A sharp oscillation criterion for a linear delay differential equation

A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay

“…First, Lemma 2 infers that function A from Equation (11) is slowly varying. As already noted after Theorem 4 of [15], the slowly varying property together with continuity implies uniform continuity. Hence, p and τ are uniformly continuous, so Theorem 3 applies, which finishes the proof.…”

“…In a subsequent paper, Pituk, Stavroulakis, and the present author [15] generalized the above result and gave a class of functions p-broader than τ-periodic-for which Condition (6) is 'almost sharp'. More precisely, the following theorem was proved.…”

“…From Theorem 1, it is apparent that condition lim sup t→∞ A(t) ≥ 1/e is necessary for the oscillation of all solutions, so Theorem 3 is sharp in this sense. Example 9 of [15] showed that the slowly varying assumption is important: even in the constant delay case, the theorem does not hold if we omit that assumption.…”

“…, p n (t) := p(t n + t) and τ n (t) := τ(t n + t) for all t ≥ −κ and n ∈ N. (15) Then, applying (1) leads to the equation…”

A sharp oscillation criterion for a difference equation with constant delay

“…e as k → ∞, and therefore conditions (5) and (6) can be interpreted as the discrete analogue of (19) and 20 Our aim in this paper is to extend this result to the discrete case. To this end, assume that the sequence A(n) = n-1 i=n-k p(i) is slowly varying at infinity (see [11]), i.e., for every λ ∈ N, lim n→∞ [A(n + λ) -A(n)] = 0. It should be mentioned that the idea how to obtain sharp oscillation conditions in the continuous time case by considering slowly varying coefficients originated from Pituk [22].…”