On Fixed Point Results of Suzuki-Type Contractions on Controlled Metric Spaces with Applications

John Pamba; Isaac Daniel Tembo

doi:doi:10.11648/j.ijsts.20251305.14

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On Fixed Point Results of Suzuki-Type Contractions on Controlled Metric Spaces with Applications

John Pamba^*

, Isaac Daniel Tembo

Published in International Journal of Science, Technology and Society (Volume 13, Issue 5)

Received: 21 January 2025 Accepted: 11 April 2025 Published: 25 September 2025

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Abstract

In this paper, we introduce the concept of Suzuki-type contractions on controlled metric spaces and prove a fixed point theory. This extends and generalises the already existing results of Suzuki-type contractions on b−metric spaces and extended b−metric spaces to controlled metric spaces. Some illustrative examples are presented in order to amplify our findings. It is shown that Suzuki-type contractions in the setting of controlled metric spaces provide greater generality and flexibility compared to the setting of metric spaces. We do this by constructing an example where a Suzuki-type contraction does not guarantee a fixed point in a standard metric space but does in a controlled metric space. In this setting, the control function in the controlled metric helps to stabilise iterative sequences in proving the fixed point theory and indeed in the application. Finally, our main result is applied to show the existence of a solution for the fredholm type integral equation. The results obtained in this paper contribute to the broader study of fixed point theory and its applications in mathematical analysis and applied sciences.

Published in	International Journal of Science, Technology and Society (Volume 13, Issue 5)
DOI	10.11648/j.ijsts.20251305.14
Page(s)	205-210
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Controlled Metric, b−Metric Space, Extended b−Metric Space, Fixed Point, Fredholm Integral Equation, Suzuki-type Contractions

References

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[2]	Ali, B.; Ali, H.; Nazir, T.; Ali, Z. Existence of Fixed Points of Suzuki-Type Contractions of Quasi-Metric Spaces. Mathematics 2023, 11, 4445. https://doi.org/10.3390/math11214445
[3]	Al-Mazrooei, A. E.; Ahmad, J. Fixed Point Results in Controlled Metric Spaces with Applications. Mathematics 2022, 10, 490. https://doi.org/10.3390/math10030490
[4]	Boriceanu, M. Strict fixed point theorems for multivalued operators in b-metric spaces, Int. J. Modern Math., 4 (2009), 285-301. 1.
[5]	Ciric, L., A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc., 45 (1974) 267-273.
[6]	Czerwik, S. Contraction mappings in b-metric spaces, Acta Math. Univ. Ostrav., 1 (1993) 5-11.
[7]	Czerwik, S. Nonlinear set-valued contraction mappings in b-metric spaces, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia., 46 (2) (1998) 263-276.
[8]	Geraghty, M. On contractive mappings, Proc. Amer. Math. Soc., 40 (1973) 604-608.
[9]	Kamran, T., Samreen, M. and UL Ain, Q. A generalisation of b−metric space and some fixed point theorems. Mathematics, vol. 5, no. 2, p. 19, 2017.
[10]	Kikkawa, M., Suzuki, T. Three fixed point theorems for generalized contractions with constants in complete metric spaces. Nonlinear Anal. Theory, Methods Appl. 2008, 69, 2942-2949.
[11]	Latif, A., Parvaneh, V., Salimi, P., Al-Mazrooei, A. E. Various Suzuki-type theorems in b−metric spaces. J. Nonlinear Sci. Appl. 8 2015, 363-377.
[12]	Leyew, B. T., Abbas, M. Fixed Point Results of Generalized Suzuki-Geraghty Contractions on f-Orbitally complete b-Metric Spaces. U. P. B. Sci. Bull., Series A, Vol. 79, Iss. 2, (2017) ISSN 1223-7027.
[13]	Mlaiki, N., Aydi, H., Souayah, N., Abdeljawad, T.: Controlled metric type spaces and the related contraction principle. Mathematics 6, 194 (2018).
[14]	Romaguera, S. Basic Contractions of Suzuki-Type on Quasi-Metric Spaces and Fixed Point Results. Mathematics 2022, 10, 3931. https://doi.org/10.3390/math10213931
[15]	Roshan, J. R., Hussain, N., Sedghi, S., Shobkolaei, N. Suzuki-type fixed point results in b−metric spaces, Math Sci, 2015. 9.153-160. https://doi.org/10.1007/s40096-015-0162-9
[16]	Roshan, J. R., Parvaneh, V., Sedghi, S., Shobkolaei, N., Shatanawi, W. Common fixed points of almost generalized (ψ,φ)s−contractive mappings in ordered b-metric spaces, Fixed Point Theory Appl., 2013 (2013), 23 pages.
[17]	Shahkoohi, R. J., Razani, A. Some fixed point theorems for rational Geraghty contractive mappings in ordered b-metric spaces, J. Inequal. Appl., 2014 (2014), 23 pages. 1.
[18]	Souayah, N. and Mrad, M., On Fixed Point Results in Controlled Partial Metric Type Spaces with a Graph, MDPI, Besel, Switzerland. 2019.
[19]	Souayah, N.; Hidri, L. New Fixed-Point Results in Controlled Metric Type Spaces with Applications. Axioms 2025, 14, 85. https://doi.org/10.3390/axioms14020085
[20]	Suzuki, T. A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc., 136 (5) (2008) 1861-1869.
[21]	Suzuki, T. A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009) 5313-5317.
[22]	K. Zoto, V. Šešum-Cavić, M. Pantović, V. Todorčević, M. Zoto, and S. Radenović, A Unified Approach and Related Fixed-Point Theorems for Suzuki Contractions, Symmetry, vol. 16, no. 6, p. 739, 2024. https://doi.org/10.3390/sym16060739

