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 ∫t (r(t)(z∆ (t))γ )∆ + where 0 a(t, s) f (s, x(s))∆s + n∑ qi (t) max s∈[τi (t),ξi (t)]xα (s) = 0, z(t) = x(t) + p1 (t)x(η1 (t)) + p2 (t)x(η2 (t)), t ∈ [0, +∞)T. i=1 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].