Continuum Limits for Adaptive Network Dynamics

The paper proves well-posedness, positivity (under A5), and discrete-to-continuum convergence (under A6–A7) for the adaptive Kuramoto-type continuum system via a fixed-point map on pairs (φ,η), showing an iterate A^n is a contraction, then extending globally and handling positivity and discretization rigorously. The candidate solution reaches the same conclusions with a different Picard scheme that first solves the linear measure equation explicitly (variation-of-constants) for η given φ, then contracts a map Γ on φ and patches short-time windows, and treats the discrete scheme via a perturbative fixed-point comparison. This approach is valid and essentially equivalent in scope. Two minor issues: (i) the candidate implicitly treats η_t as absolutely continuous with respect to μ_X even when (A4) alone does not guarantee this (the paper works in the weak/measure form); this is easily corrected by formulating the η-variation-of-constants at the measure level, as done in the paper; and (ii) a suboptimal bound for a double integral yields an unnecessary Δ^2 factor. These do not alter the conclusions.