A Short Survey on the Recent Fixed Point Results on $b$-Metric Spaces

Year 2018, , 15 - 44, 15.09.2018

Erdal Karapınar

https://doi.org/10.33205/cma.453034

Cited By: 32

Abstract

The aim of this short survey is to collect and combine basic notions and results in the fixed point theory in the context of $b$-metric spaces. It is also aimed to show that there are still enough rooms for several researchers in this interesting direction and a huge application potential.

Keywords

$b$-Metric space, Fixed point

References

• [1] H. Afshari, H. Aydi, E.Karapinar, Existence of Fixed Points of Set-Valued Mappings in b-Metric Spaces, East Asian mathematical journal , volume 32, issue 3, 2016, Pages 319 332,
• [2] H. Afshari, H. Aydi and E. Karapinar, On generalized alpha-phi-geraghty contractions on b-metricspaces, Georgian Math. Journal,
• [3] R. P. Agarwal, H. Alsulami, E.Karapınar and F.Khojasteh, Remarks on some recent fixed point results in quaternion-valued metric spaces, Abstract and Applied Analysis, (2014) Article Id: 171624
• [4] U.Aksoy, E. Karapinar and I. M. Erhan, Fixed points of generalized alpha-admissible contractions on b-metric spaces with an application to boundary value problems,Journal of Nonlinear and Convex Analysis, (2016). Volume 17,Number 6, 1095-1108
• [5] Ali and Kamran, On alpha*-phi-contractive multi-valued mappings, Fixed Point Theory and Appl. 2013 2013:137.
• [6] M. U. Ali, T. Kamran, E. Karapinar, On (alpha-phi-xi)-contractive multi-valued mappings, Fixed Point Theory Appl., 2014, 2014:7
• [7] H. Alsulami, S.Gulyaz, E. Karapinar, I. Erhan, An Ulam stability result on quasi-b-metric-like spaces, Open Mathematics, Volume 14, Issue 1 (Jan 2016) DOI 10.1515/math-2016-0097,
• [8] H. Alsulami, S. Almezel, E. Karapinar, F. Khojasteh, A note on fixed point results in complex valued metric spaces, Journal of Inequalities and Applications, 2015, 2015:33
• [9] H. H. Alsulami, E. Karapınar, F. Khojasteh, A. F. Roldán-López-de-Hierro, A proposal to the study of contractions in quasi-metric spaces, Discrete Dynamics in Nature and Society2014, Article ID 269286, 10 pages.
• [10] H. H. Alsulami, E. Karapnar, V. Rakocevic Ciric Type Nonunique Fixed Point Theorems onb-Metric Spaces, Filomat 31:11 (2017), 3147-3156
• [11] P. Amiri, S. Rezapour, N. Shahzad, Fixed points of generalized alpha-phi-contractions, Revista de laReal Academia de Ciencias Exactas, Fisicas y Naturales Serie A Mate.,
• [12] H. Aydi, M-F. Bota, E. Karapınar and S. Mitrovic, A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory Appl. 2012, 2012:88.
• [13] H. Aydi, M-F. Bota, E. Karapınar and S. Moradi, A common fixed point for weak phi-contractions on b-metric spaces, Fixed Point Theory, 13(2012), No 2, 337-346.
• [14] H. Aydi, E. Karapınar, B.Samet, Fixed points for generalized (alpha-phi)-contractions on generalized metric spaces, Journal of Inequalities and Applications 2014, 2014:229
• [15] I.A. Bakhtin, The contraction mapping principle in quasimetric spaces, Funct. Anal., Unianowsk Gos. Ped. Inst. 30(1989), 26-37.
• [16] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math. 3 (1922) 133–181.
• [17] V. Berinde, Generalized contractions in quasimetric spaces, Seminar on Fixed Point Theory, Preprint no. 3(1993), 3-9.
