A Survey on Branciari Metric Spaces

Authors

DOI:

https://doi.org/10.26713/cma.v14i2.2083

Keywords:

Metric, v-generalized metric, 2-generalized metric, 3-generalized metric, rectangular b-metric, cone rectangular metric, rectangular S-metric, partial rectangular metric, rectangular M-metric, complex valued rectangular metri, fixed point, compatible topology

Abstract

The motive of this review article is to collect most of the results on v-generalized metric space and its upto date various generalizations. We try to update the literature for continuous development on the results of v-generalized metric and its generalizations.

Downloads

Download data is not yet available.

References

M. Aamri and D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, Journal of Mathematical Analysis and Applications 270(1) (2002), 181 – 188, DOI: 10.1016/S0022-247X(02)00059-8.

M. Abbas, V. C. Rajic, T. Nazir and S. Radenovic, Common fixed point of mappings satisfying rational inequalities in ordered complex valued generalized metric spaces, Afrika Matematika 26 (2013), 17 – 30, DOI: 10.1007/s13370-013-0185-z.

M. Abbas, V. Rakochevich and Z. Noor, Perov multivalued contraction pair inrectangular cone metric spaces, Vestnik of Saint Petersburg University - Mathematics, Mechanics, Astronomy 8 (2021), 484 – 501, DOI: 10.21638/spbu01.2021.310.

O. K. Adewale and C. Iluno, Fixed point theorems on rectangular S-metric spaces, Scientific African 16 (2022), e01202, DOI: 10.1016/j.sciaf.2022.e01202.

J. Ahmad, M. Arshad and C. Vetro, On a theorem of khan in a generalized metric space, International Journal of Analysis 2013 (2013), Article ID 852727, 6 pages, DOI: 10.1155/2013/852727.

M. Akram, A. A. Zafar and A. A. Siddiqui, A general class of contractions: A contractions, Novi Sad Journal of Mathematics 38(1) (2008), 25 – 33, URL: https://sites.dmi.uns.ac.rs/nsjom/Papers/38_1/NSJOM_38_1_025_033.pdf.

B. Alamri, T. Suzuki and L. A. Khan, Caristi’s fixed point theorem and Subrahmanyam’s fixed point theorem in v-generalized metric spaces, Journal of Function Spaces 2015 (2015), Article ID 709391, 6 pages, DOI: 10.1155/2015/709391.

A. Al-Bsoul, A. Fora and A. Bellour, Some properties of generalized metric space and fixed point theory, Matematychni Studii 33(1) (2010), 85 – 91, URL: http://matstud.org.ua/texts/2010/33_1/85-91.pdf.

N. Alharbi, H. Aydi, A. Felhi, C. Özel and S. Sahmim, α-contractive mappings on rectangular b-metric spaces and an application to integral equations, Journal of Mathematical Analysis 9(3) (2018), 47 – 60, URL: http://www.ilirias.com/jma/repository/docs/JMA9-3-5.pdf.

Z. Al-Muhaiameed, Z. Mostefaoui and M. Bousselsal, Coincidence and common fixed point theorems for (ψ,φ)-weakly contractive mappings in rectangular b-metric spaces, Electronic Journal of Mathematical Analysis and Applications 6(2) (2018), 211 – 220, DOI: 10.21608/EJMAA.2018.312562.

T. V. An, L. Q. Tuyen and N. V. Dung, Stone-type theorem on b-metric spaces and applications, Topology and its Applications 185–186 (2015), 50 – 64, DOI: 10.1016/j.topol.2015.02.005.

M. Arshad, J. Ahmad and E. Karapınar, Some common fixed point results in rectangular metric spaces, International Journal of Analysis 2013 (2013), Article ID 307234, 7 pages, DOI: 10.1155/2013/307234.

M. Asadi, E. Karapınar and A. Kumar, α-ψ-Geraghty contractions on generalized metric spaces, Journal of Inequalities and Applications 2014 (2014), Article number: 423, DOI: 10.1186/1029-242X-2014-423.

M. Asadi, E. Karapınar and P. Salimi, New extension of p-metric spaces with some fixed-point results on M-metric spaces, Journal of Inequalities and Applications 2014 (2014), Article number: 18, DOI: 10.1186/1029-242X-2014-18.

M. Asim and Meenu, Fixed point theorem via Meir-Keeler contraction in rectangular Mb-metric spaces, Korean Journal of Mathematics 30(1) (2022), 161 – 173, DOI: 10.11568/kjm.2022.30.1.161.

M. Asim, M. Imdad and S. Radenovic, Fixed point results in extended rectangular b-metric spaces with an application, UPB Scientific Bulletin, Series A 81(2) (2019), 11 – 20, URL: https://www.scientificbulletin.upb.ro/rev_docs_arhiva/fulldfa_249654.pdf.

M. Asim, M. Imdad and S. Shukla, Fixed point results for Geraghty-weak contractions in ordered partial rectangular b-metric spaces, Afrika Matematika 32 (2021), 811 – 827, DOI: 10.1007/s13370-020-00862-6.

M. Asim, S. Mujahid and I. Uddin, Meir-Keeler contraction in rectangular M-metric space, Topological Algebra and its Applications 9 (2021), 96 – 104, DOI: 10.1515/taa-2021-0106.

