Second Order Unstable Type Difference Equations with Deviating Arguments: New Asymptotic Results

Differential equations with deviating arguments serve as a fundamental cornerstone in mathematical modeling, offering a robust framework for characterizing the dynamics of various systems across multiple disciplines. Meanwhile, oscillatory theorems are essential in analyzing the intrinsic vibrational patterns within dynamic systems, providing critical insights into their stability and periodicity. This study focuses on examining the behavior of half-linear second-order difference equations of an unstable type when their arguments are altered from the form <img src=image/13440270_01.gif>. Utilizing the summation averaging method alongside the generalized Riccati transformations, we initiate new properties of monotonic to non-oscillatory solutions, enabling us to establish conditions that eliminate specific types of non-oscillatory behavior. These findings lead to novel oscillation criteria applicable to second-order difference equations with both advanced and delayed arguments. A key application of these results is the analysis of the oscillatory nature of difference equations arising in the Thomas–Fermi (T–F) model, a fundamental equation in physics. The T–F equation represents the simplest formulation for modeling the screened electrostatic Coulomb potential around a highly charged nucleus and its surrounding electron cloud. Beyond atomic physics, this equation finds broad applicability across numerous physical domains. The examples presented at the conclusion of this work illustrate the enhanced results achieved, demonstrating improvements over previously established findings.