Interpolative Fixed Point Theory in Perturbed Metric Spaces via Exact Metric Decomposition

Perturbed metric spaces arise when measured distances are affected by errors or structural distortions and admit a decomposition into an exact metric and a perturbation term. Working with the induced exact metric, we develop a unified interpolative fixed point framework in complete perturbed metric spaces. We introduce a generalized activated scheme, called the generalized interpolative perturbed contractive mapping (GIPCM), which combines interpolative control with a Suzuki-type activation condition. Within this setting, we establish existence, uniqueness, and global convergence of Picard iterates. As consequences, fixed point theorems are obtained for several strict interpolative families, including strict Kannan-type (sIPKC), Reich-Rus-Čirić-type (IPRRC), and strict Suzuki-triggered (sPIST) contractions. The classical interpolative perturbed Kannan contraction (IPKC), involving complementary exponents, is treated separately and proved via a direct geometric decay argument. A nonlinear decay lemma is developed to justify convergence under recursive estimates. Nonlinear examples and applications to a nonlinear integral equation and a Bellman-type dynamic programming equation illustrate the effectiveness of the proposed theory.