New fixed point theorems by applying Singh's framework to iterated $ \rho $-Ćirić contractions

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

doi:10.3934/math.2026069

AIMS Mathematics

2026, Volume 11, Issue 1: 1653-1674. doi: 10.3934/math.2026069

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Research article

New fixed point theorems by applying Singh's framework to iterated $ \rho $-Ćirić contractions

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 Mathematics and Statistics, College of Science, King Faisal University, Al Ahsa 31982, Saudi Arabia
5.
Department of Quantitative Methods, School of Business, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Received: 27 November 2025 Revised: 28 December 2025 Accepted: 30 December 2025 Published: 19 January 2026
MSC : 47H09, 47H10, 54H25

This paper develops new unified fixed point theorems by extending the classical Ćirić contractions through the iterative framework introduced by Singh in his generalization of Kannan-type mappings. We first revisit the two well-known definitions of Ćirić contractions in their standard forms and then introduce the concepts of $ \rho $-Ćirić contraction and generalized $ \rho $-Ćirić contraction, in which the contractive condition is imposed on higher iterates of the mapping. These extended classes retain the intrinsic structure of Círic-type mappings while significantly enlarging the family of operators that admit a unique fixed points. For each class, we establish comprehensive fixed point theorems ensuring existence, uniqueness, and global convergence of the associated Picard iteration. Our results unify and generalize several classical fixed-point theorems, including those of Banach, Kannan, Chatterjea, and the original Ćirić contractions, which arise as special cases within this broader framework. Illustrative examples are provided to demonstrate the applicability and sharpness of the proposed generalizations.
- fixed point theory,
- Singh contraction,
- Ćirić contraction,
- $ p $-iterated mappings,
- complete metric spaces,
- Banach contraction principle,
- generalized contractions
Citation: Zouaoui Bekri, Nicola Fabiano, Abdulaziz Khalid Alsharidi, Mohammed Ahmed Alomair. New fixed point theorems by applying Singh's framework to iterated $ \rho $-Ćirić contractions[J]. AIMS Mathematics, 2026, 11(1): 1653-1674. doi: 10.3934/math.2026069

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Abstract

This paper develops new unified fixed point theorems by extending the classical Ćirić contractions through the iterative framework introduced by Singh in his generalization of Kannan-type mappings. We first revisit the two well-known definitions of Ćirić contractions in their standard forms and then introduce the concepts of $ \rho $-Ćirić contraction and generalized $ \rho $-Ćirić contraction, in which the contractive condition is imposed on higher iterates of the mapping. These extended classes retain the intrinsic structure of Círic-type mappings while significantly enlarging the family of operators that admit a unique fixed points. For each class, we establish comprehensive fixed point theorems ensuring existence, uniqueness, and global convergence of the associated Picard iteration. Our results unify and generalize several classical fixed-point theorems, including those of Banach, Kannan, Chatterjea, and the original Ćirić contractions, which arise as special cases within this broader framework. Illustrative examples are provided to demonstrate the applicability and sharpness of the proposed generalizations.

References

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