Behavior of the Solutions of Functional Equations

摘要: In the last decades the oscillation theory of delay differential equations has been extensively developed. The oscillation theory of discrete analogues of delay differential equations has also attracted growing attention in the recent years. Consider the first-order delay differential equation, $$\displaystyle \begin{aligned} x'(t) + p(t) \, x(\tau(t)) = 0, \,\,\,\,\,\, t \ge t_0, \end{aligned} $$ where , τ(t) is nondecreasing, τ(t) < t for t ≥ t 0 and $$\displaystyle \begin{aligned} \varDelta x(n) + p(t) \, x(\tau(n)) = 0, \,\,\,\,\,\, n = 0, 1, 2, \ldots , \end{aligned} $$ , and the (discrete analogue) difference equation, where 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 . In this review chapter, a survey of the most interesting oscillation conditions is …