An Oscillation criteria for second order functional equations

Abstract

This paper is concerned with the oscillation of second order linear functional equations of the form x(g(t)) = P(t)x(t) + Q(t)x(g2 (t)), where P,Q,g : [t0, ∞) → R+ = [0, ∞) are given real valued functions such that g(t) ≢ t, limt→∞g(t) = ∞. It is proved here that when 0 ≤ m : = lim inft→∞Q(t)P(g(t)) ≤ 1/4 all solutions of this equation oscillate if the condition lim sup t→∞Q(t)P(g(t))>(1+1−4m2)2 (*)is satisfied. It should be emphasized that the condition (*) can not be improved in some sense.