H. Aydi, A. Felhi, T. Kamran, E. Karapınar and M. U. Ali, On nonlinear contractions in new extended b-metric spaces, Applications and Applied mathematics 14(1) (2019), 537 – 547, URL: https://digitalcommons.pvamu.edu/aam/vol14/iss1/37.

H. Aydi, E. Karapınar and B. Samet, Fixed points for generalized (α-ψ)-contractions on generalized metric spaces, Journal of Inequalities and Applications 2014 (2014), Article number: 229, DOI: 10.1186/1029-242X-2014-229.

H. Aydi, E. Karapinar and H. Lakzian, Fixed point results on a class of generalized metric spaces, Mathematical Sciences 6 (2012), Article number: 46, DOI: 10.1186/2251-7456-6-46.

H. Aydi, Z. D. Mitrovic, S. Radenovic and M. de la Sen, On a common Jungck type fixed point result in extended rectangular b-metric spaces, Axioms 9(1) (2019), 4, DOI: 10.3390/axioms9010004.

A. Azam and M. Arshad, Kannan fixed point theorems on generalized metric spaces, The Journal of Nonlinear Sciences and its Applications, 1(1) (2008), 45 – 48, URL: https://www.emis.de/journals/TJNSA/article/TJNSA_07.pdf.

A. Azam, B. Fisher and M. Khan, Common fixed point theorems in complex valued metric spaces, Numerical Functional Analysis and Optimization 32(3) (2011), 243 – 253, DOI: 10.1080/01630563.2011.533046.

A. Azam, M. Arshad and I. Beg, Banach contraction principle on cone rectangular metric spaces, Applicable Analysis and Discrete Mathematics 3(2) (2009), 236 – 241, DOI: 10.2298/AADM0902236A.

M. Balaiah, A fixed point theorem in generalized metric spaces, International Journal of Scientific and Innovative Mathematical Research 3(12) (2015), 24 – 26, URL: https://www.arcjournals.org/pdfs/ijsimr/v3-i12/7.pdf.

S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundamenta Mathematicae 3(1) (1922), 133 – 181, URL: https://eudml.org/doc/213289.

P. Baradol and D. Gopal, Note on recent fixed point results in graphical rectangular b-metric spaces, Science & Technology Asia 25(4) (2020), 1 – 11, DOI: 10.14456/scitechasia.2020.44.

P. Baradol, J. Vujakovic, D. Gopal and S. Radenovic, On some new results in graphical rectangular b-metric spaces, Mathematics 8(4) (2020), 488, DOI: 10.3390/math8040488.

C. D. Bari and P. Vetro, Common fixed points in generalized metric spaces, Applied Mathematics and Computation 218(13) (2012), 7322 – 7325, DOI: 10.1016/j.amc.2012.01.010.

A. Beraž, H. Garai, B. Damjanovic and A. Chanda, Some interesting results on F-metric spaces, Filomat 33(10) (2019), 3257 – 3268, DOI: 10.2298/FIL1910257B.

J. D. Bhutia and K. Tiwary, Common fixed point on cone rectangular metric space, International Journal of Mathematics Trends and Technology 56(5) (2018), 335 – 343, URL: https://ijmttjournal.org/archive/ijmtt-v56p545.

R. Bianchini, Su un problema di S. Reich riguardante la teori dei punt i fessi, Boll. Un. Mat. Ital. 5 (1972), 103 – 108.

N. Bilgili, E. Karapınar and D. Turkoglu, A note on common fixed points for (ψ,α,β)-weakly contractive mappings in generalized metric spaces, Fixed Point Theory and Applications 2013 (2013), Article number: 287, DOI: 10.1186/1687-1812-2013-287

D. W. Boyd and J. S. W. Wong, On nonlinear contraction, Proceedings of the American Mathematical Society 20 (1969), 458 – 464, DOI: 10.1090/S0002-9939-1969-0239559-9.

A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, International Journal of Mathematics and Mathematical Sciences 29 (2002), Article ID 641824, 6 pages, DOI: 10.1155/S0161171202007524.

A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publicationes Mathematicae Debrecen 57(1-2) (2000), 31 – 37, URL: https://publi.math.unideb.hu/load_doc.php?p=617&t=pap.

L. Budhia, M. Kir, D. Gopal and H. Kiziltunç, New fixed point results in rectangular metric space and application to fractional calculus, Tbilisi Mathematical Journal 10(1) (2017), 91 – 104, DOI: 10.1515/tmj-2017-0006.

J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Transactions of the American Mathematical Society 215 (1976), 241 – 251, DOI: 10.2307/1999724.

C.-M. Chen, Common fixed-point theorems in complete generalized metric spaces, Journal of Applied Mathematics 2012 (2012), Article ID 945915, 14 pages, DOI: 10.1155/2012/945915.

C.-M. Chen and W. Y. Sun, Periodic points and fixed points for the weaker (φ,ψ)-contractive mappings in complete generalized metric spaces, Journal of Applied Mathematics 2012 (2012), Article ID 856974, 7 pages, DOI: 10.1155/2012/856974.

L. Chen, N. Yang and Y. Zhao, Fixed point theorems for the Mann’s iteration scheme in convex graphical rectangular b-metric spaces, Optimization 70 (2021), 1359 – 1373, DOI: 10.1080/02331934.2021.1887180.

Lj. B. Ciric, A generalization of Banach’s contraction principle, Proceedings of the American Mathematical Society 45 (1974), 267 – 273, DOI: 10.1090/S0002-9939-1974-0356011-2.

