Asian Research Journal of Mathematics

This study examines the monotonic properties of nonoscillatory solutions of a second-order linear delay difference equation with positive variable coefficients and a fixed even delay. The analysis focuses on eventually positive solutions of degree zero, also referred to as Kneser-type solutions. By introducing suitable auxiliary sequences and applying repeated summation arguments, decreasing behaviour is established for a weighted transformation of the solution. This property yields estimates for delayed solution terms and leads to sufficient conditions under which degree-zero nonoscillatory solutions cannot exist. A further family of auxiliary sequences is then constructed to obtain a complementary monotonicity property for another weighted transformation. Combining the decreasing and increasing transformations produces an additional nonexistence criterion for Kneser-type solutions. Corollaries are derived from the principal estimates, including conditions expressed through bounds on the auxiliary sequences. Two examples involving constant coefficients are presented to demonstrate how the criteria can be evaluated for specific delay equations and to identify parameter ranges that exclude degree-zero nonoscillatory solutions. The results extend the qualitative analysis of second-order delay difference equations by providing refined monotonic estimates and related oscillation conditions within the stated assumptions. The approach is analytical and preserves the variable-coefficient framework while relying on positivity, convergence, and boundedness requirements imposed on the constructed sequences.