Various versions of distances on different metric spaces and fixed point theory; a discussion of Rahimi et al.'s works
Subject Areas : Functional analysis
1 - Department of Mathematics, Faculty of Basic Sciences, Ilam University, P.O. Box 69315-516, Ilam, Iran
Keywords: $w$-distance, $c$-distance, $wt$-distance, $ct$-distance, Banach algebra,
Abstract :
The main purpose of this article is to investigate the relation between different kinds of distances on divers metric spaces. Also, we consider some fixed point theorems with respect to a specified distance and show the validity of them regarding other distances. This helps researchers avoid constructing new theorems and additional proofs, which are indeed an illogical task. In particular, we discussed the fixed point theorems of Rahimi et al. with respect to $w$-distances ($c$-distances) and $wt$-distances ($ct$-distances).
References:
[1] A. Arabnia Firozjah, H. Rahimi, M. De la Sen, G. Soleimani Rad, Fixed point results under generalized c-distance in cone b-metric spaces over Banach algebras, Axioms. 9 (2020), 1:31.
[2] A. Arabnia Firozjah, H. Rahimi, G. Soleimani Rad, Fixed and periodic point results in cone b-metric spaces over Banach algebras; a survey., Fixed Point Theory. 22 (1) (2021), 157-168.
[3] L. Aryanpour, H. Rahimi, G. Soleimani Rad, Common fixed points of two mappings regarding a generalized c-distance over a Banach algebra, J. Function Spaces. 2021 (2021), 1:6623606.
[4] L. Aryanpour, H. Rahimi, G. Soleimani Rad, Fixed point results for HardyRogers type contractions with respect to a c-distance in graphical cone metric spaces, Probl. Anal. Issues Anal. 9(27) (1) (2020), 27-37.
[5] L. Aryanpour, H. Rahimi, G. Soleimani Rad, The existence of fixed points for Hardy-Rogers contractive mappings with respect to a wt-distance in b-metric spaces, Facta Uni. Ser. Math. Inform. 37 (5) (2022), 839-848.
[6] R. Babaei, H. Rahimi, M. De La Sen, G. S. Rad, w-b-cone distance and its related results: a survey, Symmetry. 12 (2020), 1:171.
[7] R. Babaei, H. Rahimi, G. Soleimani Rad, Fixed point results with respect to a wt-distance in partially ordered b-metric spaces and its application to nonlinear fourth-order differential equation, Appl. Gen. Topol. 23 (1) (2022), 121-133.
[8] R. Babaei, H. Rahimi, G. Soleimani Rad, Weakly compatible mappings with respect to a generalized c-distance and common fixed point results, Cogent Mathematics & Statistics. (2020), 7:1.
[9] S. Banach, Sur les opérations dans les ensembles abstraits et leur application auxéquations intégrales, Fund. Math. J. 3 (1922), 133-181.
[10] B. Bao, S. Xu, L. Shi, V. Cojbasic Rajic, Fixed point theorems on generalized c-distance in ordered cone b-metric spaces, Int. J. Nonlinear Anal. Appl. 6 (1) (2015), 9-22.
[11] Y. J. Cho, R. Saadati, S. H. Wang, Common fixed point theorems on generalized distance in ordered cone metric spaces, Comput. Math. Appl. 61 (2011), 1254-1260.
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[13] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav. 1 (1) (1993), 5-11.
[14] M. Djordevic, D. Doric, Z. Kadelburg, S. Radenovic, D. Spasic, Fixed point results under c-distance in tvs-cone metric spaces, Fixed Point Theory Appl. 29 (2011), 1-9.
[15] W. S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010), 2259-2261.
[16] W. S. Du, E. Karapinar, A note on b-cone metric and its related results: Generalizations or equivalence?, Fixed Point Theory Appl. (2013), 2013:210.
[17] K. Fallahi, A. Petru¸ sel, G. Soleimani Rad, Fixed point results for pointwise Chatterjea type mappings with respect to a c-distance in cone metric spaces endowed with a graph, UPB Sci. Bull. Ser. A. 80 (2018), 47-54.
