Kwong-wong-type Integral Equation on Time Scales

Consider the second-order nonlinear dynamic equation [r(t)x(ρ(t))] + 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)x∆(t) f(xσ(t)) = P(t) + Z ∞ σ(t) rσ(s)[ R 1 0 f (xh(s))dh][x ∆(s)]2 f(x(s))f(xσ(s)) ∆s is satisfied for t ≥ T0, where Pσ(t) = R∞ σ(t) p(s)∆s, and xh(s) = x(s) + hμ(s)x∆(s). As an application, we show that the superlinear dynamic equation [r(t)x(ρ(t))] + p(t)f(x(t)) = 0, is oscillatory, under certain conditions.