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APA Style

Pamba, J., Tembo, I. D. (2025). On Fixed Point Results of Suzuki-Type Contractions on Controlled Metric Spaces with Applications. International Journal of Science, Technology and Society, 13(5), 205-210. https://doi.org/10.11648/j.ijsts.20251305.14

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ACS Style

Pamba, J.; Tembo, I. D. On Fixed Point Results of Suzuki-Type Contractions on Controlled Metric Spaces with Applications. Int. J. Sci. Technol. Soc. 2025, 13(5), 205-210. doi: 10.11648/j.ijsts.20251305.14

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AMA Style

Pamba J, Tembo ID. On Fixed Point Results of Suzuki-Type Contractions on Controlled Metric Spaces with Applications. Int J Sci Technol Soc. 2025;13(5):205-210. doi: 10.11648/j.ijsts.20251305.14

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@article{10.11648/j.ijsts.20251305.14,
  author = {John Pamba and Isaac Daniel Tembo},
  title = {On Fixed Point Results of Suzuki-Type Contractions on Controlled Metric Spaces with Applications
},
  journal = {International Journal of Science, Technology and Society},
  volume = {13},
  number = {5},
  pages = {205-210},
  doi = {10.11648/j.ijsts.20251305.14},
  url = {https://doi.org/10.11648/j.ijsts.20251305.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsts.20251305.14},
  abstract = {In this paper, we introduce the concept of Suzuki-type contractions on controlled metric spaces and prove a fixed point theory. This extends and generalises the already existing results of Suzuki-type contractions on b−metric spaces and extended b−metric spaces to controlled metric spaces. Some illustrative examples are presented in order to amplify our findings. It is shown that Suzuki-type contractions in the setting of controlled metric spaces provide greater generality and flexibility compared to the setting of metric spaces. We do this by constructing an example where a Suzuki-type contraction does not guarantee a fixed point in a standard metric space but does in a controlled metric space. In this setting, the control function in the controlled metric helps to stabilise iterative sequences in proving the fixed point theory and indeed in the application. Finally, our main result is applied to show the existence of a solution for the fredholm type integral equation. The results obtained in this paper contribute to the broader study of fixed point theory and its applications in mathematical analysis and applied sciences.
},
 year = {2025}
}

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TY  - JOUR
T1  - On Fixed Point Results of Suzuki-Type Contractions on Controlled Metric Spaces with Applications

AU  - John Pamba
AU  - Isaac Daniel Tembo
Y1  - 2025/09/25
PY  - 2025
N1  - https://doi.org/10.11648/j.ijsts.20251305.14
DO  - 10.11648/j.ijsts.20251305.14
T2  - International Journal of Science, Technology and Society
JF  - International Journal of Science, Technology and Society
JO  - International Journal of Science, Technology and Society
SP  - 205
EP  - 210
PB  - Science Publishing Group
SN  - 2330-7420
UR  - https://doi.org/10.11648/j.ijsts.20251305.14
AB  - In this paper, we introduce the concept of Suzuki-type contractions on controlled metric spaces and prove a fixed point theory. This extends and generalises the already existing results of Suzuki-type contractions on b−metric spaces and extended b−metric spaces to controlled metric spaces. Some illustrative examples are presented in order to amplify our findings. It is shown that Suzuki-type contractions in the setting of controlled metric spaces provide greater generality and flexibility compared to the setting of metric spaces. We do this by constructing an example where a Suzuki-type contraction does not guarantee a fixed point in a standard metric space but does in a controlled metric space. In this setting, the control function in the controlled metric helps to stabilise iterative sequences in proving the fixed point theory and indeed in the application. Finally, our main result is applied to show the existence of a solution for the fredholm type integral equation. The results obtained in this paper contribute to the broader study of fixed point theory and its applications in mathematical analysis and applied sciences.

