KWONG-WONG-TYPE INTEGRAL EQUATION ON TIME SCALES.

Consider the second-order nonlinear dynamic equation [r(t)xΔ(ρ(t))]Δ + p(t)f(x(t)) = 0, where p(t) is the backward jump operator. We obtain a Kwong-Wong-type integral equation, that is: If x(t) is a nonoscillatory solution of the above equation on [T0,∞), then the integral equation rσ(t)xΔ(t)/f(xσ(t)) = Ρσ(t) + ∫∞ σ(t) rσ(s)[∫¹⁰ f′(xh(s))dh][xΔ(s)]²/f(x(s))f(xσ(s)) Δs is satisfied for t ≥ T⁰, where Ρσ(t) = ∫∞σ(t) p(s)Δs, and xh(s) = x(s) + hµ(s)xΔ(s). As an application, we show that the superlinear dynamic equation [r(t)xΔ(ρ(t))]Δ + p(t)f(x(t)) = 0, is oscillatory, under certain conditions. [ABSTRACT FROM AUTHOR]