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


Hassan Ahmed Hassan Aqwa, Mokhtar Ahmed Abdel Naby, Heba Mohamed Arafa




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