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Oscillation criteria for second-order delay, difference, and functional equations. (English) Zbl 1207.34082

Consider the second-order linear delay differential equation

x '' (t)+p(t)x(τ(t))=0,tt 0 ,

where pC([t 0 ,), + ), τC([t 0 ,),), τ(t) is nondecreasing, τ(t)t for tt 0 and lim t τ(t)=, the (discrete analogue) second-order difference equation

Δ 2 x(n)+p(n)x(τ(n))=0,

where Δx(n)=x(n+1)-x(n), Δ 2 =ΔΔ, p: + , τ:, τ(n)n-1, and lim n τ(n)=+, and the second-order functional equation

x(g(t))=P(t)x(t)+Q(t)x(g 2 (t)),tt 0 ,

where the functions P,QC([t 0 ,), + ), gC([t 0 ,),), g(t)¬t for tt 0 , lim t g(t)=, and g 2 denotes the second iterate of the function g, that is, g 0 (t)=t, g 2 (t)=g(g(t)), tt 0 . The most interesting oscillation criteria for the second-order linear delay differential equation, the second-order difference equation and the second-order functional equation, especially in the case where lim inf t τ(t) t τ(s)p(s)ds1/e and lim sup t τ(t) t τ(s)p(s)ds<1 for the second-order linear delay differential equation, and 0<lim inf t {Q(t)P(g(t))}1/4 and lim sup t {Q(t)P(g(t))}<1, for the second-order functional equation, are presented.

34K11Oscillation theory of functional-differential equations
39A21Oscillation theory (difference equations)
39B22Functional equations for real functions