Best Proximity Points for Geraghty-Type Non-Self Mappings with a Registration-Inspired Alignment Model
Fatemeh Fogh, Sara Behnamian
TL;DR
This work extends best proximity point theory to non-self mappings by introducing $S$-proximal Geraghty and $S$-proximal Kannan--Geraghty frameworks, proving the existence and uniqueness of a best proximity point $x^*$ in $A$ satisfying $d(Sx^*,STx^*)=d(A,B)$ under an auxiliary function $S$ that is continuous, one-to-one, and subsequentially convergent with $S(A_0) subseteq A_0$, $S(B_0) subseteq B_0$. It unifies proximal and Kannan-type contractions under Geraghty-type control with $\beta\in\Gamma$ and $d^*$-augmented variants, providing counterexamples showing the necessity of the subsequential convergence of $S$. It then applies the theory to a rigorously checkable image-registration toy model, where $A$ and $B$ encode feature curves and the unique alignment anchor is guaranteed and can be identified via the iterative scheme. These results offer rigorous guarantees and constructive methods for alignment problems in image processing and more generally for non-self mapping scenarios.
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.