b-Metric Like Spaces: A Survey of Concepts and Applications
Sažetak
This paper provides a comprehensive survey of fixed point results in metric-like spaces, with a particular focus on b-metric-like spaces. The notion of b-metric-like spaces generalizes several important concepts, including partial metric spaces, metric-like spaces, and b-metric spaces. The paper highlights significant contributions from the early stages of research to the present, offering an extensive compilation of foundational and recent results. Finally, we provide affirmative answers to some open problems arising from best proximity results in orthogonal 0-complete b-metric-like spaces, further advancing the understanding of fixed point theory in these generalized settings. This survey aims to equip researchers with valuable insights and references, facilitating further exploration and development in the study of metric-like spaces and their generalizations.
Reference
Abbas, M. and Jungck, J., 2008. Common fixed point results for noncommuting mappings without continuity in cone metric spaces. textit{Journal of Mathematical Analysis and Applications}, 341(1), pp. 416–420. Available at: url{https://doi.org/10.1016/j.jmaa.2007.09.070}.
Abkar, A. and Gabeleh, M., 2013. A note on some best proximity point theorems proved under P-property. textit{Abstract and Applied Analysis}, Vol. 2013, Article ID 189567. Available at: url{https://doi.org/10.1155/2013/189567}.
Aleksić, S., Mitrović, Z.D. and Radenović, S., 2020. Picard sequences in b-metric spaces. textit{Fixed Point Theory}, 21(1), pp. 35-46. Available at: url{https://doi.org/10.24193/fpt-ro.2020.1.03}.
Alghamdi, M.A., Hussain, N., Salimi, P., 2013. Fixed point and coupled fixed point theorems on b-metric-like spaces. textit{Journal of Inequalities and Applications}, 2013, Article 402. Available at: url{https://doi.org/10.1186/1029-242X-2013-402}.
Amini-Harandi, A., 2012. Fixed point theorems for monotone operators in partially ordered metric-like spaces and application to integral equations. textit{Journal of Nonlinear and Convex Analysis}. Available at: url{https://content.iospress.com/articles/journal-of-nonlinear-and-convex-analysis/jnca180}.
Amini-Harandi, A., 2012. Metric-like spaces, partial metric spaces and fixed points. textit{Fixed Point Theory and Applications}, 2012. Available at: url{https://doi.org/10.1186/1687-1812-2012-204}.
Aage, C.T., Salunke, J.N., 2008. The results on fixed points in dislocated and dislocated quasi-metric space. textit{Applied Mathematical Sciences}, 2(59), pp. 2941-2948. Available at: url{https://www.m-hikari.com/ams/ams-2008/ams-59-2008/ageams59-2941-2948.pdf}.
Aydi, H., Felhi, A. and Afshari, H., 2017. New Geraghty type contractions on metric-like spaces. textit{Journal of Nonlinear Science and Applications}, 10, pp. 780–788. Available at: url{http://dx.doi.org/10.22436/jnsa.010.02.38}.
Aydi, H., Lakzian, H., Mitrovic, Z.D. and Radenović, S., 2020. Best proximity points of MT-cyclic contractions with property UC. textit{Numerical Functional Analysis and Optimization}, 41, pp. 871–882. Available at: url{https://doi.org/10.1080/01630563.2020.1796959}.
Berinde, V. and Păcurar, M., 2022. The early developments in fixed point theory on b-metric spaces: a brief survey and some important related aspects. textit{ Carpathian journal of Mathematics }, 38(3), pp. 523–538. Available at: url{https://doi.org/10.37193/CJM.2022.03.01}.
Chen, C., Dong, J. and Zhu, C., 2015. Some fixed point theorems in b-metric-like spaces. textit{Fixed Point Theory and Applications}, 2015, Article ID 122. Available at: url{https://doi.org/10.1186/s13663-015-0369-3}.
Choudhury, B., Metiya, N., Postolache, M. and Konar, P., 2015. A discussion on best proximity point and coupled best proximity point in partially ordered metric spaces. textit{Fixed Point Theory and Applications}, 2015, Article 170. Available at: url{https://doi.org/10.1186/s13663-015-0397-3}.
Ćirić, L.B., 1974. A generalization of Banach's contraction principle. textit{Proceedings of the American Mathematical Society}, 45(2), pp. 267–273. Available at: url{https://doi.org/10.1090/S0002-9939-1974-0356011-2}.
