Oscillation and asymptotic behavior of neutral differential equations with deviating arguments

Abstract

Consider the neutral differential equation

where q≠0, p, τ, and σ are real numbers. Let y(t) be a nonoscillatory solution of Eq. (1). Then limt_t→∞y(t) is determined for all cases, except:

. Two conjectures (as well as evidence indicating their possible validity) are given to cover the missing cases i), ii), and iii). It is also shown that if qτ≧0, or if qτ<0 and p≧0, then each of the following conditions implies that every solution of Eq. (1) is oscillatory:

.

AMS (MOS)::

• 34K15
• 34G10