Monotonic and Oscillatory Solutions of a Linear Neutral Delay Equation with Infinite Lag

This paper is devoted to the discussion of monotonic and oscillatory solutions of the linear neutral delay equation \[ y'(t) = Ay(t) + \sum\limits_{i = 1}^M {B_i y\left( {\lambda _i t} \right)} + \sum\limits_{i = 1}^N {C_i y'} \left( {\eta _i t} \right),\] where $0 < \lambda _i < 1$ for $i = 1, \cdots ,M$, and $0 < \eta _i < 1$ for $i = 1, \cdots ,N$. Under one set of conditions, all nontrivial solutions are absolutely monotone. Under a different set of conditions, all nontrivial solutions oscillate unboundedly. This resolves most parts of the conjecture recently made by Feldstein and Jackiewicz. Some existence, uniqueness, and analyticity results are also included.