Fixed Point Theorems Analysis

A fixed point of a function is a value that the function maps to itself, and fixed point theorems establish conditions under which such points are guaranteed to exist. In mathematics, these results underpin large swaths of analysis, from proving that certain differential equations have solutions to guaranteeing convergence in iterative algorithms, because existence of a fixed point often translates directly into existence of a solution. Research in this area has expanded well beyond the classical settings, examining how fixed point guarantees behave when the underlying space has a cone metric structure, when mappings are set-valued rather than single-valued, or when the geometry of the space allows only approximate fixed points—so-called best proximity points—rather than exact ones. Active open questions concern how far the ordering and contraction conditions can be weakened while preserving existence and uniqueness, and how these generalizations extend to fuzzy and probabilistic metric spaces where distance itself is not a crisp real number.

Works

37,587

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289,553

Keywords

Fixed Point TheoremsMetric SpacesContractive MappingsPartial OrderingGeneralized ContractionsBest Proximity Points