International Journal of Mathematics and Mathematical Sciences 
Volume 15 (1992), Issue 1, Pages 119-128
doi:10.1155/S0161171292000140

On the oscillatory properties of the solutions of a class of integro-differential equations of neutral type

D. D. Bainov,1 A. D. Myshkis,2 and A. I. Zahariev3

 1P.O. Box 45, Sofia 1504, Bulgaria
2Plovdiv University “Paissii Hilendarski”, Bulgaria
3Moscow Institute of Railway Transport Engineering, Russia
Received 17 January 1990; Revised 6 September 1990

Abstract

In the present paper the oscillatory properties of the solutions of the equation[(Lx)(t)](n)+∫ItK(t,s,x(s))ds=0are investigated where n≥1, L is an operator of the difference type, It⊂ℝ, K:DK→ℝ, DK⫅ℝ3, x:[αx,∞]→ℝ. Under natural conditions imposed on L, It and K it is proved that for n even all ultimately nonzero solutions oscillate and for n odd they either oscillate or tend to zero as t→∞.