A functional-analytic framework for nonlinear adaptive memory: hierarchical kernels, state-dependent sensitivity, and memory-dependent functionals

In this work, we developed a systematic framework for nonlinear adaptive memory within nonlinear functional analysis, where the influence of past events depends on both elapsed time and the state values along a trajectory. The framework comprises three hierarchical layers. First, memory kernels were classified into mathematically admissible, regular (uniformly bounded, normalized, Lipschitz), and generalized (bounded variation, possibly sign-changing) classes. Second, adaptive sensitivity functions $ \Lambda(s, f(s)) $ were introduced, satisfying natural conditions; a concrete construction based on historical deviation accumulation interpolated continuously between instantaneous response and history-dependent sensitivity, with an explicit Lipschitz estimate $ \|\Lambda_f-\Lambda_g\|_\infty\le L_\Lambda\|f-g\|_\infty $. Third, an adaptive memory-dependent functional

$ S_{\kappa,\Lambda}(f) = \sup\limits_{t\in I} \left( |f(t)| + \int_0^t \Lambda(s,f(s))\,\kappa(t-s)\,|f(s)|\,ds \right) $

and the associated set $ \mathscr{M}_{\kappa, \Lambda}(I) = \left\{ f : S_{\kappa, \Lambda}(f) < \infty \right\} $ were constructed. Fundamental properties of the framework were established, including absolute convergence, measurability, uniform boundedness, positive definiteness, and comparison with the classical supremum norm. It was shown that $ \mathcal{C}(I) \subset \mathscr{M}_{\kappa, \Lambda}(I) $ strictly, with discontinuous functions (e.g., indicator functions of subintervals) belonging to the set, capturing abrupt signal changes such as on-off switching in nonlinear systems. When the maximum of $ |f| $ was attained in $ (0, T] $, a strict inequality $ S_{\kappa, \Lambda}(f) > \|f\|_\infty $ was proved, demonstrating the nontrivial contribution of the memory component. The resulting framework could be viewed as a mathematical description of nonlinear adaptive phenomena, including habituation, state-dependent weighting, and selective retention, within a rigorous functional-analytic setting. By separating temporal weighting from state-dependent modulation, the construction offered a modular methodology applicable to adaptive phenomena in fields such as neuroscience, adaptive control, and machine learning, where memory is time-dependent and content-sensitive, and may be of interest across other mathematical disciplines, as a step toward a broader understanding of nonlinear phenomena from a mathematical perspective.