The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type ∆x(n) + ∑ q k=−p ak(n)x(n + k) = 0, n > n0, where ∆x(n) = x(n + 1) − x(n) is the difference operator and {ak(n)} are sequences of real numbers for k = −p, . . . , q, and p > 0, q > 0. We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.
We consider preservation of exponential stability for the scalar nonoscillatory linear equation with several delays x˙ (t) + ∑m k=1 ak(t)x(hk(t)) = 0, ak(t) ≥ 0 under the addition of new terms and a delay perturbation. We assume that the original equation has a positive fundamental function; our method is based on Bohl-Perron type theorems. Explicit stability conditions are obtained.