On the oscillation of the solutions to delay and difference equations

Abstract

Consider the first-order linear delay differential equationwherewhere Δx(n) = x(n + 1) − x(n), p(n) is a sequence of nonnegative real numbers and τ (n) is a nondecreasing sequence of integers such that τ (n) ≤ n − 1 for all n ≥ 0 and lim n→∞ τ (n) = ∞. Optimal conditions for the oscillation of all solutions to the above equations are presented. 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: 39A11. K e y w o r d s: delay equation, difference equation, oscillatory solution, nonoscillatory solution.

New oscillation tests and some refinements for first-order delay dynamic equations

“…Proof Assume the contrary that (1) admits a nonoscillatory solution x. It is obvious that if (17) or (18) holds, then we arrive at a contradiction (see Remark 1). Hence, we only consider the case where (19) holds but ( 17) and ( 18) do not hold.…”

“…It is obvious that if (17) or (18) holds, then we arrive at a contradiction (see Remark 1). Hence, we only consider the case where (19) holds but (17) and (18) do not hold. It follows from Lemma 2 and Lemma 3 that…”

“…Readers are also referred to [17,21,22] for some other interesting results/discussions on this topic.…”