L. Ciric, A new fixed-point theorem for contractive mappings, Publications del’Institut Mathématique 30(44) (1981), 25 – 27, URL: https://www.emis.de/journals/PIMB/044/n044p025.pdf.

S. Czerwik, Contraction mappings in b-metric spaces, Acta Mathematica et Informatica Universitatis Ostraviensis 1(1) (1993), 5 – 11, URL: https://dml.cz/bitstream/handle/10338.dmlcz/120469/ActaOstrav_01-1993-1_2.pdf.

S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti del Seminario Matematico e Fisico dell’Università di Modena 46 (1998), 263 – 276.

P. Das, A fixed point theorem on a class of generalized metric space, Korean Journal of Mathematical Science 9 (2002), 29 – 33.

P. Das and L. Dey, Fixed point of contractive mappings in generalized metric spaces, Mathematica Slovaca 59(4) (2009), 499 – 504, DOI: 10.2478/s12175-009-0143-2.

P. Das and L. K. Dey, A fixed point theorem in generalized metric spaces, Soochow Journal of Mathematics 33(1) (2007), 33 – 39.

A. Das and T. Bag, A generalization to parametric metric spaces, International Journal of Nonlin-ear Analysis and Applications 14(1) (2023), 229 – 244, DOI: 10.22075/IJNAA.2022.26832.3420.

A. Das and T. Bag, A study on parametric S-metric spaces, Communications in Mathematics and Applications 13(3) (2022), 921 – 933, DOI: 10.26713/cma.v13i3.1789.

A. Das and T. Bag, Some fixed point theorems in extended cone b-metric spaces, Communications in Mathematics and Applications 13(2) (2022), 1 – 13, DOI: 10.26713/cma.v13i2.1768.

M. B. Devi, B. Khomdram and Y. Rohen, Fixed point theorems of generalised α-rational contractive mappings on rectangular b-metric spaces, Journal of Mathematical and Computational Science 11(1) (2021), 991 – 1010, DOI: 10.28919/jmcs/5255.

B. C. Dhage, Generalized metric spaces mappings with fixed point, Bulletin of Calcutta Mathematical Society 84 (1992), 329 – 336.

W. S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Analysis: Theory, Methods & Applications 72(5) (2010), 2259 – 2261, DOI: 10.1016/j.na.2009.10.026.

N. V. Dung, A new approach to fixed point theorems in compact 2-generalized metric spaces, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 115 (2021), Article number: 30, DOI: 10.1007/s13398-020-00972-7.

N. V. Dung, The metrization of rectangular b-metric spaces, Topology and its Applications 261 (2019), 22 – 28, DOI: 10.1016/j.topol.2019.04.010.

N. V. Dung and V. T. L. Hang, A note on partial rectangular metric spaces, Mathematica Moravica 18(1) (2014), 1 – 8, URL: http://elib.mi.sanu.ac.rs/files/journals/mm/25/Math.%20Moravican25p1-8.pdf.

N. V. Dung and V. T. L. Hang, On the metrization problem of v-generalized metric spaces, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 112 (2018), 1295 – 1303, DOI: 10.1007/s13398-017-0425-4.

P. N. Dutta and B. S. Choudhury, A generalisation of contraction principle in metric spaces, Fixed Point Theory and Applications 2008 (2008), Article number: 406368, DOI: 10.1155/2008/406368.

M. Edelstein, An extension of Banach’s contraction principle, Proceedings of the American Mathematical Society 12 (1961), 7 – 10, DOI: 10.1090/S0002-9939-1961-0120625-6.

O. Ege, Complex valued rectangular b-metric spaces and an application to linear equations, Journal of Nonlinear Sciences and Applications 8(6) (2015), 1014 – 1021, DOI: 10.22436/jnsa.008.06.12.

˙I. M. Erhan, E. Karapınar and T. Sekulic, Fixed points of (ψ,φ)-contractions on rectangular metric spaces, Fixed Point Theory and Applications 2012 (2012), Article number: 138, DOI: 10.1186/1687-1812-2012-138.

B. Fisher, Related fixed point on two metric spaces, Mathematics Seminar Notes – Kobe University 10 (1983), 17 – 26, URL: http://www.math.kobe-u.ac.jp/jmsj/kjm/s10.html.

A. Fora, A. Bellour and A. Al-Bsoul, Some results in fixed point theory concerning generalized metric spaces, Matematiˇcki Vesnik 61(3) (2009), 203 – 208, URL: http://www.vesnik.math.rs/landing.php?p=mv093.cap&name=mv09303.

M. M. Fréchet, Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo 22 (1906), 1 – 72, DOI: 10.1007/BF03018603.

S. P. Franklin, Spaces in which sequences suffice, Fundamenta Mathematicae 57(1) (1965), 107 – 115, URL: https://eudml.org/doc/213854.

G. Gadkari, M. S. Rathore and N. Singh, Some common fixed point theorems for weakly compatible mappings in complex valued rectangular metric space, International Journal of Innovation in Science and Mathematics 7(1) (2019), 1 – 18, URL: https://www.ijism.org/administrator/components/com_jresearch/files/publications/IJISM_801_FINAL.pdf.

S. Gähler, 2-metrische Räume und ihre topologische Struktur, Mathematische Nachrichten 26 (1-4) (1963), 115 – 148, DOI: 10.1002/mana.19630260109.

