Oscillation criteria for delay and difference equations with non-monotone arguments

Abstract

Consider the first-order linear delay differential equation(1)x′(t)+p(t)x(τ(t))=0,t⩾t0,where p,τ∈C([t0,∞),R+), τ(t)<t for t⩾t0 and limt→∞τ(t)=∞, and the (discrete analogue) difference equation(1′)Δx(n)+p(n)x(τ(n))=0,n=0,1,2,…,where Δ denotes the forward difference operator Δx(n)=x(n+1)-x(n),p(n) is a sequence of nonnegative real numbers and τ(n) is a sequence of integers such that τ(n)⩽n-1 for all n⩾0 and limn→∞τ(n)=∞. The state-of-the-art on the oscillation of all solutions to these equations are established especially in the case of non-monotone arguments. Examples illustrating the results are given.