Oscillations of difference equations with several deviated arguments

Oscillations of difference equations with non-monotone retarded arguments

Keyword: Difference equation Non-monotone argument Retarded argument Oscillatory solutions Nonoscillatory solutions Consider the first-order retarded difference equation DxðnÞ þ pðnÞx sðnÞ ð Þ 1⁄4 0; n 2 N0 where ðpðnÞÞnP0 is a sequence of nonnegative real numbers, and ðsðnÞÞnP0 is a sequence of integers such that sðnÞ 6 n 1, n P 0; and limn!1 sðnÞ 1⁄4 1. Under the assumption that the retarded ...

Oscillations of Advanced Difference Equations with Variable Arguments

Consider the first-order advanced difference equation of the form ∇x(n)− p(n)x(μ(n)) = 0, n ≥ 1, [∆x(n)− p(n)x(ν(n)) = 0, n ≥ 0], where ∇ denotes the backward difference operator ∇x(n) = x(n) − x(n − 1), ∆ denotes the forward difference operator ∆x(n) = x(n+ 1)− x(n), {p(n)} is a sequence of nonnegative real numbers, and {μ(n)} [{ν(n)}] is a sequence of positive integers such that μ(n) ≥ n+ 1 f...

Necessary and Sufficient Conditions for Oscillations of Neutral Delay Difference Equations with Several Coefficients

Minimal and maximal solutions to first-order differential equations with state-dependent deviated arguments

We prove some new results on existence of solutions to first-order ordinary differential equations with deviated arguments. Delay differential equations are included in our general framework, which even allows deviations to depend on the unknown solutions. Our existence results lean on new definitions of lower and upper solutions introduced in this article, and we show with an example that simi...

Linearized Oscillation of Nonlinear Difference Equations with Advanced Arguments

This paper is concerned with the nonlinear advanced difference equation with constant coefficients