Autors: Agarwal R.P., Konstantinov M.M., Madamlieva, E. B.
Title: Ulam-Type Stability and Krasnosel’skii’s Fixed Point Approach for φ-Caputo Fractional Neutral Differential Equations with Iterated State-Dependent Delays
Keywords: Caputo derivative with respect to another functions, fractional neutral equations, iterated state-dependent delays, Krasnosel’skii fixed point theorem, Ulam–Hyers stability

Abstract: This work analyses the existence, uniqueness, and Ulam-type stability of neutral fractional functional differential equations with recursively defined state-dependent delays. Employing the Caputo fractional derivative of order (Formula presented.) with respect to a strictly increasing function (Formula presented.), the analysis extends classical results to nonuniform memory. The neutral term and delay chain are defined recursively by the solution, with arbitrary continuous initial data. Existence and uniqueness of solutions are established using Krasnosel’skii’s fixed point theorem. Sufficient conditions for Ulam–Hyers stability are obtained via the Volterra-type integral form and a (Formula presented.) -fractional Grönwall inequality. Examples illustrate both standard and nonlinear time scales, including a Hopfield neural network with iterated delays, which has not been previously studied even for integer-order equations. Fractional neural networks with iterated state-dependent delays provide a new and effective model for the description of AI processes—particularly machine learning and pattern recognition—as well as for modelling the functioning of the human brain.

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Issue

Fractal and Fractional, vol. 9, 2025, Switzerland, https://doi.org/10.3390/fractalfract9120753

Вид: статия в списание, публикация в издание с импакт фактор, публикация в реферирано издание, индексирана в Scopus и Web of Science