On some properties of solutions of second-order linear functional differential equations

Abstract

The properties of solutions of the equationu″(t) =p ₁(t)u(τ₁(t)) +p ₂(t)u′(τ₂(t)) are investigated wherep _i :a, + ∞[→R (i=1,2) are locally summable functions τ₁ :a, + ∞[→R is a measurable function, and τ₂ :a, + ∞[→R is a nondecreasing locally absolutely continuous function. Moreover, τ _i (t) ≥t (i = 1,2),p ₁(t)≥0,p ²₂ (t) ≤ (4 - ɛ)τ ^′₂ (t)p ₁(t), ɛ =const > 0 and$\int {_a^{ + \infty } } (\tau _1 (t) - t)p_1 (t)dt< + \infty$. In particular, it is proved that solutions whose derivatives are square integrable on [α,+∞] form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that$\int {_a^{ + \infty } } tp_1 (t)dt = + \infty$.