Best Proximity Points for Geraghty-Type Non-Self Mappings with a Registration-Inspired Alignment Model

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Abstract
We study Geraghty-type non-self mappings within the framework of best proximity point theory. By introducing auxiliary functions with subsequential convergence, we establish general conditions ensuring the existence and uniqueness of best proximity points. Our results extend and unify earlier work on proximal and Kannan-type contractions under a Geraghty setting, and we provide counterexamples showing that the auxiliary assumptions are essential. To demonstrate applicability, we construct a registration-inspired alignment model in which all hypotheses can be explicitly verified. This example illustrates how the theoretical framework guarantees a unique and well-defined alignment anchor, thereby highlighting the relevance of best proximity theory in registration problems.
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