Oscillatory and asymptotic behavior of solutions for second-order mixed nonlinear integro-dynamic equations with maxima on time scales

Abstract

This paper is concerned with the oscillatory and asymptotic behavior for solutions of the following second-order mixed nonlinear integro-dynamic equations with maxima on time scales (r(t)(z?(t))?)? + ?t0 a(t,s) f(s, x(s))?s + ?n,i=1 qi(t) max s?[?i(t),?i(t)] x?(s) = 0, where z(t) = x(t) + p1(t)x(?1(t)) + p2(t)x(?2(t)), t ? [0,+?)T. The oscillatory behavior of this equation hasn?t been discussed before, also our results improve and extend some results established by Grace et al. [2] and [8].