R. George and K. P. Reshma, Common coupled fixed points of some generalised T-contractions in rectangular b-metric space and application, Advances in Fixed Point Theory 10 (2020), Article ID 18, DOI: 10.28919/afpt/4873.

R. George, H. A. Nabwey, K. P. Reshma and R. Rajagopalan, Generalized cone b-metric spaces and contraction principles, Matematioki Vesnik 67(4) (2015), 246 – 257, URL: https://www.emis.de/journals/MV/154/mv15402.pdf.

R. George, S. Radenovic, K. P. Reshma and S. Shukla, Rectangular b-metric space and contraction principles, Journal of Nonlinear Sciences and Applications 8(6) (2015), 1005 – 1013, DOI: 10.22436/jnsa.008.06.11.

M. A. Geraghty, On contractive mappings, Proceedings of the American Mathematical Society 40 (1973), 604 – 608, DOI: 10.2307/2039421.

G. Gruenhage, Generalized metric spaces, Chapter 10, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (editors), North-Holland, 423 – 501 (1984), DOI: 10.1016/B978-0-444-86580-9.50013-6.

F. Hausdorff, Grundzuge der Mengenlehre (Fundamentals of Set Theory), Veit and Company, Leipzig, viii + 476 pages (1914).

L. G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, Journal of Mathematical Analysis and Applications 332(2) (2007), 1468 – 1476, DOI: 10.1016/j.jmaa.2005.03.087.

N. Hussian and M. H. Shah, KKM mappings in cone b-metric spaces, Computers & Mathematics with Applications 62(4) (2011), 1677 – 1684, DOI: 10.1016/j.camwa.2011.06.004.

N. Hussain, S. Khaleghizadeh, P. Salimi and A. A. N. Abdou, A new approach to fixed point results in triangular intuitionistic fuzzy metric spaces, Abstract and Applied Analysis 2014 (2014), Article ID 690139, 16 pages, DOI: 10.1155/2014/690139.

J. Jachymski, Equivalent conditions and the Meir-Keeler type theorems, Journal of Mathematical Analysis and Applications 194(1) (1995), 293 – 303, DOI: 10.1006/jmaa.1995.1299.

S. Jain and P. Chaubey, Contraction principle in rectangular M-metric spaces with a binary relation, Advances in Mathematics: Scientific Journal, 9(12) (2020), 10171 – 10179, DOI: 10.37418/amsj.9.12.9.

S. Jain and S. Jain, Some results on a cone rectangular metric space, Jordan Journal of Mathematics and Statistics 8(3) (2015), 239 – 255.

M. Jleli and B. Samet, A new generalization of the Banach contraction principle, Journal of Inequalities and Applications 2014 (2014), Article number: 38, DOI: 10.1186/1029-242X-2014-38.

M. Jleli and B. Samet, On a new generalization of metric spaces, Journal of Fixed Point Theory and Applications 20 (2018), Article number: 128, DOI: 10.1007/s11784-018-0606-6.

M. Jleli and B. Samet, The Kannan fixed point theorem in a cone rectangular metric space, Journal of Nonlinear Sciences and Applications 2(3) (2009), 161 – 167, DOI: 10.22436/jnsa.002.03.03.

M. Jleli, E. Karapınar and B. Samet, Further generalizations of the Banach contraction principle, Journal of Inequalities and Applications 2014 (2014), Article number: 439, DOI: 10.1186/1029-242X-2014-439.

G. Jungck, Compatible mappings and common fixed points, International Journal of Mathematics and Mathematical Sciences 9 (1986), Article ID 531318, 9 pages, DOI: 10.1155/S0161171286000935.

G. Jungck and B. E. Rhoades, Fixed points for set valued functions without continuity, Indian Journal of Pure and Applied Mathematics 29(3) (1998), 227 – 238.

Z. Kadelburg and S. Radenovic, Pata-type common fixed point results in b-metric and b-rectangular metric spaces, Journal of Nonlinear Sciences and Applications 8(6) (2015), 944 – 954, DOI: 10.22436/jnsa.008.06.05.

Z. Kadelburg and S. Radenovic, Fixed point results in generalized metric spaces without Hausdorff property, Mathematical Sciences 8 (2014), Article number: 125, DOI: 10.1007/s40096-014-0125-6.

Z. Kadelburg and S. Radenovic, On generalized metric spaces: A survey, Turkic World Mathematical Society (TWMS) Journal of Pure and Applied Mathematics 5(1) (2014), 3 – 13, URL: http://www.twmsj.az/Abstract.aspx?Id=114.

Z. Kadelburg, S. Radenovic and V. Rakocevic, A note on the equivalence of some metric and cone metric fixed point results, Applied Mathematics Letters 24(3) (2011), 370 – 374, DOI: 10.1016/j.aml.2010.10.030.

O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12(3) (1984), 215 – 229, DOI: 10.1016/0165-0114(84)90069-1.

R. Kannan, Some results on fixed points – II, The American Mathematical Monthly 76(4) (1969), 405 – 408, DOI: 10.2307/2316437.

E. Karapınar, Discussion on (α,ψ)-contractions in generalized metric spaces, Abstract and Applied Analysis 2014 (2014), Article ID 962784, 7 pages, DOI: 10.1155/2014/962784.

E. Karapınar, Fixed point results for α-admissible mapping of integral type on generalized metric space, Abstract and Applied Analysis 2015 (2015), Article ID 141409, 11 pages, DOI: 10.1155/2015/141409.

