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In: Tatra Mountains Mathematical Publications, vol. 43, no. 2
Khadija Niri - J. P. Stavroulakis

On the oscillation of the solutions to delay and difference equations

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Year, pages: 2009, 173 - 187
Keywords: delay equation, difference equation, oscillatory solution, nonoscillatory solution.

Consider the first-order linear delay differential equation \begin{equation*} x\prime(t)+p(t)x\bl(τ (t)\br)=0,   t≥ t0,\eqno(1) \end{equation*} where $p,τ \in C\bl([t0,∞ ), \mathbb{R}+\br)$, $τ (t)$ is nondecreasing, $τ (t)0$ and $\limt\rightarrow ∞τ (t)=∞$, and the (discrete analogue) difference equation \begin{equation*} Δ x(n)+p(n)x\bl(τ (n)\br)=0,    n=0,1,2,…, \eqno(1)\prime \end{equation*} where $Δ x(n)=x(n+1)-x(n)$, $p(n)$ 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 $\limn\rightarrow ∞τ (n)=∞$. Optimal conditions for the oscillation of all solutions to the above equations are presented.