Study of Mappings and Metric Spaces in Fixed Point Theory: A Review

Authors

DOI:

https://doi.org/10.26713/cma.v15i3.2674

Keywords:

Fixed points, Mappings, Metric spaces

Abstract

Fixed point theory is a rich, fascinating, and intriguing subject of mathematics. Though it is a completely established field of research, but very fresh. It has been a hot topic of research since its inception. In this work, we present an overview of the major branches of fixed-point theory along with the key results. The focus of this article is to investigate different types of mappings and their variants. Furthermore, a survey of several extensions of metric space has also been carried out.

Downloads

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.

T. Abdeljawad, N. Mlaiki, H. Aydi and N. Souayah, Double controlled metric type spaces and some fixed point results, Mathematics 6(12) (2018), 320, DOI: 10.3390/math6120320.

Y. I. Alber and S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert spaces, in: New Results in Operator Theory and Its Applications, Operator Theory: Advances and Applications, I. Gohberg and Y. Lyubich (editors), Vol. 98, Birkhäuser, Basel, Switzerland, DOI: 10.1007/978-3-0348-8910-0_2.

M. A. Alghamdi, N. Hussain and P. Salimi, Fixed point and coupled fixed point theorems on b-metric-like spaces, Journal of Inequalities and Applications 2013 (2013), Article number: 402, DOI: 10.1186/1029-242X-2013-402.

M. A. Al-Thagafi and N. Shahzad, Generalized I-nonexpansive selfmaps and invariant approximations, Acta Mathematica Sinica, English Series 24 (2008), 867 – 876, DOI: 10.1007/s10114-007-5598-x.

A. Amini-Harandi, Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory and Applications 2012 (2012), Article number: 204, DOI: 10.1186/1687-1812-2012-204.

H. Aydi, A. Felhi and H. Afshari, New Geraghty type contractions on metric-like spaces, Journal of Nonlinear Science and Applications 10(2) (2017), 780 – 788, DOI: 10.22436/jnsa.010.02.38.

I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Functional Analysis 30 (1989), 26 – 37.

S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundamenta Mathematicae 3(1) (1922), 133 – 181, DOI: 10.4064/fm-3-1-133-181.

A. E. Bashirov, E. M. Kurpınar and A. Özyapıcı, Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications 337(1) (2008), 36 – 48, DOI: 10.1016/j.jmaa.2007.03.081.

R. M. T. Bianchini, Su un problema di S. Reich riguardonte la teoria dei punti fissi, Bollettino dell’Unione Matematica Italiana 5 (1972), 103 – 108.

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, DOI: 10.5486/pmd.2000.2133.

L. E. J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Mathematische Annalen 71 (1911), 97 – 115, DOI: 10.1007/BF01456931.

J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Transactions of the American Mathematical Society 215 (1976), 241 – 251, DOI: 10.1090/S0002-9947-1976-0394329-4.

B. K. Dass and S. Gupta, An extension of Banach contraction principle through rational expression, Indian Journal of Pure and Applied Mathematics 6(12) (1975), 1455 – 1458.

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.

I. Ekeland, On the variational principle, Journal of Mathematical Analysis and Applications 47(2) (1974), 324 – 353, DOI: 10.1016/0022-247X(74)90025-0.

M. M. Fréchet, Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo (1884-1940) 22 (1906), 1 – 72, DOI: 10.1007/BF03018603.

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

K. Goebel, A coincidence theorem for three system of transformations, Bulletin L’Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques 16 (1968), 733 – 735.

D. S. Jaggi, Some unique fixed point theorems, Indian Journal of Pure and Applied Mathematics 8(2) (1977), 223 – 230.

J. A. Jiddah, M. S. Shagari, M. Noorwali, S. Kanwal, H. Aydi and M. De La Sen, Fixed point results of Jaggi-Suzuki-type hybrid contractions with applications, Journal of Inequalities and Applications 2023 (2023), Article number: 46, DOI: 10.1186/s13660-023-02947-9.

M. Jleli and B. Samet, A generalized metric space and related fixed point theorems, Fixed Point Theory and Applications 2015 (2015), Article number: 61, DOI: 10.1186/s13663-015-0312-7.

G. Jungck, Commuting mappings and fixed points, The American Mathematical Monthly 83(4) (1976), 261 – 263, DOI: 10.2307/2318216.

G. Jungck, Compatible mappings and common fixed points, International Journal of Mathematics and Mathematical Sciences 9(4) (1986), 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.

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.

T. Kamran, M. Samreen and Q. Ul Ain, A generalization of b-metric space and some fixed point theorems, Mathematics 5(2) (2017), 19, DOI: 10.3390/math5020019.

R. Kannan, Some results on fixed points–II, The American Mathematical Monthly 76(4) (1969), 405 – 408, DOI: 10.1080/00029890.1969.12000228.

E. Karapınar and A. Fulga, Discussion on the hybrid Jaggi-Meir-Keeler type contractions, AIMS Mathematics 7(7) (2022), 12702 – 12717, DOI: 10.3934/math.2022703.

E. Karapınar and A. Fulga, New hybrid contractions on b-metric spaces, Mathematics 7(7) (2019), 578, DOI: 10.3390/math7070578.

E. Karapınar and F. Khojasteh, Super metric spaces, Filomat 36(10) (2022), 3545 – 3549, DOI: 10.2298/FIL2210545K.

E. Karapınar, Revisiting the Kannan type contractions via interpolation, Advances in the Theory of Nonlinear Analysis and its Application 2(2) (2018), 85 – 87, DOI: 10.31197/atnaa.431135.

