Long-Term Behavior of Non-Oscillating Solutions in
High-Order Forced and Disturbed Fractional Difference
Systems

This paper develops a comprehensive asymptotic framework for analyzing the long-term behavior of non-oscillatory solutions in high-order forced and disturbed fractional difference systems of the Caputo type. We consider a general nonlinear model in which memory effects, external forcing, and nonlinear disturbance terms interact within a higher-order nabla fractional operator. Despite the growing interest in discrete fractional calculus, existing results primarily address first-order or unforced systems, leaving a significant gap in understanding the asymptotic dynamics of complex high-order models with multiple nonlinearities. To address this gap, we derive new sufficient conditions ensuring that all eventually non-oscillatory solutions remain bounded within an explicit asymptotic envelope of the form
[
|Ψ(ι)| = O!\left((ι^{n-1})^{1/υ} R(ι,c)\right),
]
where (R(ι,c)) is a summable fractional kernel depending on the system’s coefficients. The established criteria incorporate delicate growth restrictions on the forcing term, the nonlinear damping functions, and their relative exponents, thereby generalizing earlier theorems and offering sharper bounds.

The analysis further yields refined exponential-type bounds under additional summability conditions, highlighting how fractional memory and nonlinear perturbations shape the qualitative behavior of solutions. Two detailed numerical examples validate the theoretical findings and illustrate the precision of the derived envelopes. The results significantly extend the current theory of fractional difference equations, providing new analytical tools for models arising in discrete population dynamics, engineering, and other applications where memory and external disturbances play essential roles.