Generalized Enriched Cyclic Contractions in Controlled Metric Spaces: Fixed Point Results, Iterated Function Systems, and a Physics-Informed Neural Network Realization
DOI:
https://doi.org/10.66972/iscis21202620Keywords:
Controlled Metric Space, Enriched Cyclic Contraction, Convex Structure, Iterated Function System, Controlled Pompeiu--Hausdorff Metric, Physics-Informed Neural NetworkAbstract
This research introduces the class of generalized enriched cyclic contractions in the setting of controlled metric spaces equipped with a convex structure. A unified fixed point theory for this class is developed. Enrichment, cyclicity, and point-dependent controlled distances have not been treated together. Besides, no single principle recovered the classical results while also supporting fractal and neural applications. By combining the Krasnoselskij averaging device with a cyclic representation of the underlying space, we prove existence, uniqueness, and convergence theorems together with explicit a priori and a posteriori error estimates. Also, we recover Banach, Kannan, Reich, and Edelstein type principles as particular cases. As an application, we construct a generalized iterated function system driven by a finite family of enriched cyclic contractions. We study the associated Hutchinson operator on the hyperspace of nonempty compact subsets endowed with the controlled Pompeiu--Hausdorff metric. We further give a physics-informed neural network realization. The solution of an initial value problem is identified with the unique fixed point of a Picard operator that is a contraction in a Bielecki weighted metric. The framework strictly enlarges several results in the literature, connecting fixed point theory, controlled metric geometry, fractal analysis, and scientific machine learning.
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