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

Consider the first-order linear delay differential equation(1)x^'(t)+p(t)x(@t(t))=0,t>=t"0,where p,@[email protected]?C([t"0,~),R^+), @t(t)=t"0 and lim"t"->"[email protected](t)=~, and the (discrete analogue) difference equation(1')@Dx(n)+p(n)x(@t(n))=0,n=0,1,2,...,where @D denotes the forward difference operator @Dx(n)=x(n+1)-x(n),p(n) is a sequence of nonnegative real numbers and @t(n) is a sequence of integers such that @t(n)==0 and lim"n"->"[email protected](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.

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