F. Khojasteh, S. Shukla and S. Radenovi´c, A new approach to the study of fixed point theory for simulation functions, Filomat 29(6) (2015), 1189 – 1194, DOI: 10.2298/FIL1506189K.

G. N. V. Kishore, K. P. R. Rao, H. IsIk, B. S. Rao and A. Sombabu, Covarian mappings and coupled fixed point results in bipolar metric spaces, International Journal of Nonlinear Analysis and Applications 12(1) (2021), 1 – 15, DOI: 10.22075/ijnaa.2021.4650.

S. Kumar, S. Vashistha and B. Singh, Compatible mappings and its variants for generalized ψ-;-weak contraction condition, Journal of Mathematical and Computational Science 10(6) (2020), 2686 – 2712, DOI: 10.28919/jmcs/4840.

F. Hausdorff, Grundzüge der Mengenlehre, Vol. 7, Verlag von Veit & Company, Leipzig (1914).

P. Hitzler and A. K. Seda, Dislocated topologies, Journal of Electrical Engineering 51 (2000), 3 – 7.

J. Liouville, Solution d’un Problème d’Analyse, Journal de Mathématiques Pures et Appliquées (1837), 1 – 2, URL: http://eudml.org/doc/234990.

Y. Liu, J. Wu and Z. Li, Common fixed points of single-valued and multivalued maps, International Journal of Mathematics and Mathematical Sciences 2005(19) (2005), 3045 – 3055, DOI: 10.1155/IJMMS.2005.3045.

S. G. Matthews, 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.

A. Meir and E. Keeler, A theorem on contraction mappings, Journal of Mathematical Analysis and Applications 28(2) (1969), 326 – 329, DOI: 10.1016/0022-247X(69)90031-6.

K. Menger, Statistical metrics, Proceedings of the National Academy of Sciences 28(12) (1942), 535 – 537, DOI: 10.1073/pnas.28.12.535.

Z. D. Mitrovic, H. Aydi, M. S. M. Noorani and H. Qawaqneh, The weight inequalities on Reich type theorem in b-metric spaces, Journal of Mathematics and Computer Science 19(1) (2019), 51 – 57, DOI: 10.22436/jmcs.019.01.07.

N. Mlaiki, H. Aydi, N. Souayah and T. Abdeljawad, Controlled metric type spaces and the related contraction principle, Mathematics 6(10) (2018), 194, DOI: 10.3390/math6100194.

N. Mlaiki, T. Abdeljawad, W. Shatanawi, H. Aydi and Y. U. Gaba, On complex-valued triple controlled metric spaces and applications, Journal of Function Spaces 2021(1) (2021), 5563456, 7 pages, DOI: 10.1155/2021/5563456.

Z. Mustafa and B. Sims, A new approach to generalized metric spaces, Journal of Nonlinear and Convex Analysis 7(2) (2006), 289 – 297.

A. Mutlu and U. Gürdal, Bipolar metric spaces and some fixed point theorems, Journal of Nonlinear Sciences and Applications 9(9) (2016), 5362 – 5373, DOI: 10.22436/jnsa.009.09.05.

A. Mutlu, K. Ozkan and U. Gürdal, Coupled fixed point theorems on bipolar metric spaces, European Journal of Pure and Applied Mathematics 10(4) (2017), 655 – 667, URL: https://ejpam.com/index.php/ejpam/article/view/3019.

R. P. Pant, Common fixed points of noncommuting mappings, Journal of Mathematical Analysis and Applications 188(2) (1994), 436 – 440, DOI: 10.1006/jmaa.1994.1437.

H. K. Pathak, Y. J. Cho and S. M. Kang, Remarks on R-weakly commuting mappings and common fixed point theorems, Bulletin of the Korean Mathematical Society 34(2) (1997), 247 – 257.

E. Petrov, Fixed point theorem for mappings contracting perimeters of triangles, Journal of Fixed Point Theory and Applications 25 (2023), article number 74, DOI: 10.1007/s11784-023-01078-4.

E. Picard, Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives, Journal de Mathématiques Pures et Appliquées 6 (1890), 145 – 210, URL: http://eudml.org/doc/235808.

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.

J. Schauder, Der Fixpunktsatz in Funktionalraümen, Studia Mathematica 2(1) (1930), 171 – 180, URL: http://eudml.org/doc/217247.

W. Sintunavarat and P. Kumam, Common fixed points for R-weakly commuting in fuzzy metric spaces, Annali Dell’universita’ Di Ferrara Sezione Vii - Scienze Matematiche 58(2) (2012), 389 – 406, DOI: 10.1007/s11565-012-0150-z.

N. Souayah and M. Mrad, On fixed-point results in controlled partial metric type spaces with a graph, Mathematics 8(1) (2019), 33, DOI: 10.3390/math8010033.

A. Tarski, A lattice-theoretical fixpoint theorem and its applications, Pacific Journal of Mathematics 5 (1955), 285 – 309.

A. Tychonoff, Ein Fixpunktsatz, Mathematische Annalen 111 (1935), 767 – 776, DOI: 10.1007/BF01472256.

W. A. Wilson, On quasi-metric spaces, American Journal of Mathematics 53(3) (1931), 675 – 684, DOI: 10.2307/2371174.

Downloads

Published

30-11-2024
CITATION

How to Cite

Rani, A., & Bharti. (2024). Study of Mappings and Metric Spaces in Fixed Point Theory: A Review. Communications in Mathematics and Applications, 15(3), 1289–1304. https://doi.org/10.26713/cma.v15i3.2674

Issue

Vol. 15 No. 3 (2024)

Section

Review Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.