W-Distance Spaces, Taylor & Francis Group, 2022.
[32] S. Romaguera, Basic contractions of Suzuki-type on quasi-metric spaces and fixed point results, Mathematics. (2022), 2022:3931.
[33] S. Romaguera, Remarks on the fixed point theory for quasi-metric spaces, Results Nonlinear Anal. 7 (2024), 70-74.
[34] S. Romaguera, P. Tirado, A characterization of Smyth complete quasi-metric spaces via Caristi’s fixed point theorem, Fixed Point Theory Appl. (2015), 2015:183.
[35] S. Romaguera, P. Tirado, α−ψ-contractive type mappings on quasi-metric spaces, Filomat. 35 (2021), 1649-1659.
[36] S. Romaguera, P. Tirado, Fixed point theorems that characterize completeness of G-metric spaces, J. Nonlinear Convex Anal. 23 (2022), 1525-1536.
[37] S. Romaguera, P. Tirado, Partial metrics viewed as w-distances: Extending some powerful fixed-point theorems, Mathematics. (2024), 2024:3991.
[38] S. Romaguera, P. Tirado, Presymmetric w-distances on metric spaces, Mathematics. (2024), 2024:305.
[39] S. Romaguera, P. Tirado, The property of presymmetry for w-distances on quasi-metric spaces, Filomat. 38 (2024), 8937-8945.
[40] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α−ψ-contractive type mappings, Nonlinear Anal. 75 (2012), 2154-2165.
[41] M. Schellekens, The Smyth completion: a common foundation for denonational semantics and complexity analysis. Electron. Notes Theor. Comput. Sci. 1 (1995), 535-556.
[42] N. Shioji, T. Suzuki, W. Takahashi, Contractive mappings, Kannan mappings and metric completeness, Proc. Amer. Math. Soc. 126 (1998), 3117-3124.
[43] P. V. Subrahmanyam, Completeness and fixed-points, Mh. Math. 80 (1975), 325-330.
[44] T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136 (2008), 1861-1869.
[45] T. Suzuki, Several fixed point theorems in complete metric spaces, Yokohama Math. 44 (1997), 61-72.
[46] T. Suzuki, W. Takahashi, Fixed point theorems and characterizations of metric completeness, Topol. Methods Nonlinear Anal. 8 (1996), 371-382.
[47] W. A. Wilson, On quasi-metric spaces, Amer. J. Math. 53 (1931), 675-684.
[1] O. Acar, I. Altun, S. Romaguera, Caristi’s type mappings on complete partial metric spaces, Fixed Point Theory. 14 (2013), 3-10.
[2] R. P. Agarwal, E. Karapinar, Remarks on some coupled fixed point theorems in G-metric spaces, Fixed Point Theory Appl. (2013), 2013:2.
[3] R. P. Agarwal, E. Karapınar, D. O’Regan, A. F. Roldán-López-de-Hierro, Fixed Point Theory in Metric Type Spaces, Springer, 2015.
[4] R. P. Agarwal, E. Karapınar, A. F. Roldán-López-de-Hierro, Fixed point theorems in quasimetric spaces and applications to multidimensional fixed points on G-metric spaces, J. Nonlinear Convex Anal. 16 (2015), 1787-1816.
[5] E. S. Ahmed, A. Fulga, The Górnicki-Proinov type contraction on quasi-metric spaces, AIMS Math. 6 (2021), 8815-8834.
[6] C. Alegre, H. Dag, S. Romaguera, P. Tirado, Characterizations of quasi-metric completeness in terms of Kannan-type fixed point theorems, Hacet. J. Math. Stat. 46 (2017), 67-76.
[7] S. Al-Homidan, Q. H. Ansari, J. C. Yao, Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory, Nonlinear Anal. 69 (2008) 126-139.
[8] B. Ali, H. Ali, T. Nazir, Z. Ali, Existence of fixed points of Suzuki-type contractions of quasi-metric spaces, Mathematics. (2023), 2023:4445.
[9] I. Altun, S. Romaguera, Characterizations of partial metric completeness in terms of weakly contractive mappings having fixed point, Appl. Anal. Discr. Math. 6 (2012), 247-256.
[10] T. V. An, N. V. Dung, V. T. L. Hang, A new approach to fixed point theorems on G-metric spaces, Topol. Appl. 160 (2013), 1486-1493.
[11] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241-251.
[12] S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Birkhaüser, 2013.
[13] E. H. Connell, Properties of fixed point spaces, Proc. Amer. Math. Soc. 10 (1959), 974-979.
