Fixed point results of weakly polynomial contractions

Authors

  • Mohammed Shehu Shagari
    Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Zaria, Nigeria
  • Ishaq Sadiq Abubakar
    Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Zaria, Nigeria

Keywords:

Fixed point, Weak contraction, Polynomial contraction

Abstract

In this paper, a new family of weakly polynomial-type contractions defined on a metric space is presented. Under suitable hypotheses, it is shown that such contractive operators possess unique fixed points (FPs). Owing to the polynomial nature of the higher-order terms in the contractions, several significant particular cases, including existing results, are highlighted and discussed. In contrast to many existing Lipschitz-type inequalities, the proposed family of contractive inequalities does not force the mappings to be continuous.

Dimensions

[1] T. G. Bhaskar & V. Lakshmikantham, ``Fixed point theorems in partially ordered metric spaces and applications'', Nonlinear Analysis: Theory, Methods & Applications 65 (2006) 1379. https://doi.org/10.1016/j.na.2005.10.017.

[2] I. A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, Romania, 2001. https://search.worldcat.org/title/Generalized-contractions-and-applications/oclc/690385579.

[3] B. S. Choudhury, N. Metiya & M. Postolache, ``A generalized weak contraction principle with applications to coupled coincidence point problems'', Fixed Point Theory and Applications 2013 (2013) 152. https://doi.org/10.1186/1687-1812-2013-152.

[4] D. W. Boyd & J. S. W. Wong, ``On nonlinear contractions'', Proceedings of the American Mathematical Society 20 (1969) 458. https://doi.org/10.1090/S0002-9939-1969-0239559-9.

[5] J. Caristi, ``Fixed point theorems for mappings satisfying inwardness conditions'', Transactions of the American Mathematical Society 215 (1976) 241. https://doi.org/10.1090/S0002-9947-1976-0394329-4.

[6] G. E. Hardy & T. D. Rogers, ``A generalization of a fixed point theorem of Reich'', Canadian Mathematical Bulletin 16 (1973) 201. https://doi.org/10.4153/CMB-1973-036-0.

[7] R. Kannan, ``Some results on fixed points--II'', The American Mathematical Monthly 76 (1969) 405. https://doi.org/10.2307/2316437.

[8] R. Kannan, ``Some results on fixed points'', Bulletin of the Calcutta Mathematical Society 60 (1968) 71.

[9] L. B. '{C}iri'{c}, ``A generalization of Banach's contraction principle'', Proceedings of the American Mathematical Society 45 (1974) 267. https://doi.org/10.1090/S0002-9939-1974-0356011-2.

[10] J. M. Moreno, W. Sintunavarat & Y. J. Cho, ``Common fixed point theorems for Geraghty's type contraction mappings using the monotone property with two metrics'', Fixed Point Theory and Applications 2015 (2015) 174. https://doi.org/10.1186/s13663-015-0426-y.

[11] A. Meir & E. Keeler, ``A theorem on contraction mappings'', Journal of Mathematical Analysis and Applications 28 (1969) 326. https://doi.org/10.1016/0022-247X(69)90031-6.

[12] F. Khojasteh, S. Shukla & S. Radenovi'{c}, ``A new approach to the study of fixed point theory for simulation functions'', Filomat 29 (2015) 1189. https://doi.org/10.2298/FIL1506189K.

[13] M. A. Kutbi, A. Rold'{a}n, W. Sintunavarat, J. Mart'{i}nez-Moreno & C. Rold'{a}n, ``F-closed sets and coupled fixed point theorems without the mixed monotone property'', Fixed Point Theory and Applications 2013 (2013) 330. https://doi.org/10.1186/1687-1812-2013-330.

[14] M. Jleli & B. Samet, ``A generalized metric space and related fixed point theorems'', Fixed Point Theory and Applications 2015 (2015) 61. https://doi.org/10.1186/s13663-015-0312-7.

[15] S. Banach, ``Sur les op'erations dans les ensembles abstraits et leur application aux '{e}quations int'{e}grales'', Fundamenta Mathematicae 3 (1922) 133. https://doi.org/10.4064/fm-3-1-133-181.

[16] Y. I. Alber & S. Guerre-Delabriere, ``Principle of weakly contractive maps in Hilbert spaces'', in New Results in Operator Theory and Its Applications, I. Gohberg & Y. Lyubich (Eds.), Birkh"{a}user, Basel, Switzerland, 1997, pp. 7--22. https://doi.org/10.1007/978-3-0348-8910-0_2.

[17] B. E. Rhoades, ``Some theorems on weakly contractive maps'', Nonlinear Analysis: Theory, Methods & Applications 47 (2001) 2683. https://doi.org/10.1016/S0362-546X(01)00388-1.

[18] M. Jleli, C. M. Pu{a}curar & B. Samet, ``Fixed point results for contractions of polynomial type'', Demonstratio Mathematica 58 (2025) 20250098. https://doi.org/10.1515/dema-2025-0098.

[19] A. Moumen, H. N. Saleh, H. Albala, K. Aldwoah, H. Saber, E. I. Hassan & T. S. Hassan, ``On polynomial $phi$-contractions with applications to fractional logistic growth equations'', Fractal and Fractional 9 (2025) 366. https://doi.org/10.3390/fractalfract9060366.

[20] P. Shahi, ``Polynomial contractions in $G$-metric spaces'', Letters in Nonlinear Analysis and its Applications 2 (2024) 138. Available online: https://lettersinnonlinearanalysis.com/index.php/lnaa/article/view/38.

[21] S. Reich, ``Some remarks concerning contraction mappings'', Canadian Mathematical Bulletin 14 (1971) 121. https://doi.org/10.4153/CMB-1971-024-9.

[22] S. K. Chatterjea, ``Fixed point theorems'', Comptes Rendus de l’Acad´emie Bulgare des Sciences 25 (1972) 727.

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Published

2026-06-08

How to Cite

Fixed point results of weakly polynomial contractions. (2026). Proceedings of the Nigerian Society of Physical Sciences, 3, 252. https://doi.org/10.61298/pnspsc.2026.3.252

Issue

Volume 3, NSPS-ABU Conference, Feb. 9 - 14, 2026

Section

Articles
Copyright & Licensing

Copyright (c) 2026 Mohammed Shehu Shagari, Ishaq Sadiq Abubakar (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Fixed point results of weakly polynomial contractions. (2026). Proceedings of the Nigerian Society of Physical Sciences, 3, 252. https://doi.org/10.61298/pnspsc.2026.3.252