Interpolative Modulated Contractions and Common Fixed Point Theory in Complete Fuzzy Cone Metric Spaces with Applications

Gyanrao Dhote; Rashmi Jaisawal; Sheetal Panwar; Aditi malviya; Priyanshi Chandrawanshi

doi:10.69968/ijisem.2026v5i351-62

Authors

Gyanrao Dhote Head, Department of Computer Science and Mathematics Rai Saheb Bhanwar Singh College, Nasrullaganj, Madhya Pradesh, India
Rashmi Jaisawal Govt. Dr Shyama Prasad Mukherjee Science and Commerce College Bhopal MP Affiliation: - Barkartullah University, Bhopal MP
Sheetal Panwar Department of Computer Science and Mathematics Rai Saheb Bhanwar Singh College, Nasrullaganj, Madhya Pradesh, India
Aditi malviya Department of Computer Science and Mathematics Rai Saheb Bhanwar Singh College, Nasrullaganj, Madhya Pradesh, India
Priyanshi Chandrawanshi Department of Computer Science and Mathematics Rai Saheb Bhanwar Singh College, Nasrullaganj, Madhya Pradesh, India

DOI:

https://doi.org/10.69968/ijisem.2026v5i351-62

Keywords:

Fixed Point Theorem, Common Fixed Point, Interpolative Contraction, Modulated Contraction, Fuzzy Cone Metric Space, Weakly Compatible Mappings, Nonlinear Integral Equation

Abstract

In this paper, we establish a common fixed point theory for an interpolative modulated contractions class in the setup of fuzzy cone metric spaces. We outline a setting that blends the order structure of cones, the fuzziness sensitive nearness function of fuzzy metric spaces with a scalar modulation system that reshapes fuzzy nearness into an adjustable dial. This paper presents a generalized contractive condition for two self-mappings on a complete fuzzy cone metric space based on five fuzzy metric terms; here, the modulation function plays an important role. The condition encompasses Banach-type, Kannan-type, Chatterjea-type and F-type reductions with appropriate specialization of the modulation parameters. Based on explicit scalarization, completeness and orbital continuity assumptions, we demonstrate the existence of a point of coincidence via an alternating sequence in the sense of (P) along with a geometric Cauchy estimate. To show the uniqueness part, a standalone theorem of uniqueness follows immediately after which weak compatibility is invoked to transform the unique coincidence point into a unique common fixed point. The hypotheses are stated in order that convergence, Cauchy behavior, uniqueness and compatibility are not obscured by the notation. For illustration, two examples are included in this paper: one is a finite fuzzy cone metric example that must satisfy the new condition while failing ordinary Banach and Kannan pair conditions. It is applied to a nonlinear Fredholm integral equation and a dynamic programming equation. We provide also a numerical iteration table and a convergence curve showing the rate predicted by the proof.

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