NUMERICAL DYNAMICS OF INTEGRODIFFERENCE EQUATIONS: FORWARD DYNAMICS AND PULLBACK ATTRACTORS

The paper proves existence and convergence of pullback attractors for the collocation schemes (Theorem 3.8) and existence plus convergence of forward ω-limit sets under strong A–asymptotic compactness (Theorem 3.10). The candidate solution establishes the same two claims: (A) constructs a common pullback absorbing family via the standard a priori estimate and complete continuity to get pullback asymptotic compactness, then uses a finite-horizon tracking bound (via H5 and one-step convergence) and a triangle inequality to show A*_n → A*_0 on bounded time windows; (B) under forward assumptions (including (3.5), (3.10), strong A–asymptotic compactness with compact K_n, and (i)–(iv)) it shows ω^+_{A,n} is nonempty and compact and converges to ω^+_{A,0}. These align precisely with Theorem 3.8 and Theorem 3.10, respectively. The paper’s proofs are organized through abstract theorems (2.1, 2.2, 2.6, 2.7) and verification of their hypotheses, while the model’s solution gives a direct but equivalent tracking/triangle-inequality argument and a diagonal compactness argument. Minor differences: the model implicitly assumes a uniform-in-time one-step consistency bound, whereas the paper proves the needed uniform-in-τ, finite-horizon estimate more delicately (see the verification of Theorem 2.6(iii)). Overall, both are correct; the model’s proof follows a slightly different route but reaches the same results. Citations: existence of absorbing sets and radii R_τ (Prop. 3.1 and (3.4)/(3.5)) , existence and convergence of pullback attractors (Thm. 3.8) , definitions and existence results for pullback/forward attractors (Sec. 2.1–2.2; Thm. 2.1 and Thm. 2.2) , Lipschitz-on-balls estimate from H5 (Lemma 3.3) , the finite-horizon convergence estimate used in verifying Thm. 2.6(iii) , positive invariance under (3.10) (Cor. 3.9) , and forward ω-limit sets and their convergence (Thm. 3.10) .