[18] K. Fallahi, G. Soleimani Rad, The Banach type contraction for mappings on algebraic cone metric spaces associated with an algebraic distance and endowed with a graph, IJMSI. 15 (1), (2020), 41-52.
[19] M. M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Matem. Palermo, 22 (1906), 1-72.
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[21] H. Huang, G. Deng, S. Radenovic, Some topological properties and fixed point results in cone metric spaces over Banach algebras, Positivity. 23 (2019), 23:21.
[22] H. Huang, S. Hu, B. Z. Popovic, S. Radenovic, Common fixed point theorems for four mappings on cone b-metric spaces over Banach algebras, J. Nonlinear Sci. Appl. 9 (2016), 3655-3671.
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[27] O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon. 44 (1996), 381-391.
[28] Z. Kadelburg, Lj. Paunovic, S. Radenovic, G. Soleimani Rad, Non-normal cone metric and cone b-metric spaces and fixed point results, Ser. A. Appl. Math. Intorm. Mech. 8 (2) (2016), 177-186.
[29] E. Karapinar, R. P. Agarwal, Fixed Point Theory in Generalized Metric Spaces, Synthesis Lectures on Mathematics & Statistics, Springer, 2023.
[30] E. Karapınar, C. Chifu, Results in wt-distance over b-metric spaces, Mathematics. 8 (2020), 8:220.
[31] S. S. Karimizad, G. Soleimani Rad, Pre-symmetric c-distances and characterization of complete cone metric spaces, Appl. Gen. Topol. 26 (1) (2025), 529-542.
[32] S. S. Karimizad, G. Soleimani Rad, Pre-symmetric w-cone distances and characterization of TV S-cone metric completeness, Mathematics. 12 (2024), 12:1833.
[33] M. A. Khamsi, N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal. 73 (2010), 3123-3129.
[34] A. Kostic, H. Rahimi, G. Soleimani Rad, wt0-distance and best proximity points involving b-simulation functions, 113 (127) (2023), 113-127.
[35] A. Kostic, V. Rakocevic, S. Radenovic, Best proximity points involving simulation functions with w0-distance, RACSAM. 113 (2) (2019), 715-727.
[36] H. Lakzian, V. Rakocevic, H. Aydi, Extensions of Kannan contraction via w-distances, Aequat. Math. 93 (2019), 1231-1244.
[37] H. Liu, S. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl. (2013), 2013:320.
[38] H. Rahimi, G. Soleimani Rad, Common fixed-point theorems and c-distance in ordered cone metric spaces, Ukrain. Math. J. 65 (2014), 1845-1861.
[39] H. Rahimi, Gh. Soleimani Rad, Fixed Point Theory in Various Spaces, Lambert Academic Publishing, Germany, 2013.
[40] V. Rakocevic, Fixed Point Results in W-Distance Spaces, Monographs and Research Notes in Mathematics, Taylor & Francis Group, CRC Press, USA, 2022.
[41] A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. Amer. Math. Soc. 132 (2004), 1435-1443.
[42] S. Romaguera, P. Tirado, Presymmetric w-distances on metric spaces, Mathematics. 12 (2024), 2:305.
[43] W. Sintunavarat, Y. J. Cho, P. Kumam, Common fixed point theorems for c-distance in ordered cone metric spaces, Comput. Math. Appl. 62 (2011), 1969-1978.
[44] G. Soleimani Rad, K. Fallahi, Z. Kadelburg, New fixed point results under generalized c-distance in tvs-cone b-metric spaces with an application to systems of Fredholm integral equations, J. Math. Ext. 12 (2018), 1-19.
[45] G. Soleimani Rad, H. Rahimi, C. Vetro, Fixed point results under generalized c-distance with application to nonlinear fourth-order differential equation, Fixed Point Theory. 20 (2019), 635-648.
[46] F. Yousefi, H. Rahimi, G. Soleimani Rad, A discussion on fixed point theorems of´Ciri´ c and Fisher on w-distance, J. Nonlinear Convex Anal. 23 (2022), 1409-1418.
[47] P. P. Zabrejko, K-metric and K-normed linear spaces: survey, Collect. Math. 48 (1997), 825-859.