• [18] V. Berinde, Sequences of operators and fixed points in quasimetric spaces, Stud. Univ. ”Babes-Bolyai”, Math., 16(4)(1996), 23-27.
• [19] V. Berinde, Contractii generalizatesi aplicatii, Editura Club Press 22, Baia Mare, 1997.
• [20] N. Bilgili, E. Karapınar, A note on “Common fixed points for (alpha-phi,beta)-weakly contractive mappings in generalized metric spaces”, Fixed Point Theory Appl. 2013, 2013:287.
• [21] R.M. Bianchini, M. Grandolfi, Transformazioni di tipo contracttivo generalizzato in uno spazio metrico, Atti Acad. Naz. Lincei, VII. Ser., Rend., Cl. Sci. Fis. Mat. Natur. 45 (1968), 212-216.
• [22] M. F. Bota-Boriceanu, A. Petru¸ sel, Ulam-Hyers stability for operatorial equations, Analel Univ. Al. I. Cuza, Ia¸ si, 57(2011), 65-74.
• [23] M.-F. Bota, E. Karapinar and O. Mlesnite, Ulam-Hyers stability results for fixed point problems via alpha-psi-contractive mapping in b-metric space, Abstract and Applied Analysis, 2013 Article Id: 825293
• [24] M. Boriceanu, A. Petru¸ sel, I.A. Rus, Fixed point theorems for some multivalued generalized contractions in b-metric spaces, International Journal of Mathematics and Statistics, 6(2010), 65-76.
• [25] M. Boriceanu, Strict fixed point theorems for multivalued operators in b-metric spaces, IJMM, Vol 4, 3(2009), 285-301.
• [26] M. Boriceanu, Fixed point theory for multivalued generalized contraction on a set with two b-metrics, Studia Univ. Babes-Bolyai, Mathematica, 3(2009), 3-14.
• [27] M. Bota, Dynamical Aspects in the Theory of Multivalued Operators, Cluj University Press, 2010.
• [28] N. Bourbaki, Topologie G´ en´ erale, Herman, Paris, 1974.
• [29] J. Brzdek, J. Chudziak, and Z. Pales, A fixed point approach to stability of functional equations, Nonlinear Analysis: Theory, Methods and Applications, vol. 74, no. 17, pp. 6728-6732, 2011.
• [30] J. Brzdek and K. Ciepliski, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Analysis: Theory, Methods and Applications, vol. 74, no. 18, pp. 6861-6867, 2011.
• [31] J. Brzdek and K. Cieplinski, A fixed point theorem and the Hyers-Ulam stability in non-Archimedean spaces, Journal of Mathematical Analysis and Applications, vol. 400, no. 1, pp. 68-75, 2013.
• [32] L.B. Ciric, On some maps with a nonunique fixed point. Publ. Inst. Math., 17, 52–58 (1974).
• [33] S. Czerwik, Contraction mappings in b-metric spaces, Acta Mathematica et Informatica Universitatis Ostraviensis 1(1993), 5-11.
• [34] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Univ. Modena, 46(1998), 263-276.
• [35] T.Do^ senovic, M. Postolache and S. Radenovic, On multiplicative metric spaces: survey FixedPoint Theory and Applications20162016:92
• [36] W.-S. Du, A note on cone metric fixed point theory and its equivalence Nonlinear Analysis72 (2010), no : 5, 2259-2261.
• [37] M. Frechet Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat. Palermo 22 (1906)1-74.
• [38] S.Gupta and B. Ram, Non-unique fixed point theorems of Ciric type, (Hindi) VijnanaParishad Anusandhan Patrika 41( 4), 217–231(1998).
• [39] F. Hausdorff. Mengenle hre . W. de Gruyter & Co. , 1927.
• [40] J. Hasanzade Asl, Sh. Rezapour and N. Shahzad, On fixed points of alpha-phi- contractivemultifunctions, Fixed Point Theory and Applications, 2012(2012), doi:10.1186/1687-1812-2012-212.
