Abstract
With a proper metrizability approach that preserves completeness of a metric space, Hardy–Rogers contraction may be observed as a Banach contraction. Consequently, the same conclusion holds for Kannan, Reich and Chatterjea contractions along with several their modifications. Theoretical results are substantiated with several examples which additionally validate the independence between some contractive conditions. Rate and factor of convergence of iterative process in a newly defined metric space is studied along with a numerical example claiming a slight advantage of the proposed approach.
Similar content being viewed by others
Data Availability
No datasets were generated or analysed during the current study.
References
Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math. 3, 133–181 (1922)
Berinde, V., Păcurar, M.: Approximating fixed points of enriched Chatterjea contractions by Krasnoselskij iterative algorithm in Banach spaces. J. Fixed Point Theory Appl. 23, 1–16 (2022). https://doi.org/10.1007/s11784-021-00904-x
Collaço, P., Silva, J.C.E.: A complete comparison of 25 contraction conditions. Nonlinear Anal. Theory Methods Appl. 30(1), 471–476 (1997). https://doi.org/10.1016/S0362-546X(97)00353-2
Chatterjea, S.K.: Fixed-point theorems. C. R. Acad. Bulg. Sci. 25, 727–730 (1972)
Ćirić, L.B.: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 45, 267–273 (1974). https://doi.org/10.2307/2040075
Chaib, R., Merghadi, F., Mouhoubi, Z.: Improvement of fixed point theorems for Hardy–Rogers contraction type in b-metric spaces without F-contraction assumption. Rend. Circ. Mat. Palermo II Ser. 72, 4209–4237 (2023). https://doi.org/10.1007/s12215-023-00892-6
Debnath, P., Neog, M., Radenović, S.: Set valued Reich type G-contractions in a complete metric space with graph. Rend. Circ. Mat. Palermo 2(69), 917–924 (2020). https://doi.org/10.1007/s12215-019-00446-9
Derouiche, D., Ramoul, H.: New fixed point results for F-contractions of Hardy–Rogers type in b-metric spaces with applications. J. Fixed Point Theory Appl. (2020). https://doi.org/10.1007/s11784-020-00822-4
Hardy, G.E., Rogers, T.D.: A generalization of a fixed point theorem of Reich. Can. Math. Bull. 16(2), 201–206 (1973). https://doi.org/10.4153/CMB-1973-036-0
Jabeen, S., Koksal, M.E., Younis, M.: Convergence results based on graph-Reich contraction in fuzzy metric spaces with application. Nonlinear Anal. Model. Control 29(1), 71–95 (2023). https://doi.org/10.15388/namc.2024.29.33668
Kannan, R.: Some remarks on fixed points. Bull. Calcutta Math. Soc. 60, 71–76 (1960). https://doi.org/10.2307/2316437
Kincses, J., Totik, V.: Theorems and counter-examples on contractive map**s. Math. Balc. 4, 69–90 (1990)
Nashine, H.K., Kadelburg, Z.: Common fixed point theorems under weakly Hardy–Rogers-type contraction conditions in ordered orbitally complete metric spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 108, 377–395 (2014). https://doi.org/10.1007/s13398-012-0106-2
Picard, E.: Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives. J. Math. 6, 145–210 (1890)
Popescu, O.: Fixed-point results for convex orbital operators. Demonstr. Math. 56(1), 20220184 (2023). https://doi.org/10.1515/dema-2022-0184
Rasham, T., Shoaib, A., Arshad, M.: Fixed point results for locally Hardy Rogers-type contractive map**s for dislocated cone metric spaces. TWMS J. Pure Appl. Math. 10, 76–82 (2019). https://doi.org/10.13140/RG.2.2.15980.41602
Reich, S.: Some remarks concerning contraction map**s. Can. Math. Bull. 14, 121–124 (1971). https://doi.org/10.4153/CMB-1971-024-9
Reich, S.: Kannan’s fixed point theorem. Boll. Un. Mat. Ital. 4(4), 1–11 (1971)
Reich, S.: Fixed point of contractive functions. Boll. Un. Mat. Ital. 4(5), 26–42 (1972)
Rhoades, B.E.: A comparison of various definitions of contractive map**s. Trans. Am. Math. Soc. 226, 257–290 (1977). https://doi.org/10.1090/S0002-9947-1977-0433430-4
Rhoades, B.E.: A collections of contractive definitions. Math. Semin. Notes 7, 229–235 (1979)
Rhoades, B.E.: Contractive definitions revisited. Topol. Methods NonLinear Anal. Contemp. Math. AMS 21, 189–205 (1983). https://doi.org/10.1090/conm/021
Vetro, F.: F-contractions of Hardy–Rogers type and application to multistage decision. Nonlinear Anal. Model. Control 21(4), 531–546 (2016). https://doi.org/10.15388/NA.2016.4.7
Funding
The author is supported by the Ministry of Science, Technological Development and Innovation, Grant No. 451-03-47/2023-01/200124.
Author information
Authors and Affiliations
Contributions
The author confirms sole responsibility for the following: study conception and design, data collection, analysis and interpretation of results, and manuscript preparation.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no Conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Cvetković, M. Results on Hardy–Rogers Contraction. Mediterr. J. Math. 21, 140 (2024). https://doi.org/10.1007/s00009-024-02686-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-024-02686-1
Keywords
- Hardy–Rogers contraction
- Banach contraction
- Kannan contraction
- Chatterjea contraction
- Reich contraction
- metrizability