On p-Hardy–Rogers and p-Zamfirescu Contractions in Complete Metric Spaces: Existence and Uniqueness Results

Bekri, Zouaoui; Fabiano, Nicola; Alomair, Mohammed Ahmed; Alsharidi, Abdulaziz Khalid

doi:10.3390/math13244011

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On p-Hardy–Rogers and p-Zamfirescu Contractions in Complete Metric Spaces: Existence and Uniqueness Results

¹

Laboratory of Fundamental and Applied Mathematics, University of Oran 1, Ahmed Ben Bella, Es-Senia 31000, Algeria

²

Department of Sciences and Technology, Institute of Sciences, Nour-Bachir University Center, El-Bayadh 32000, Algeria

³

“Vinča” Institute of Nuclear Sciences—National Institute of the Republic of Serbia, University of Belgrade, 11351 Belgrade, Serbia

⁴

Department of Quantitative Methods, School of Business, King Faisal University, Al-Ahsa 31982, Saudi Arabia

⁵

Department of Mathematics and Statistics, College of Science, King Faisal University, Al Ahsa 31982, Saudi Arabia

^*

Authors to whom correspondence should be addressed.

Mathematics 2025, 13(24), 4011; https://doi.org/10.3390/math13244011

Submission received: 18 November 2025 / Revised: 9 December 2025 / Accepted: 15 December 2025 / Published: 16 December 2025

(This article belongs to the Section C: Mathematical Analysis)

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Abstract

In this paper, we introduce and investigate two generalized forms of classical contraction mappings, namely the p-Hardy–Rogers and p-Zamfirescu contractions. By incorporating the integer parameter

p \geq 1

, these new definitions extend the traditional Hardy–Rogers and Zamfirescu conditions to iterated mappings

ℏ^{p}

. We establish fixed-point theorems, ensuring both existence and uniqueness of fixed points for continuous self-maps on complete metric spaces that satisfy these p-contractive conditions. The proofs are constructed via geometric estimates on the iterates and by transferring the fixed point from the p-th iterate

ℏ^{p}

to the original mapping ℏ. Our results unify and broaden several well-known fixed-point theorems reported in previous studies, including those of Banach, Hardy–Rogers, and Zamfirescu as special cases.

Keywords: fixed point theory; p-Hardy–Rogers contraction; p-Zamfirescu contraction; iterated mappings; complete metric spaces; Banach contraction principle; nonlinear analysis

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MDPI and ACS Style

Bekri, Z.; Fabiano, N.; Alomair, M.A.; Alsharidi, A.K. On p-Hardy–Rogers and p-Zamfirescu Contractions in Complete Metric Spaces: Existence and Uniqueness Results. Mathematics 2025, 13, 4011. https://doi.org/10.3390/math13244011

AMA Style

Bekri Z, Fabiano N, Alomair MA, Alsharidi AK. On p-Hardy–Rogers and p-Zamfirescu Contractions in Complete Metric Spaces: Existence and Uniqueness Results. Mathematics. 2025; 13(24):4011. https://doi.org/10.3390/math13244011

Chicago/Turabian Style

Bekri, Zouaoui, Nicola Fabiano, Mohammed Ahmed Alomair, and Abdulaziz Khalid Alsharidi. 2025. "On p-Hardy–Rogers and p-Zamfirescu Contractions in Complete Metric Spaces: Existence and Uniqueness Results" Mathematics 13, no. 24: 4011. https://doi.org/10.3390/math13244011

APA Style

Bekri, Z., Fabiano, N., Alomair, M. A., & Alsharidi, A. K. (2025). On p-Hardy–Rogers and p-Zamfirescu Contractions in Complete Metric Spaces: Existence and Uniqueness Results. Mathematics, 13(24), 4011. https://doi.org/10.3390/math13244011

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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On p-Hardy–Rogers and p-Zamfirescu Contractions in Complete Metric Spaces: Existence and Uniqueness Results

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