ON $\Psi_{\alpha,\beta}$--EXPANSIVE MAPPINGS WITH DISPLACEMENT CONTROL AND FIXED POINT CONSEQUENCES

In this paper, we introduce a new class of nonlinear expansive mappings governed by a rational displacement--distance gauge $\Psi_{\alpha,\beta}$, which simultaneously depends on the interpoint distance and the individual self--displacements of the operator. This framework extends classical Wang--type expansive models that are based solely on interpoint distances. Under a natural domination condition linking displacement and distance, we establish the existence, uniqueness, and global convergence of fixed points for $\Psi_{\alpha,\beta}$--expansive mappings in complete metric spaces. The proposed approach yields a displacement--sensitive expansive mechanism that enables the treatment of operators not covered by classical expansive conditions, thereby overcoming limitations of existing theories and providing a more flexible framework for applications in nonlinear analysis. Several nontrivial examples are presented to illustrate the applicability, strength, and novelty of the proposed theory.