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Oscillation criteria for delay, difference and functional equations. (English) Zbl 1063.34057

The paper is a survey on the most interesting oscillation criteria for the following type of equations:

1) the first-order linear delay differential equation x ' (t)+p(t)x(τ(t))=0, t>t 0 , with emphasis to the case

0<lim inf t τ(t) t p(s)ds1 eandlim sup t t-τ t p(s)ds<1;

2) the difference equation x n+1 -x n +p n x n-k =0, n=0,1,2,, with emphasis to the case

lim inf n i=n-k n-1 k k+1 k+1 andlim sup n i=n-k n p i <1;

3) the functional equation x(g(t))=P(t)x(t)+Q(t)x(g 2 (t)), tt 0 , with emphasis to the case

0<lim inf t {Q(t)P(g(t))}1 4andlim sup t {Q(t)P(g(t))}<1·

34K11Oscillation theory of functional-differential equations
39A12Discrete version of topics in analysis
39B22Functional equations for real functions