Oscillations of Delay and Difference Equations with Variable Coefficients and Arguments

Abstract

Consider the first-order linear differential equation with several deviating arguments:

$$\displaystyle{ x^{{\prime}}(t) +\sum _{ i=1}^{m}p_{ i}(t)x(\tau _{i}(t)) = 0\ \left [x^{{\prime}}(t) -\sum _{ i=1}^{m}p_{ i}(t)x(\sigma _{i}(t)) = 0\right ]\mbox{, }t \geq t_{0} }$$

and the discrete analogue difference equation

$$\displaystyle{ \Delta x(n)+\sum _{i=1}^{m}p_{ i}(n)x(\tau _{i}(n)) = 0\mbox{, }n \geq 0\ \left [\nabla x(n) -\sum _{i=1}^{m}p_{ i}(n)x(\sigma _{i}(n)) = 0\mbox{, }n \geq 1\right ]\text{ } }$$

where the functions $p_{i},\tau _{i},\sigma _{i} \in C([t_{0,}\infty ), \mathbb{R}^{+})$ and τ _i (t) [σ _i (t)] are retarded (τ _i (t) ≤ t) [advanced (σ _i (t) ≥ t)] arguments, for every i = 1, 2, …, m, lim _{t → ∞} τ _i (t) = ∞, and (p _i (n)), 1 ≤ i ≤ m are sequences of nonnegative real numbers, τ _i (n) [σ _i (n)], 1 ≤ i ≤ m are retarded (τ _i (n)  ≤ n − 1) [advanced (σ _i (n) ≥ n + 1)] arguments, $\lim \limits _{n\rightarrow \infty }\tau _{i}(n) = \infty$, and $\Delta$ [∇] denotes the forward [backward] difference operator $\Delta x(n) = x(n + 1) - x(n)$ [∇x(n) = x(n) − x(n − 1)]. A survey on the oscillation of all solutions to these equations is presented in the case of several deviating arguments and especially when well-known oscillation conditions are not satisfied. Examples illustrating the results are given.

Dedicated to the memory of George Roger Sell an excellent scientist and colleague.