M. A. Khamsi and N. Hussain, KKM mappings in metric type spaces, Nonlinear Analysis: Theory, Methods & Applications 73(9) (2010), 3123 – 3129, DOI: 10.1016/j.na.2010.06.084.

L. Kikina and K. Kikina, A fixed point theorem in generalized metric spaces, Demonstratio Mathematica 46 (2013), 181 – 190, DOI: 10.1515/dema-2013-0432.

L. Kikina and K. Kikina, Fixed point theorems on generalized metric spaces for mappings in a class of almost φ-contractions, Demonstratio Mathematica 48(3) (2015), 440 – 451, DOI: 10.1515/dema-2015-0031.

L. Kikina and K. Kikina, Fixed points on two generalized metric spaces, International Journal of Mathematical Analysis 5(30) (2011), 1459 – 1467.

L. Kikina and K. Kikina, On fixed point of a Ljubomir Ciric quasi-contraction mapping in generalized metric spaces, Publications Mathematicae – Debrecen 83(3) (2013), 353 – 358, URL: https://publi.math.unideb.hu/load_doi.php?pdoi=10_5486_PMD_2013_5528.

M. Kir and H. Kiziltunc, On some well known fixed point theorems in b-metric spaces, Turkish Journal of Analysis and Number Theory 1(1) (2013), 13 – 16, DOI: 10.12691/tjant-1-1-4.

W. A. Kirk and N. Shahzad, Generalized metrics and Caristi’s theorem, Fixed Point Theory and Applications 2013 (2013), Article number: 129, DOI: 10.1186/1687-1812-2013-129.

M. Kuczma, B. Choczewski and R. Ger, Iterative Functional Equations, Cambridge University Press, Cambridge, UK (1990), DOI: 10.1017/CBO9781139086639.

P. Kumam and N. V. Dung, Some remarks on generalized metric spaces of Branciari, Sarajevo Journal of Mathematics 10(23) (2014), 209 – 219, DOI: 10.5644/SJM.10.2.07.

M. Kumar, P. Kumar and S. Kumar, Some common fixed point theorems in generalized metric spaces, Journal of Mathematics 2013 (2013), Article ID 719324, 7 pages, DOI: 10.1155/2013/719324.

M. Kumar, S. Araci and P. Kumam, Fixed point theorems for generalized (α-ψ)-expansive mappings in generalized metric space, Communication in Mathematics and Applications 7(3) (2016), 227 – 240, DOI: 10.26713/cma.v7i3.431.

M. Kumar, S. Araci, A. Dahiya, A. Rani and P. Singh, Common fixed point for generalized-(ψ-α-β)-weakly contractive mappings in generalized metric space, Global Journal of Pure and Applied Mathematics 12 (2016), 3021 – 3035.

B. K. Lahiri and P. Das, Fixed point of a Ljubomir Ciric’s quasi-contraction mapping in a generalized metric space, Publicationes Mathematicae Debrecen 61(3-4) (2002), 589 – 594, DOI: 10.5486/PMD.2002.2677.

H. Lakzian and B. Samet, Fixed points for (ψ,ϕ)-weakly contractive mapping in generalized metric spaces, Applied Mathematics Letters 25(5) (2012), 902 – 906, DOI: 10.1016/j.aml.2011.10.047.

S. K. Malhotra, J. B. Sharma and S. Shukla, g-Weak contraction in ordered cone rectangular metric spaces, The Scientific World Journal, 2013 (2013), Article ID 810732, DOI: 10.1155/2013/810732.

S. K. Malhotra, S. Shukla and R. Sen, Some fixed point theorems for ordered Reich type contractions in cone rectangular metric spaces, Acta Mathematica Universitatis Comenianae 82(2) (2013), 165 – 175, URL: http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/iew/737/492.

S. G. Mathews, Partial metric topology, Annals of the New York Academy of Sciences 728(1) (1994), 183 – 197, DOI: 10.1111/j.1749-6632.1994.tb44144.x.

J. Matkowski, Fixed point theorems for contractive mappings in metric spaces (English), Casopis pro pestování matematiky 105(4) (1980), 341 – 344, URL: https://dml.cz/handle/10338.dmlcz/108246.

G. Meena, Best proximity and fixed point results in complex valued rectangular metric spaces, Global Journal of Pure and Applied Mathematics 14 (2018), 689 – 698.

D. Mihe¸t, On Kannan fixed point principle in generalized metric spaces, Journal of Nonlinear Sciences and Applications 2(2) (2009), 92 – 96, DOI: 10.22436/jnsa.002.02.03.

Z. D. Mitrovic and S. Radenovic, The Banach and Reich contractions in bv(s)-metric spaces, Journal of Fixed Point Theory and Applications 19 (2017), 3087 – 3095, DOI: 10.1007/s11784-017-0469-2.

Z. D. Mitrovic, On an open problem in rectangular b-metric space, The Journal of Analysis 25 (2017), 135 – 137, DOI: 10.1007/s41478-017-0036-7.

Z. D. Mitrovic, R. George and N. Hussain, Some remarks on contraction mappings in rectangular b-metric spaces, Boletim da Sociedade Paranaense de Matemática 39(6) (2021), 147 – 155, DOI: 10.5269/bspm.41754.