• [41] N.Hussain, J.R. Roshan, V. Parvaneh and M.Abbas, Common fixed point results for weakcontractive mappings in ordered b-dislocated metric spaces with applications Journal of In-equalities and Applications (2013) 2013:486
• [42] N. Hussain, Z. Kadelburg, S. Radenovi´ c, and F.Al-Solamy, Comparison Functions and FixedPoint Results in Partial Metric Spaces, Abstract and Applied Analysis, vol. 2012, Article ID605781, 15 pages, 2012.
• [43] J. Heinonen, Lectures on Analysis on Metric Spaces, Springer Berlin, 2001.
• [44] D. H. Hyers, On the stability of the linear functional equation, Proceedings of the NationalAcademy of Sciences of the United States of America, vol. 27, no. 4, pp. 222-224, 1941.
• [45] M. Jleli and B. Samet, Remarks on G-metric spaces and fixed point theorems, Fixed PointTheory Appl. 2012, 2012:210, 7 pages.
• [46] E. Karapinar, H. Piri and H. AlSulami, Fixed Points of Generalized F-Suzuki Type Contrac-tion in Complete b-Metric Spaces,,” Discrete Dynamics in Nature and Society, 2015 (2015),Article ID 969726, 8 pages
• [47] E. Karapinar, H.Piri and H.H. AlSulami, Fixed Points of Generalized F-Suzuki Type Con-traction in Complete b-Metric Spaces” Discrete Dynamics in Nature and Society, 2015 (2015),Article ID 969726, 8 pages
• [48] E. Karapinar, A New Non-Unique Fixed Point Theorem, J. Appl. Funct. Anal. , 7 (2012),no:1-2, 92-97.
• [49] E. Karapinar, Some Nonunique Fixed Point Theorems of Ciric type on Cone Metric Spaces,Abstr. Appl. Anal., vol. 2010, Article ID 123094, 14 pages (2010).
• [50] E. Karapinar, H. Piri and H. AlSulami, Fixed Points of Generalized F-Suzuki Type Contrac-tion in Complete b-Metric Spaces, Discrete Dynamics in Nature and Society, 2015 (2015),Article ID 969726, 8 pages
• [51] E. Karapınar, P. Kuman, P. Salimi, On alpha-phi-Meri-Keeler contractive mappings, Fixed PointTheory Appl. 2013:94 (2013)
• [52] E. Karapınar, H.H. Alsulami and M. Noorwali, Some extensions for Geragthy type contractivemappings Journal of Inequalities and Applications 2015:303 (2015)
• [53] E. Karapınar, B.Samet, Generalized alpha-phi-Contractive Type Mappings and Related FixedPoint Theorems with Applications Abstract and Applied Analysis Volume 2012, Article ID793486, 17 pages
• [54] E. Karapinar and W.-S. Du, A note on b-cone metric and its related results: Generalizationsor equivalence? , Fixed Point Theory and Applications, (2013), 2013:210
• [55] F. Khojasteh, S. Shukla, S. Radenovi´ c, A new approach to the study of fixed point theoremsvia simulation functions, Filomat 29:6 (2015), 1189–1194.
• [56] M.A. Kutbi , E. Karapinar, J. Ahmed, A. Azam, Some fixed point results for multi-valuedmappings in b-metric spaces , Journal of Inequalities and Applications 2014, 2014:126
• [57] A. Latif, M. E. Gordji, E. Karapınar, W. Sintunavarat, Fixed point results for generalized(alpha-phi)-Meir-Keeler contractive mappings and applications, J. Ineq. Appl. 2014, 2014:68.
• [58] V. La Rosa, P. Vetro, Common fixed points for ?-?-?-contractions in generalized metricspaces, Nonlinear Anal. Model. Control 19 (2014), no. 1, 43-54
