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Kamenev-type oscillation criteria for second order nonlinear dynamic equations on time scales. (English) Zbl 1211.34115

Summary: The purpose of this paper is to establish oscillation criteria for the second order nonlinear dynamic equation

(r(t)(x Δ (t)) γ ) Δ +f(t,x(g(t)))=0,

on an arbitrary time scale 𝕋, where γ is a quotient of odd positive integers and r is a positive rd-continuous function on 𝕋. The function g:𝕋𝕋 satisfies g(t)t and lim t g(t)= and fC(𝕋×,). We establish some new sufficient conditions under which the above equation is oscillatory by using the generalized Riccati transformation. Our results in the special cases when 𝕋= and 𝕋= involve and improve some oscillation results for second-order differential and difference equations; and when 𝕋=h, 𝕋=q 0 and 𝕋= 2 our oscillation results are essentially new. Some examples illustrating the importance of our results are included.

MSC:
34N05Dynamic equations on time scales or measure chains
34K11Oscillation theory of functional-differential equations
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