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September 10, 2025

Computational Approaches to Fixed-Point Problems in Convex Metric Spaces: A Generalization of the Contraction Principle

Key Points

The study generalizes the contraction principle for broader classes of mappings in convex metric spaces, improving applicability.
Computational approaches, including Picard and Mann iterations, are utilized to solve generalized fixed-point problems effectively.
Convex metric spaces enhance the development of sophisticated algorithms, capitalizing on geometric properties for better convergence.
Non-expansive mappings are explored, providing solutions even when strict contraction conditions do not apply.

Abstract

The concept of a fixed point, a point that remains unchanged under a given transformation, lies at the heart of numerous mathematical disciplines, from topology and analysis to economics and engineering. The renowned Banach Contraction Principle provides an elegant and powerful tool for guaranteeing the existence and uniqueness of fixed points for contraction mappings in complete metric spaces. However, the strict contractive condition often limits its applicability in real-world scenarios. This has driven extensive research into generalizing the contraction principle to broader classes of mappings and more complex spaces, particularly convex metric spaces. The computational approaches to solving these generalized fixed-point problems in such spaces represent a vibrant and crucial area of modern mathematics, offering robust frameworks for addressing diverse practical challenges. Convex metric spaces, which include normed linear spaces and CAT(0) spaces as special cases, provide a richer geometric structure than general metric spaces. This convexity allows for the development of more sophisticated iterative algorithms that leverage the "averaging" or "midpoint" properties of the space. One of the most significant generalizations of the contraction principle in this context involves extending the notion of contraction to various non-expansive or quasi-nonexpansive mappings. While these mappings do not necessarily shrink distances between all points, they possess properties that still allow for the convergence of iterative sequences to a fixed point. Examples include firmly nonexpansive mappings, Suzuki generalized non-expansive mappings, and mappings satisfying more abstract contractive conditions, all of which are instrumental in modeling phenomena where strict contraction might not hold. We used the Python language. Picard iteration was used to find a fixed point of a mapping. Mann iteration was used in convex space.

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