On Some Properties of Solutions of Second Order Linear Functional Differential Equations

The properties of solutions of the equation u′′(t) = p1(t)u(τ1(t)) + p2(t)u′(τ2(t)) are investigated where pi : [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 one. Moreover, τi(t) ≥ t (i = 1, 2), p1(t) ≥ 0, p2(t) ≤ (4− ε)τ ′ 2(t)p1(t), ε = const > 0 and ∫ +∞ a (τ1(t)− t)p1(t)dt < +∞. In particular, it is proved that solutions whose derivatives are square integrable on [a, +∞[ form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that ∫ +∞ a tp1(t)dt = +∞. Consider the differential equation u′′(t) = p1(t)u(τ1(t)) + p2(t)u(τ2(t)), (1) where pi : [a, +∞[→ R (i = 1, 2) are locally summable functions, τi : [a, +∞[→ R (i = 1, 2) are measurable functions and τi(t) ≥ t for t ≥ a (i = 1, 2). (2) We say that a solution u of the equation (1) is a Kneser-type solution if it satisfies the inequality u′(t)u(t) ≤ 0 for t ≥ a0 for some a0 ∈ [a, +∞[. A set of such solutions is denoted by K. By W we denote a space of solutions of (1) that satisfy ∫ +∞ a u ′(t)dt < +∞. The results of [1, 2] imply that if p1(t) ≥ 0 for t ≥ a and the condition (i) τi(t) ≡ t, (i = 1, 2), +∞ ∫