Non-uniform ISS small-gain theorem for infinite networks

The paper’s main theorem (Theorem III.5) rigorously establishes non-uniform ISS for infinite networks under (i) uniform K-bounds on subsystem transients and external gains, (ii) MBI for id−Γ⊗ together with Assumption 1, and (iii) finite in-degree, and it carefully derives the required uniform-in-initial-state attractivity via the dyadic decomposition B(r,k) and a limit-superior argument that yields uniform times τ_i(ε,r) independent of x, as demanded by Proposition III.2 (UGS + uniform attractivity ⇔ non-uniform ISS) . The candidate solution’s Step 1 (UGS via MBI) is acceptable, but Step 2 only bounds limsup along individual trajectories and then asserts a bounded-input uniform asymptotic gain without producing times τ_i(ε,r) uniform over all initial states in a bounded set; this misses the uniformization that the paper achieves through (20)–(27). Hence the model’s proof is incomplete where the paper’s is complete and correct .