[14] H. Dag, S. Romaguera, P. Tirado, A characterization of quasi-metric completeness, Proc. WATS’17. (2017), 27-32.
[15] P. Fletcher, W. F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, 1982.
[16] A. Fulga, E. Karapinar, G. Petrusel, On hybrid contractions in the context of quasi-metric spaces, Mathematics. (2020), 2020:675.
[17] T. K. Hu, On a fixed point theorem for metric spaces, Amer. Math. Monthly. 74 (1967), 436-437.
[18] O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Japon. 44 (1996), 381-391.
[19] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71-76.
[20] E. Karapinar, R. P. Agarwal, Fixed Point Theory in Generalized Metric Spaces, Springer, 2022.
[21] E. Karapinar, S. Romaguera, P. Tirado, Characterizations of quasi-metric and G-metric completeness involving w-distances and fixed points, Demonstr. Math. 55 (2022), 939-951.
[22] S. S. Karimizad, G. Soleimani Rad, Pre-symmetric w-cone distances and characterization of TVS-cone metric completeness, Mathematics. (2024), 2024: 1833.
[23] S. S. Karimizad, G. Soleimani Rad, Pre-symmetric c-distances and characterization of complete cone metric spaces, Appl. Gen. Topol. 26 (2025), 529-542.
[24] W. A. Kirk, Caristi’s fixed point theorem and metric convexity, Colloq. Math. 36 (1976), 81-86.
[25] A. Latif, S. A. Al-Mezel, Fixed point results in quasimetric spaces, Fixed Point Theory Appl. (2011), 2011:178306.
[26] S. G. Matthews, Partial metric topology, Proc. 8th Summer Conference on General Topology and Appl., Ann. New York Acad. Sci. 728 (1994), 183-197.
[27] Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal. 7 (2006), 289-297.
[28] Z. Mustafa, B. Sims, Fixed point theorems for contractive mappings in complete G-metric spaces, Fixed Point Theory Appl. (2009), 2009:917175.
[29] D. Paesano, P. Vetro, Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces, Topol. Appl. 159 (2012), 911-920.
[30] S. Park, On generalizations of the Ekeland-type variational principles, Nonlinear Anal. 39 (2000), 881-889.
[31] V. Rakocevic, Fixed Point Results in W-Distance Spaces, Taylor & Francis Group, 2022.
[32] S. Romaguera, Basic contractions of Suzuki-type on quasi-metric spaces and fixed point results, Mathematics. (2022), 2022:3931.
[33] S. Romaguera, Remarks on the fixed point theory for quasi-metric spaces, Results Nonlinear Anal. 7 (2024), 70-74.
[34] S. Romaguera, P. Tirado, A characterization of Smyth complete quasi-metric spaces via Caristi’s fixed point theorem, Fixed Point Theory Appl. (2015), 2015:183.
[35] S. Romaguera, P. Tirado, α−ψ-contractive type mappings on quasi-metric spaces, Filomat. 35 (2021), 1649-1659.
[36] S. Romaguera, P. Tirado, Fixed point theorems that characterize completeness of G-metric spaces, J. Nonlinear Convex Anal. 23 (2022), 1525-1536.
[37] S. Romaguera, P. Tirado, Partial metrics viewed as w-distances: Extending some powerful fixed-point theorems, Mathematics. (2024), 2024:3991.
[38] S. Romaguera, P. Tirado, Presymmetric w-distances on metric spaces, Mathematics. (2024), 2024:305.
[39] S. Romaguera, P. Tirado, The property of presymmetry for w-distances on quasi-metric spaces, Filomat. 38 (2024), 8937-8945.
[40] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α−ψ-contractive type mappings, Nonlinear Anal. 75 (2012), 2154-2165.
[41] M. Schellekens, The Smyth completion: a common foundation for denonational semantics and complexity analysis. Electron. Notes Theor. Comput. Sci. 1 (1995), 535-556.
[42] N. Shioji, T. Suzuki, W. Takahashi, Contractive mappings, Kannan mappings and metric completeness, Proc. Amer. Math. Soc. 126 (1998), 3117-3124.
[43] P. V. Subrahmanyam, Completeness and fixed-points, Mh. Math. 80 (1975), 325-330.
[44] T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136 (2008), 1861-1869.
[45] T. Suzuki, Several fixed point theorems in complete metric spaces, Yokohama Math. 44 (1997), 61-72.
[46] T. Suzuki, W. Takahashi, Fixed point theorems and characterizations of metric completeness, Topol. Methods Nonlinear Anal. 8 (1996), 371-382.
[47] W. A. Wilson, On quasi-metric spaces, Amer. J. Math. 53 (1931), 675-684.