Z. Mustafa and B. Sims, A new approach to generalized metric spaces, Journal of Nonlinear and Convex Analysis 7 (2006), 289 – 297, URL: https://carma.edu.au/brailey/Research_papers/A%20new%20Approach%20to%20Generalized%20Metric%20Spaces.pdf.

Z. Mustafa, H. Obiedat and F. Awawdeh, Some fixed point theorem for mapping on complete G-metric spaces, Fixed Point Theory and Applications 2008 (2008), Article number: 189870, DOI: 10.1155/2008/189870.

Z. Mustafa, V. Parvaneh, M. M. M. Jaradat and Z. Kadelburg, Extended rectangular b-metric spaces and some fixed point theorems for contractive mappings, Symmetry 11(4) (2019), 594, DOI: 10.3390/sym11040594.

M. Nazam, A. Arif, H. Mahmood and S. O. Kim, Fixed point problems in cone rectangular metric spaces with applications, Journal of Function Spaces 2020 (2020), Article ID 8021234, DOI: 10.1155/2020/8021234.

A. Ninsri and W. Sintunavarat, Fixed point theorems for partial α-φ contractive mappings in generalized metric spaces, Journal of Nonlinear Science and Applications 9 (2016), 83 – 91, URL: http://www.kurims.kyoto-u.ac.jp/EMIS/journals/TJNSA/includes/files/articles/Vol9_Iss1_83--91_Fixed_point_theorems_for_partial_al.pdf.

B. Nurwahyu, M. S. Khan, N. Fabiano and S. Radenovic, Common fixed point on generalized weak contraction mappings in extended rectangular b-metric spaces, Filomat 35(11) (2021), 3621 – 3633, URL: https://www.pmf.ni.ac.rs/filomat-content/2021/35-11/35-11-6-14382.pdf.

J. O. Olaleru and B. Samet, Some fixed point theorems in conerectangular metric spaces, Journal of the Nigerian Mathematical Society 33 (2014), 145 – 158, URL: https://ojs.ictp.it/jnms/index.php/jnms/article/view/713/181.

N. Y. Özgür, N. Mlaiki, N. Ta¸s and N. Souayah, A new generalization of metric spaces: rectangular M-metric spaces, Mathematical Sciences 12 (2018), 223 – 233, DOI: 10.1007/s40096-018-0262-4.

V. Parvaneh, F. Golkarmanes and R. George, Fixed points of Wardowski-Ciri c-Presic type contractive mappings in a partial rectangular b-metric space, Journal of Mathematical Analysis 8(1) (2017), 183 – 201, URL: http://www.ilirias.com/jma/repository/docs/JMA8-1-15.pdf.

S. R. Patil and J. N. Salunke, Common fixed point theorems in complex valued rectangular metric spaces, South Asian Journal of Mathematics 6(1) (2016), 10 – 23, URL: http://www.sajm-online.com/uploads/sajm6-1-2.pdf.

S. Patil and J. Salunke, Fixed point theorems for expansion mappings in cone rectangular metric spaces, General Mathematics Notes 29(1) (2015), 30 – 39.

J. Patil, B. Hardan, A. A. Hamoud, A. Bachhav and H. Günerhan, Generalization contractive mappings on rectangular b-metric space, Advances in Mathematical Physics 2022 (2022), Article ID 7291001, 10 pages, DOI: 10.1155/2022/7291001.

V. Popa, Fixed points on two complete metric spaces, Zb. Rad. Prirod. – Mat. Fak. (N.S.) Ser. Mat. 21 (1991), 83 – 93, URL: https://www.emis.de/journals/NSJOM/Papers/21_1/NSJOM_21_1_083_093.pdf.

K. Rana and A. K. Garg, Kannan-type fixed point results in extended rectangular b-metric spaces, Advances in Mathematics: Scientific Journal 9(8) (2020), 5491 – 5499, DOI: 10.37418/amsj.9.8.19.

M. Rangamma and P. M. Reddy, A common fixed point theorem for four self maps in cone rectangular metric space under Kannan type contractions, International Journal of Pure and Applied Mathematics 103(2) (2015), 281 – 293, DOI: 10.12732/ijpam.v103i2.13.

M. Rangamma and P. M. Reddy, A common fixed point theorem for three self maps in cone rectangular metric space, Asian Journal of Fuzzy and Applied Mathematics 3(2) (2015), 62 – 69, URL: https://ajouronline.com/index.php/AJFAM/article/view/2379/1369.

R. A. Rashwan and S. M. Saleh, Some fixed point theorems in cone rectangular metric spaces, Mathematica Aeterna 2(6) (2012), 573 – 587, URL: https://www.longdom.org/articles-pdfs/some-fixed-point-theorems-in-cone-rectangular-metric-spaces.pdf.

S. Rathee, K. Dhingra and A. Kumar, Various contractions in generalized metric space, Boletim da Sociedade Paranaense de Matemática 39(4) (2021), 111 – 130, DOI: 10.5269/bspm.41092.

M. P. Reddy and M. Rangamma, A unique common fixed point theorem for four self maps under Reich type contractive conditions in cone rectangular metric space, Journal of Advanced Studies in Topology 6(4) (2015), 143 – 151, URL: http://www.m-sciences.com/index.php/jast/article/view/186.

S. Reich, Kannan’s fixed point theorem, Boll. Un. Mat. Ital. 4(4) (1971), 1 – 11.

Sh. Rezapour, M. Derafshpour and R. Hamlbarani, A review on topological properties of cone metric spaces, in: Proceedings of the International Conference on Analysis, Topology and Applications, Vol. 13 (2008), 163 – 171.

