Ciric Fixed Point Theorems in Metric Spaces with Binary Operation
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Abstract
This study extends recent developments in operational metric spaces, metric spaces endowed with an arbitrary binary operation, by introducing and proving new Ćirić fixed point theorems and Banach-like fixed point theorems within this framework. Operational metric spaces, recently formalized by Adewale et al. (2025), generalized classical metric spaces by allowing diverse binary operations, leading to broader applicability in fixed point theory. While prior work established several fixed point results in such spaces, it did not address the more general Ćirić-type contractions, which encompass a wide class of mappings beyond Banach contractions.
The research employs a rigorous analytical approach, proving multiple versions of Ćirić fixed point theorems under different binary operations; addition, maximum and minimum, alongside other Banach-type results. Each theorem is established by constructing appropriate iterative sequences, demonstrating their Cauchy property via the operational metric axioms, and applying completeness to guarantee convergence to a unique fixed point. Variants of contractive conditions, including max-based, min-based, and additive formulations, are systematically addressed, with detailed uniqueness proofs.
Key findings confirm that under suitable contractive conditions and binary operations, self-maps on complete operational metric spaces possess a unique fixed point. The results generalize several known theorems in classical metric spaces, b-metric spaces, and S-metric spaces, thereby broadening the scope of fixed point theory. The study concludes that incorporating arbitrary binary operations into the metric framework not only preserves core fixed point properties but also enables more flexible and encompassing contractive mappings. These theorems unify and extend existing results, providing a foundation for further applications in nonlinear analysis, optimization, and other mathematical models where binary operations influence metric behaviour.