[1] A. Arabnia Firozjah, H. Rahimi, M. De la Sen, G. Soleimani Rad, Fixed point results under generalized c-distance in cone b-metric spaces over Banach algebras, Axioms. 9 (2020), 1:31.
[2] A. Arabnia Firozjah, H. Rahimi, G. Soleimani Rad, Fixed and periodic point results in cone b-metric spaces over Banach algebras; a survey., Fixed Point Theory. 22 (1) (2021), 157-168.
[3] L. Aryanpour, H. Rahimi, G. Soleimani Rad, Common fixed points of two mappings regarding a generalized c-distance over a Banach algebra, J. Function Spaces. 2021 (2021), 1:6623606.
[4] L. Aryanpour, H. Rahimi, G. Soleimani Rad, Fixed point results for HardyRogers type contractions with respect to a c-distance in graphical cone metric spaces, Probl. Anal. Issues Anal. 9(27) (1) (2020), 27-37.
[5] L. Aryanpour, H. Rahimi, G. Soleimani Rad, The existence of fixed points for Hardy-Rogers contractive mappings with respect to a wt-distance in b-metric spaces, Facta Uni. Ser. Math. Inform. 37 (5) (2022), 839-848.
[6] R. Babaei, H. Rahimi, M. De La Sen, G. S. Rad, w-b-cone distance and its related results: a survey, Symmetry. 12 (2020), 1:171.
[7] R. Babaei, H. Rahimi, G. Soleimani Rad, Fixed point results with respect to a wt-distance in partially ordered b-metric spaces and its application to nonlinear fourth-order differential equation, Appl. Gen. Topol. 23 (1) (2022), 121-133.
[8] R. Babaei, H. Rahimi, G. Soleimani Rad, Weakly compatible mappings with respect to a generalized c-distance and common fixed point results, Cogent Mathematics & Statistics. (2020), 7:1.
[9] S. Banach, Sur les opérations dans les ensembles abstraits et leur application auxéquations intégrales, Fund. Math. J. 3 (1922), 133-181.
[10] B. Bao, S. Xu, L. Shi, V. Cojbasic Rajic, Fixed point theorems on generalized c-distance in ordered cone b-metric spaces, Int. J. Nonlinear Anal. Appl. 6 (1) (2015), 9-22.
[11] Y. J. Cho, R. Saadati, S. H. Wang, Common fixed point theorems on generalized distance in ordered cone metric spaces, Comput. Math. Appl. 61 (2011), 1254-1260.
[12] Lj.Ciric, H. Lakzian, V. Rakocevic, Fixed point theorems for w-cone distance contraction mappings in tvs-cone metric spaces, Fixed Point Theory Appl. (2012), 2012:3.
[13] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav. 1 (1) (1993), 5-11.
[14] M. Djordevic, D. Doric, Z. Kadelburg, S. Radenovic, D. Spasic, Fixed point results under c-distance in tvs-cone metric spaces, Fixed Point Theory Appl. 29 (2011), 1-9.
[15] W. S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010), 2259-2261.
[16] W. S. Du, E. Karapinar, A note on b-cone metric and its related results: Generalizations or equivalence?, Fixed Point Theory Appl. (2013), 2013:210.
[17] K. Fallahi, A. Petru¸ sel, G. Soleimani Rad, Fixed point results for pointwise Chatterjea type mappings with respect to a c-distance in cone metric spaces endowed with a graph, UPB Sci. Bull. Ser. A. 80 (2018), 47-54.
[18] K. Fallahi, G. Soleimani Rad, The Banach type contraction for mappings on algebraic cone metric spaces associated with an algebraic distance and endowed with a graph, IJMSI. 15 (1), (2020), 41-52.
[19] M. M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Matem. Palermo, 22 (1906), 1-72.
[20] Y. Han, S. Xu, Some new theorems on c-distance without continuity in cone metric spaces over Banach algebras, J. Function Spaces. (2018), 2018:7463435.
[21] H. Huang, G. Deng, S. Radenovic, Some topological properties and fixed point results in cone metric spaces over Banach algebras, Positivity. 23 (2019), 23:21.