V. L. Rosa and P. Vetro, Common fixed points for α-ψ-φ-contractions in generalized metric space, Nonlinear Analysis: Modeling and Control 19(1) (2014), 43 – 54, DOI: 10.15388/NA.2014.1.3.

B. Samet and C. Vetro, A fixed point theorem for uniformly locally contractive mappings in a c-chainable cone rectangular metric space, Surveys in Mathematics and its Applications 6 (2011), 107 – 116, URL: https://www.utgjiu.ro/math/sma/v06/p07.pdf.

B. Samet, A fixed point theorem in a generalized metric space for mappings satisfying a contractive condition of integral type, International Journal of Mathematical Analysis 3 (2009), 1265 – 1271.

B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Analysis: Theory, Methods & Applications 75(4) (2012), 2154 – 2165, DOI: 10.1016/j.na.2011.10.014.

B. Samet, Discussion on “A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces” by A. Branciari, Publicationes Mathematicae Debrecen 76(4) (2010), 493 – 494, DOI: 10.5486/PMD.2010.4595.

K. Sarkar, Rectangular partial b-metric spaces, Journal of Mathematical and Computational Science 10(6) (2020), 2754 – 2768, DOI: 10.28919/jmcs/4995.

S. Sedghi, D. Turkoglu, N. Shobe and S. Sedghi, Common fixed point theorems for six weakly compatible mappings in D∗-metric spaces, Thai Journal of Mathematics 7(2) (2009), 381 – 391.

S. Sedghi, N. Shobe and A. Aliouche, A generalization of fixed point theorems in S-metric spaces, Matematiqki Vesnik 64(3) (2012), 258 – 266, URL: https://www.emis.de/journals/MV/123/mv12309.pdf.

S. E. Setiawan, Sunarsini and Sadjidon, Completeness and fixed point theorem in cone rectangular metric spaces, Journal of Physics: Conference Series 1490(1) (2020), 012009, DOI: 10.1088/1742-6596/1490/1/012009

M. S. Shagari and N. Ullah, Fixed point results in complex valued rectangular extended b-metric spaces with applications, Mathematical Analysis and Convex Optimization 1(2) (2020), 109 – 122, DOI: 10.29252/maco.1.2.11.

I. R. Sharma, J. M. Rao and S. S. Rao, Contractions over generalized metric spaces, Journal of Nonlinear Sciences and Applications 2(3) (2009), 180 – 182, DOI: 10.22436/jnsa.002.03.06.

S. Shukla, G-(F,τ)-contraction in partial rectangular metric spaces endowed with a graph and fixed point theorems, TWMS Journal of Applied and Engineering Mathematics 6 (2016), 342 – 353, URL: http://jaem.isikun.edu.tr/web/images/articles/vol.6.no.2/17.pdf.

S. Shukla, Partial rectangular metric spaces and fixed point theorems, The Scientific World Journal 2014 (2014), Article ID 756298, 7 pages, DOI: 10.1155/2014/756298.

D. Singh, O. P. Chauhan, N. A. Singh and V. Joshi, Complex valued rectangular metric spaces and common fixed point theorems, Bulletin of Mathematical Analysis and Applications 7(2) (2015), 1 – 13, URL: http://emis.icm.edu.pl/journals/BMAA/repository/docs/BMAA7-2-1.pdf.

W. Sintunavarat and P. Kumam, Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces, Journal of Applied Mathematics 2011 (2011), Article ID 637958, 14 pages, DOI: 10.1155/2011/637958.

A. H. Soliman, M. A. Ahmed and A. M. Zidan, A new contribution to the fixed point theory in b-generalized metric spaces, Journal of Advanced Studies in Topology 8(1) (2017), 111 – 116, URL: https://www.m-sciences.com/index.php/jast/article/view/229.

C. Suanoom, W. Khuangsatung and T. Bantaojai, On an open problem in complex valued rectangular b-metric spaces with an application, Science and Technology Asia 27(2) (2022), 78 – 83, URL: https://ph02.tci-thaijo.org/index.php/SciTechAsia/article/view/241485/.

P. V. Subrahmanyam, Remarks on some fixed point theorems related to Banach’s contraction principle, Journal of Mathematical and Physical Sciences 8 (1974), 445 – 457.

Z. Sun and M. Song, Common fixed point theorems in complex valued generalized metric spaces, Journal of Mathematical and Computational Science 7(4) (2017), 739 – 754, URL: https://scik.org/index.php/jmcs/article/view/3033.

Sunarsini, A. Biahdillah and S. D. Surjanto, Application of Banach contraction principle in complex valued rectangular b-metric space, Journal of Physics: Conference Series 1490 (2020), 012003, DOI: 10.1088/1742-6596/1490/1/012003.

T. Suzuki, Another generalization of Edelstein’s fixed point theorem in generalized metric spaces, Linear Nonlinear Analysis 2 (2016), 271 – 279.

T. Suzuki, Completeness of 3-generalized metric spaces, Filomat 30(13) (2016), 3575 – 3585, DOI: 10.2298/FIL1613575S.

T. Suzuki, Every generalized metric space has a sequentially compatible topology, Linear and Nonlinear Analysis 3(3) (2017), 393 – 399, URL: http://www.yokohamapublishers.jp/online-p/LNA/vol3/lnav3n3p393.pdf.

