G. Kvinikadze

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH PROPERTY A

The differential equation

u^{(n)}(t) + ∑_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)}u(s) d_{s}r_{i}(s, t) = 0  (1)

is considered, where n ≥ 2, σ_{i}, τ_{i} ∈ C(R_{+}; (0, ∞)), τ_{i}(t) ≤ σ_{i}(t), while the functions r_{i} : R_{+} × R_{+} → R are nondecreasing in the first argument and Lebesgue integrable in the second argument on any finite subsegment of [0, +∞).

We say that equation (1) has Property A if any of its solutions is oscillatory when n is even and either is oscillatory or satisfies

|u^{(i)}(t)| ↓ 0 for t ↑ +∞ (i = 0, …, n − 1)

when n is odd. Sufficient conditions are established for equation (1) to have Property A. The obtained results are new even for a special case of equation (1) – the differential equation with deviated arguments.