Czerwik, S., 1993. Contraction mappings in b-metric spaces. textit{Acta Mathematica et Informatica Universitatis Ostraviensis}, 1(1), pp. 5–11. Available at: url{https://doi.org/10.5817/AMUO1993-1-1}.
Edelstein, M., 1962. On fixed and periodic points under contractive mappings. textit{Journal of the London Mathematical Society}, 37, pp. 74-79. Available at: url{https://doi.org/10.1112/JLMS/S1-37.1.74}.
Fabiano, N., Dosenović, T., Rakić, D., Radenović, S., 2020. Some new results on $(s,q)$-Dass-Gupta-Jaggi type contractive mappings in b-metric-like spaces. textit{Filomat}, 34(12), pp. 4015-4026. Available at: url{https://doi.org/10.2298/FIL2012015F}.
Fallahi, K. and Eivani, Sh., 2022. Orthogonal b-metric spaces and best proximity points. textit{Journal of Mathematical Extension}, 16(6), pp. 1–17. Available at: url{https://doi.org/10.30495/JME.2022.2000}.
Fréchet, M., 1906. Sur quelques points du calcul fonctionnel. textit{Rendiconti del Circolo Matematico di Palermo}, 22, pp. 1–74. Available at: url{https://doi.org/10.1007/BF03018603}.
Fulga, A. and Proca, A.M., 2017. Fixed point for φₐ-Geraghty contraction. textit{Journal of Nonlinear Science and Applications}, 10, pp. 5125–5131. Available at: url{https://doi.org/10.22436/jnsa.010.09.5699}.
Garakoti, N., Joshi, M.C. and Kumar, R., 2014. Fixed point for $F_{perp}$-weakly commuting mappings. textit{Nonlinear Analysis}, 97, pp. 62–71. Available at: url{https://doi.org/10.1016/j.na.2013.09.002}.
Gardašević-Filipović, M., Kukić, K., Gardašević, D. and Mitrović, Z.D., 2023. Some best proximity point results in the orthogonal 0-complete b-metric-like spaces. textit{Journal of Contemporary Mathematical Analysis}, 58(2), pp. 105–115. Available at: url{https://doi.org/10.3103/S1068362323020036}.
Gardasevic-Filipovic, M., Mitrovic, Z.D., Gardasevic, D. and Radenovic, S., In press. Some best proximity point results in the orthogonal 0-complete b-metric-like spaces and rectangular b-metric-like spaces. textit{TWMS Journal of Pure and Applied Mathematics}.
Garakoti, N., Joshi, M.C. and Kumar, R., 2014. Fixed point for $F_{perp}$-weakly commuting mappings. textit{Nonlinear Analysis}, 97, pp. 62–71. Available at: url{https://doi.org/10.1016/j.na.2013.09.002}.
Gordji, M.E. and Habibi, H., 2017. Fixed point theory in generalized orthogonal metric space. textit{Journal of Linear Topology and Algebra}, 6(3), pp. 251-260. Available at: url{https://doi.org/10.1234/jlta.2017.00603}.
Gordji, M.E., Rameani, M., De La Sen, M. and Cho, Y.J., 2017. On orthogonal sets and Banach fixed point theorem. textit{Fixed Point Theory}, 18(2), pp. 569–578. Available at: url{https://doi.org/10.24193/fpt-ro.2017.2.45}.
Gregori, V.; Sapena, A., 2002. On fixed-point theorems in fuzzy metric spaces. textit{Fuzzy Sets and Systems}, 125, pp. 245–252. Available at: url{https://doi.org/10.1016/S0165-0114(01)00282-1}.
Hardy, G.E. and Rogers, T.D., 1973. A generalization of a fixed point theorem of Reich. textit{Canadian Mathematical Bulletin}, 16(2), pp. 201–206. Available at: url{https://doi.org/10.4153/CMB-1973-036-0}.
Hussain, N., Rezaei, J.R., Parvaneh, V. and Kadelburg, Z., 2014. Fixed points of contractive mappings in b-metric-like spaces. textit{The Scientific World Journal~}, 2014, Article ID 471827. Available at: url{https://doi.org/10.1155/2014/471827}.
Jain, K., Kaur, J., 2021. Some fixed point results in b-metric spaces and b-metric-like spaces with new contractive mappings. textit{Axioms}, 10, 55. Available at: url{https://doi.org/10.3390/axioms10020055}.
