AN OSCILLATION CRITERIA FOR SECOND ORDER FUNCTIONAL EQUATIONS

Abstract

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) 6≡ t, limt!1 g(t) = ∞. It is proved here that when 0 ≤ m := liminft!1 Q(t)P(g(t)) ≤ 1/4 all solutions of this equation oscillate if the condition limsup t!1 Q(t)P(g(t)) > 1 + √1 − 4m 2 2 (∗) is satisfied. It should be emphasized that the condition (∗) can not be improved in some sense.

Key words: Oscillation, nonoscillation, functional equations

CLC Number:

39B22

SHEN Jian-Hua, I.P. Stavroulakis. AN OSCILLATION CRITERIA FOR SECOND ORDER FUNCTIONAL EQUATIONS[J].Acta mathematica scientia,Series B, 2002, 22(1): 56-62.

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