Classify stability in the delay-dependent parameter regions

Determine whether the zero solution of the linear pantograph difference equation x_{n+1}=\lambda x_n+\delta x_{\lfloor an\rfloor} is asymptotically stable or unstable throughout the parameter regions where the outcome depends on the proportional-delay parameter a, and characterize the corresponding transition curves.

Background

The paper proves delay-independent sufficient stability and instability conditions for the linear recurrence x_{n+1}=\lambda x_n+\delta x_{h(n)}, but leaves intermediate regions unresolved. For proportional delays h(n)=\lfloor an\rfloor, the numerical regime diagrams indicate that stability can change as a varies, and the authors caution that finite-time computations cannot establish the asymptotic behavior in these regions.

A complete solution would determine the exact stability boundary in the (\lambda,\delta,a)-parameter space rather than relying on finite-horizon numerical diagnostics. This would resolve whether trajectories in the currently unclassified regions decay, remain bounded and nonconvergent, or grow asymptotically.