Establish sharp asymptotic and Lyapunov-type results

Derive sharp transition curves and rigorous asymptotic growth rates for linear pantograph difference equations in the delay-dependent regions, and prove Lyapunov-type results for the inhomogeneous nonlinear pantograph maps with nonstationary delays.

Background

The paper uses multi-horizon numerical scans to distinguish apparent decay, bounded behavior, and ultra-slow growth in regions not covered by the analytical theorems. The authors explicitly note that these diagnostics provide numerical evidence rather than a proof of asymptotic instability, leaving rigorous growth laws and exact transition boundaries unresolved.

For the nonlinear delayed logistic and tent maps, finite-time exponential separation is observed, but the experiments do not establish a positive asymptotic Lyapunov exponent or a chaotic attractor. The paper therefore identifies rigorous Lyapunov-type analysis for nonstationary proportional or sublinear delays as an unresolved direction.