[22] H. Huang, S. Hu, B. Z. Popovic, S. Radenovic, Common fixed point theorems for four mappings on cone b-metric spaces over Banach algebras, J. Nonlinear Sci. Appl. 9 (2016), 3655-3671.
[23] H. Huang, S. Radenovic, T. Dosenovic, Some common fixed point theorems on c-distance in cone metric spaces over Banach algebras, Appl. Comput. Math. 14 (2) (2015), 180-193.
[24] L. G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007), 1467-1475.
[25] N. Hussain, R. Saadati, R. P. Agrawal, On the topology and wt-distance on metric type spaces, Fixed Point Theory Appl. (2014), 2014:88.
[26] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (2008), 1359-1373.
[27] O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon. 44 (1996), 381-391.
[28] Z. Kadelburg, Lj. Paunovic, S. Radenovic, G. Soleimani Rad, Non-normal cone metric and cone b-metric spaces and fixed point results, Ser. A. Appl. Math. Intorm. Mech. 8 (2) (2016), 177-186.
[29] E. Karapinar, R. P. Agarwal, Fixed Point Theory in Generalized Metric Spaces, Synthesis Lectures on Mathematics & Statistics, Springer, 2023.
[30] E. Karapınar, C. Chifu, Results in wt-distance over b-metric spaces, Mathematics. 8 (2020), 8:220.
[31] S. S. Karimizad, G. Soleimani Rad, Pre-symmetric c-distances and characterization of complete cone metric spaces, Appl. Gen. Topol. 26 (1) (2025), 529-542.
[32] S. S. Karimizad, G. Soleimani Rad, Pre-symmetric w-cone distances and characterization of TV S-cone metric completeness, Mathematics. 12 (2024), 12:1833.
[33] M. A. Khamsi, N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal. 73 (2010), 3123-3129.
[34] A. Kostic, H. Rahimi, G. Soleimani Rad, wt0-distance and best proximity points involving b-simulation functions, 113 (127) (2023), 113-127.
[35] A. Kostic, V. Rakocevic, S. Radenovic, Best proximity points involving simulation functions with w0-distance, RACSAM. 113 (2) (2019), 715-727.
[36] H. Lakzian, V. Rakocevic, H. Aydi, Extensions of Kannan contraction via w-distances, Aequat. Math. 93 (2019), 1231-1244.
[37] H. Liu, S. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl. (2013), 2013:320.
[38] H. Rahimi, G. Soleimani Rad, Common fixed-point theorems and c-distance in ordered cone metric spaces, Ukrain. Math. J. 65 (2014), 1845-1861.
[39] H. Rahimi, Gh. Soleimani Rad, Fixed Point Theory in Various Spaces, Lambert Academic Publishing, Germany, 2013.
[40] V. Rakocevic, Fixed Point Results in W-Distance Spaces, Monographs and Research Notes in Mathematics, Taylor & Francis Group, CRC Press, USA, 2022.
[41] A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. Amer. Math. Soc. 132 (2004), 1435-1443.
[42] S. Romaguera, P. Tirado, Presymmetric w-distances on metric spaces, Mathematics. 12 (2024), 2:305.
[43] W. Sintunavarat, Y. J. Cho, P. Kumam, Common fixed point theorems for c-distance in ordered cone metric spaces, Comput. Math. Appl. 62 (2011), 1969-1978.
[44] G. Soleimani Rad, K. Fallahi, Z. Kadelburg, New fixed point results under generalized c-distance in tvs-cone b-metric spaces with an application to systems of Fredholm integral equations, J. Math. Ext. 12 (2018), 1-19.
[45] G. Soleimani Rad, H. Rahimi, C. Vetro, Fixed point results under generalized c-distance with application to nonlinear fourth-order differential equation, Fixed Point Theory. 20 (2019), 635-648.
[46] F. Yousefi, H. Rahimi, G. Soleimani Rad, A discussion on fixed point theorems of´Ciri´ c and Fisher on w-distance, J. Nonlinear Convex Anal. 23 (2022), 1409-1418.
[47] P. P. Zabrejko, K-metric and K-normed linear spaces: survey, Collect. Math. 48 (1997), 825-859.