Javed, K., Aydi, H., Uddin, F., Arshad, M., 2021. On orthogonal partial b-metric spaces with an application. textit{J. Math.}, 1-7.
Karapınar, E., 2014. $alpha$-$psi$-Geraghty contraction type mappings and some related fixed point results. textit{Filomat}, 28(1), pp. 37–48. Available at: url{https://doi.org/10.2298/FIL1401037K}.
Karapınar, E., Alsulami, H. and Noorwali, M., 2015. Some extensions for Geraghty type contractive mappings. textit{Journal of Inequalities and Applications}, 2015, Article ID 303. Available at: url{https://doi.org/10.1186/s13660-015-0830-1}.
Kirk, W.A., Srinivas, P.S. and Veeramani, P., 2003. Fixed points for mappings satisfying cyclical contractive conditions. textit{Fixed Point Theory}, 4(1), pp. 79-89. Available at: url{http://www.mathjournals.org/}.
Matthews, S.G., 1994. Partial metric topology. In: Proceedings of the 8th Summer Conference on General Topology and Applications. textit{Annals of the New York Academy of Sciences}, 728, pp. 183–197. Available at: url{https://doi.org/10.1111/j.1749-6632.1994.tb44144.x}.
Mihec{t}, D., 2008. Fuzzy $psi$-contractive mappings in non-archimedean fuzzy metric spaces. textit{Fuzzy Sets and Systems}, 159(6), pp. 739-744. Available at: url{https://doi.org/10.1016/j.fss.2007.07.017}.
Mirkov, N., Mitrovic, Z.D., Younis, M., Radenović, S., 2021. On Palais method in b-metric-like spaces. textit{Math. Anal. Contemp. Appl.}, 3(3), pp. 33-38. Available at: url{https://doi.org/10.30495/maca.2021.1932449.1015}.
Mongkolkeha, C., Cho, Y.J. and Kumam, P., 2013. Best proximity points for generalized proximal C-contraction mappings in metric spaces with partial orders. textit{Journal of Inequalities and Applications}, 2013, Article 94. Available at: url{https://doi.org/10.1186/1029-242X-2013-94}.
Monika, 2019. On Metric-Like Spaces: A Survey. textit{International Journal of Applied Engineering Research}, 14(6), pp. 1390-1395. Available at: url{https://www.ripublication.com/ijaer19/ijaerv14n6_19.pdf}.
Moussaoui, A., Hussain, N., Melliani, S., Hayel, N. and Imdad, M., 2022. Fixed point results via extended $mathcal{FZ}$-simulation functions in fuzzy metric spaces. textit{Journal of Inequalities and Applications}, 2022, Paper No. 69. Available at: url{https://doi.org/10.1186/s13660-022-03027-4}.
Moussaoui, A., Radenović, S., and Melliani, S., 2024. A generalization of parametric metric spaces via $mathcal{B}$-actions. textit{Filomat}, 38:15, pp. 5475–5485. Available at: url{https://doi.org/10.2298/FIL2415475M}.
Păcurar, M. and Rus, I.A., 2010. Fixed point theory for cyclic $varphi$-contractions. textit{Nonlinear Analysis}, 72(3–4), pp. 1181–1187. Available at: url{https://doi.org/10.1016/j.na.2009.08.002}.
Petrusel, G., 2005. Cyclic representations and periodic points. textit{Studia Universitatis Babes-Bolyai, Mathematica}, 50, pp. 107-112. Available at: url{http://www.studiauniversitatis.ro/}.
Radenović, S., 2016. Classical fixed point results in 0-complete partial metric spaces via cyclic-type extension. textit{Bulletin of the Allahabad Mathematical Society}, 31(1), pp. 39–55. Available at: url{https://doi.org/10.1007/s13370-015-0339-2}.
Radenović, S., 2016. Some remarks on mappings satisfying cyclical contractive conditions. textit{Afrika Matematika}, 27(1), pp. 291–295. Available at: url{https://doi.org/10.1007/s13370-015-0339-2}.
Salimi, P. and Karapinar, E., 2013. Suzuki-Edelstein type contractions via auxiliary functions. textit{ Mathematical Problems In Engineering}, 2013, Article ID 648528. Available at: url{https://doi.org/10.1155/2013/648528}.
