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

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→∞.

Keywords

• integral equations
• operator differential equations
• oscillatory and non-oscillatory solutions.