"Generalizing Fixed-Point Theorems: an Investigation Across Incomplete and Non-Convex Metric Spaces"

Fixed-point theorems form one of the cornerstones of modern mathematical analysis, underpinning significant results in functional analysis, optimization, game theory, and differential equations. Classical formulations such as the Banach Contraction Principle, Brouwer’s Fixed-Point Theorem, and Schauder’s Theorem traditionally rely on strong assumptions like completeness of the metric space or convexity of the underlying set. While these conditions ensure existence and uniqueness, they impose restrictions that limit applicability in real-world problems where data structures and domains often exhibit incompleteness or lack convexity.

This study aims to bridge these limitations by investigating generalizations of fixed-point results within incomplete metric spaces and non-convex subsets. We begin by revisiting foundational results and examining why standard assumptions break down in irregular spaces. Building on this foundation, we introduce a set of generalized contraction conditions and relaxed structural requirements that extend the applicability of fixed-point theory beyond conventional settings. Our approach considers weaker forms of completeness, such as sequential completeness and partial completeness, as well as alternative notions of convexity, including star-shapedness and proximity-based convexity.

The main contribution of this work is the formulation of two generalized fixed-point theorems that hold under minimal assumptions. The first addresses mappings on incomplete metric spaces that satisfy a modified contractive condition, while the second establishes existence results for mappings defined on non-convex sets. Rigorous proofs are provided to demonstrate validity and necessity of the proposed assumptions. Additionally, illustrative examples and counterexamples are presented to highlight both the strength and boundaries of the new results.

Beyond theoretical significance, these generalizations have practical implications for solving nonlinear equations, ensuring stability of iterative algorithms, and guaranteeing equilibrium existence in models with irregular strategy spaces. Applications are discussed in the context of optimization problems, differential equations, and certain classes of dynamic systems where classical fixed-point theorems fail to apply.