• [59] V. L. Laz^ ar, Ulam-Hyers stability for partial differential inclusions, Electronic Journal ofQualitative Theory of Differential Equations, 21 (2012), 1-19.
• [60] Liu, Z. Q.: On Ciric type mappings with a nonunique coincidence points, Mathematica (Cluj)35(58),no. 2, 221–225(1993).
• [61] Liu, Z., Guo, Z., Kang, S. M., Lee, S. K.: On Ciric type mappings with nonunique fixed andperiodic points, Int. J. Pure Appl. Math., 26(3),399–408 (2006).
• [62] B. Mohammadi, S. Rezapour, N Shahzad, Some results on fixed points of alpha-phi-Ciric general-ized multifunctions. Fixed Point Theory Appl., 2013 2013:24 doi:10.1186/1687-1812-2013-24.
• [63] J.J. Nieto, R. Rodr´ıguez-López, Contractive Mapping Theorems in Partially Ordered Setsand Applications to Ordinary Differential Equations, Order. 22 (2005) 223–239.
• [64] B. G. Pachpatte, On Ciric type maps with a nonunique fixed point, Indian J. Pure Appl.Math., 10( 8), 1039–1043 (1979).
• [65] M. Pacurar, A fixed point result for varphi-contractions on b-metric spaces without the bound-edness assumption, Fasc. Math., 43(2010), 127-137.
• [66] T. P. Petru, A. Petrusel and J.-C. Yao, Ulam-Hyers stability for operatorial equations andinclusions via nonself operators, Taiwanese Journal of Mathematics, Vol. 15, No. 5, pp.2195-2212, October 2011.
• [67] O. Popescu, Some new fixed point theorems for alpha-Geraghty-contraction type maps in metricspaces, Fixed Point Theory Appl. 2014, 2014:190
• [68] A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and someapplications to matrix equations, Proc. Amer. Math. Soc. 132 (2003) 1435–1443.
• [69] I. A. Rus, The theory of a metrical fixed point theorem: theoretical and applicative relevances,Fixed Point Theory, 9(2008), No. 2, 541-559.
• [70] I. A. Rus, Generalized contractions and applications, Cluj University Press, Cluj-Napoca,2001.
• [71] I. A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory,10(2009), No. 2, 305-320.
• [72] I. A. Rus, A. Petru¸ sel, A. Sˆınt^ am^ arian, Data dependence of the fixed points set of somemultivalued weakly Picard operators, Nonlinear Anal. 52(2003), 1947-1959.
• [73] P. Salimi, A. Latif, N. Hussain, Modified alpha-phi-contractive mappings with applications, FixedPoint Theory Appl., 2013 2013:151 doi:10.1186/1687-1812-2013-151.
• [74] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for alpha-phi-contractive type mappings, Non-linear Analysis 75 (2012), 2154-2165.
• [75] B. Samet, C.Vetro, and F.Vetro , Remarks on G-Metric Spaces, International Journal ofAnalysis, Volume 2013 (2013), Article ID 917158, 6 pages
• [76] S. Shukla, Partial b-Metric Spaces and Fixed Point Theorems, Mediterr. J. Math. 11 (2014),703711
• [77] S. M. Ulam, Problems in Modern Mathematics, John Wiley and Sons, New York, NY, USA,1964.
• [78] A. F. Roldán-López-de-Hierro, E. Karapınar, C. Roldán-López-de-Hierro, J. Mart´ınez-Moreno, Coincidence point theorems on metric spaces via simulation functions, J. Comput.Appl. Math. 275 (2015) 345–355.
• [79] S. Radenovi´ c, Z. Kadelburg, D. Jandrli´ c and A. Jandrli´ c, Some results on weak contractionmaps, Bulletin of the Iranian Mathematical Society Vol. 38 No. 3 (2012), pp 625-645.
• [80] M. Turinici, Abstract comparison principles and multivariable Gronwall-Bellman inequalities,J. Math. Anal. Appl. 117 (1986) 100–127.