VL  - 13
IS  - 5
ER  -

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Author Information

John Pamba

Department of Mathematics and Statistics, Mukuba University, Kitwe, Zambia

Contact Email

http://orcid.org/0009-0002-0931-6725
Isaac Daniel Tembo

Department of Mathematics and Statistics, University of Zambia, Lusaka, Zambia

Contact Email

http://orcid.org/0009-0001-4393-0772

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Plain Text BibTeX RIS

APA Style

Pamba, J., Tembo, I. D. (2025). On Fixed Point Results of Suzuki-Type Contractions on Controlled Metric Spaces with Applications. International Journal of Science, Technology and Society, 13(5), 205-210. https://doi.org/10.11648/j.ijsts.20251305.14

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ACS Style

Pamba, J.; Tembo, I. D. On Fixed Point Results of Suzuki-Type Contractions on Controlled Metric Spaces with Applications. Int. J. Sci. Technol. Soc. 2025, 13(5), 205-210. doi: 10.11648/j.ijsts.20251305.14

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AMA Style

Pamba J, Tembo ID. On Fixed Point Results of Suzuki-Type Contractions on Controlled Metric Spaces with Applications. Int J Sci Technol Soc. 2025;13(5):205-210. doi: 10.11648/j.ijsts.20251305.14

Copy | Download

@article{10.11648/j.ijsts.20251305.14,
  author = {John Pamba and Isaac Daniel Tembo},
  title = {On Fixed Point Results of Suzuki-Type Contractions on Controlled Metric Spaces with Applications
},
  journal = {International Journal of Science, Technology and Society},
  volume = {13},
  number = {5},
  pages = {205-210},
  doi = {10.11648/j.ijsts.20251305.14},
  url = {https://doi.org/10.11648/j.ijsts.20251305.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsts.20251305.14},
  abstract = {In this paper, we introduce the concept of Suzuki-type contractions on controlled metric spaces and prove a fixed point theory. This extends and generalises the already existing results of Suzuki-type contractions on b−metric spaces and extended b−metric spaces to controlled metric spaces. Some illustrative examples are presented in order to amplify our findings. It is shown that Suzuki-type contractions in the setting of controlled metric spaces provide greater generality and flexibility compared to the setting of metric spaces. We do this by constructing an example where a Suzuki-type contraction does not guarantee a fixed point in a standard metric space but does in a controlled metric space. In this setting, the control function in the controlled metric helps to stabilise iterative sequences in proving the fixed point theory and indeed in the application. Finally, our main result is applied to show the existence of a solution for the fredholm type integral equation. The results obtained in this paper contribute to the broader study of fixed point theory and its applications in mathematical analysis and applied sciences.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - On Fixed Point Results of Suzuki-Type Contractions on Controlled Metric Spaces with Applications

AU  - John Pamba
AU  - Isaac Daniel Tembo
Y1  - 2025/09/25
PY  - 2025
N1  - https://doi.org/10.11648/j.ijsts.20251305.14
DO  - 10.11648/j.ijsts.20251305.14
T2  - International Journal of Science, Technology and Society
JF  - International Journal of Science, Technology and Society
JO  - International Journal of Science, Technology and Society
SP  - 205
EP  - 210
PB  - Science Publishing Group
SN  - 2330-7420
UR  - https://doi.org/10.11648/j.ijsts.20251305.14
AB  - In this paper, we introduce the concept of Suzuki-type contractions on controlled metric spaces and prove a fixed point theory. This extends and generalises the already existing results of Suzuki-type contractions on b−metric spaces and extended b−metric spaces to controlled metric spaces. Some illustrative examples are presented in order to amplify our findings. It is shown that Suzuki-type contractions in the setting of controlled metric spaces provide greater generality and flexibility compared to the setting of metric spaces. We do this by constructing an example where a Suzuki-type contraction does not guarantee a fixed point in a standard metric space but does in a controlled metric space. In this setting, the control function in the controlled metric helps to stabilise iterative sequences in proving the fixed point theory and indeed in the application. Finally, our main result is applied to show the existence of a solution for the fredholm type integral equation. The results obtained in this paper contribute to the broader study of fixed point theory and its applications in mathematical analysis and applied sciences.

VL  - 13
IS  - 5
ER  -

Copy | Download