T. Suzuki, Generalized metric spaces do not have the compatible topology, Abstract and Applied Analysis 2014 (2014), Article ID 458098, 5 pages, DOI: 10.1155/2014/458098.

T. Suzuki, Meir-Keeler contractions of integral type are still Meir-Keeler contractions, International Journal of Mathematics and Mathematical Sciences 2007 (2007), Article ID 039281, 6 pages, DOI: 10.1155/2007/39281.

T. Suzuki, Nadler’s fixed point theorem in v-generalized metric spaces, Fixed Point Theory and Applications 2017 (2017), Article number: 18, DOI: 10.1186/s13663-017-0611-2.

T. Suzuki, Numbers on diameter in n-generalized metric spaces, Bulletin of the Kyushu Institute of Technology – Pure and Applied Mathematics 63 (2016), 1 – 13.

T. Suzuki, Several completeness on v-generalized metric spaces, Bulletin of the Kyushu Institute of Technology – Pure and Applied Mathematics 67 (2020), 29 – 42.

T. Suzuki, Some comments on Edelstein’s fixed point theorems in v-generalized metric spaces, Bulletin of the Kyushu Institute of Technology – Pure and Applied Mathematics 65 (2018), 23 – 42.

T. Suzuki, Some metrization problem on v-generalized metric spaces, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 113 (2019), 1267 – 1278, DOI: 10.1007/s13398-018-0544-6.

T. Suzuki, The strongest sequentially compatible topology on a ν-generalized metric space, Journal of Nonlinear and Variational Analysis 1(3) (2017), 333 – 343, URL: http://jnva.biemdas.com/issues/JNVA2017-3-6.pdf.

T. Suzuki, The strongly compatible topology on v-generalized metric spaces, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 112 (2018), 301 – 309, DOI: 10.1007/s13398-017-0380-0.

T. Suzuki and C. Vetro, Three existence theorems for weak contractions of Matkowski type, International Journal of Mathematics and Statistics 6 (2010), S110 – S120, URL: http://www.ceser.in/ceserp/index.php/ijms/article/view/2718.

T. Suzuki, B. Alamri and L. A. Khan, Some notes on fixed point theorems in v-generalized metric spaces, Bulletin of the Kyushu Institute of Technology – Pure and Applied Mathematics 62 (2015), 15 – 23.

T. Suzuki, B. Alamri and M. Kikkawa, Edelstein’s fixed point theorem in generalized metric spaces, Journal of Nonlinear and Convex Analysis 16 (2015), 2301 – 2309, URL: http://www.yokohamapublishers.jp/online-p/JNCA/vol16/jncav16n11p2301.pdf.

T. Suzuki, B. Alamri and M. Kikkawa, Only 3-generalized metric spaces have a compatible symmetric topology, Open Mathematics 13 (2015), 510 – 517, DOI: 10.1515/math-2015-0048.

N. Ta¸s and N. Y. Özgür, On parametric S-metric spaces and fixed-point type theorems for expansive mappings, Journal of Mathematics 2016 (2016), Article ID 4746732, 6 pages, DOI: 10.1155/2016/4746732.

F. Tchier, C. Vetro and F. Vetro, A coincidence-point problem of Perov type on rectangular cone metric spaces, Journal of Nonlinear Sciences and Applications 10(8) (2017), 4307 – 4317, DOI: 10.22436/jnsa.010.08.25.

N. Turan and M. Ba¸sarir, A note on quasi-statistical convergence of order α in rectangular cone metric space, Konuralp Journal of Mathematics 7(1) (2019), 91 – 96, URL: https://dergipark.org.tr/tr/download/article-file/697963.

F. Vetro and S. Radenovic, Some results of Perov type in rectangular cone metric spaces, Journal of Fixed Point Theory and Applications 20 (2018), 41, DOI: 10.1007/s11784-018-0520-y.

Z. Xue, G. Lv and F. Zhang, On fixed point for generalized Boyd-Wong type contractions in Branciari distance spaces, Journal of Mathematical Analysis 12(1) (2021), 48 – 55, URL: http://www.ilirias.com/jma/repository/docs/JMA12-1-5.pdf.

M. Younis, D. Singh and L. Shi, Revisiting graphical rectangular b-metric spaces, Asian-European Journal of Mathematics 15(4) (2022), 2250072, DOI: 10.1142/S1793557122500723.

P. Zangenehmehr, A. Farajzadeh, R. Lashkaripour and A. Karamian, On fixed point theory for generalized contractions in cone rectangular metric spaces via scalarizing, Thai Journal of Mathematics 15 (2017), 33 – 45.

M. Zare and P. Torabian, Fixed points for weak contraction mappings in complete generalized metric space, Journal of Mathematical Extension 8(3) (2014), 49 – 58, URL: https://ijmex.com/index.php/ijmex/article/viewFile/209/165.

D. Zheng, P. Wang and N. Citakovic, Meir-Keeler theorem in b-rectangular metric spaces, Journal of Nonlinear Sciences and Applications 10(4) (2017), 1786 – 1790, DOI: 10.22436/jnsa.010.04.39.

Downloads

Published

18-09-2023
CITATION

How to Cite

Das, A. ., & Bag, T. (2023). A Survey on Branciari Metric Spaces. Communications in Mathematics and Applications, 14(2), 1051–1111. https://doi.org/10.26713/cma.v14i2.2083

Issue

Section

Review Article