Sarma, I.R., Kumari, P.S., 2012. On dislocated metric spaces. textit{International Journal of Mathematical Archive}, 3(1), pp. 72-77. Available at: url{https://www.ijma.info/}.
Sawangsup, K., Sintunavarat, W. and Cho, Y.J., 2020. Fixed point theorems for orthogonal F-contraction mappings on O-complete metric spaces. textit{Journal of Fixed Point Theory and Applications}, 22(10). Available at: url{https://doi.org/10.1007/s11784-020-00785-9}.
Sintunavarat, W. and Kumam, P., 2012. Common fixed point theorem for cyclic generalized multi-valued contraction mappings. textit{Applied Mathematics Letters}, 25(11), pp. 1849-1855. Available at: url{https://doi.org/10.1016/j.aml.2012.02.045}.
Suzuki, T., 2008. A Generalized Banach Contraction Principle That Characterizes Metric Completeness. textit{Proceedings of the American Mathematical Society}, 136(5), 1861–1869. Available at: url{http://www.jstor.org/stable/20535364}.
Suzuki, T., 2009. A new type of fixed point theorem in metric spaces. textit{Nonlinear Analysis}, 71(11), pp. 5313-5317. Available at: url{https://doi.org/10.1016/j.na.2009.04.017}.
bibitem[Yang et al., 2020]{ref7} Yang, Q. and Bai, C., 2020. Fixed point theorem for orthogonal contraction of Hardy-Rogers type mapping on O-complete metric space. textit{AIMS Mathematics}, 5(6), pp. 5734–5742. Available at: url{https://doi.org/10.3934/math.2020.6.5734}.
bibitem[Yamaod & Sintunavarat, 2018]{Yamaod} Yamaod, O., Sintunavarat, W., 2018. On new orthogonal contractions in b-metric spaces. textit{International Journal of Pure and Applied Mathematics}, 5(4), pp. 37-40. Available at: url{https://doi.org/10.12732/ijpam.v5i4.6}.
bibitem[Yu, et al., 2018]{Yu2018} Yu, D., Chen, C., Wang, H., 2018. Common fixed point theorems for $(T,g)_F$-contraction in b-metric-like spaces. textit{Journal of Inequalities and Applications}, 2018, Article 222. Available at: url{https://doi.org/10.1186/s13660-018-1802-z}.
bibitem[Zoto, et al., 2018]{Zoto2018} Zoto, K., Radenović, S. and Ansari, A.H., 2018. On some fixed point results for (s, p, α)-contractive mappings in b-metric-like spaces and applications to integral equations. textit{Open Mathematics}, 16, pp. 235-249. Available at: url{https://doi.org/10.1515/math-2018-0024}.
bibitem[Zoto & Hoxha, 2012]{discolated3} Zoto, K., Hoxha, E., 2012. Fixed point theorems in dislocated and dislocated quasi-metric spaces. textit{Journal of Advanced Studies in Topology}, 3(4), pp. 119-124. Available at: url{https://www.sdiarticle2.in/prh/JAST/2012/}.
Sva prava zadržana (c) 2025 Abdelhamid Moussaoui
Ovaj rad je pod Creative Commons Autorstvo 4.0 međunarodnom licencom.
Vojnotehnički glasnik omogućava otvoreni pristup i, u skladu sa preporukom CEON-a, primenjuje Creative Commons odredbe o autorskim pravima:
Autori koji objavljuju u Vojnotehničkom glasniku pristaju na sledeće uslove:
- Autori zadržavaju autorska prava i pružaju časopisu pravo prvog objavljivanja rada i licenciraju ga Creative Commons licencom koja omogućava drugima da dele rad uz uslov navođenja autorstva i izvornog objavljivanja u ovom časopisu.
- Autori mogu izraditi zasebne, ugovorne aranžmane za neekskluzivnu distribuciju rada objavljenog u časopisu (npr. postavljanje u institucionalni repozitorijum ili objavljivanje u knjizi), uz navođenje da je rad izvorno objavljen u ovom časopisu.
- Autorima je dozvoljeno i podstiču se da postave objavljeni rad onlajn (npr. u institucionalnom repozitorijumu ili na svojim internet stranicama) pre i tokom postupka prijave priloga, s obzirom da takav postupak može voditi produktivnoj razmeni ideja i ranijoj i većoj citiranosti objavljenog rada (up. Efekat otvorenog pristupa).