On a boundary value problem for integro-differential equations on the halfline

The following boundary value problem (0.1)x^′ (t) = p (t) ∫₀^{σ (t)} q (s) x (τ (s)) d s,(0.2)x (t) = φ (t) for t ∈ [τ₀, 0), under(lim inf, t → + ∞) | x (t) | < + ∞ is considered, where p ∈ C (R₊ ; (0, + ∞)), q ∈ L_loc (R₊ ; R₊), 0 ≤ σ (t) ≤ t, τ (t) < t for t ∈ R₊, lim_{t → + ∞} σ (t) = lim_{t → + ∞} τ (t) = + ∞, τ₀ = inf {τ (t) : t ∈ R₊}, φ ∈ C ([τ₀, 0)) and sup {{divides} φ (t) {divides} : t ∈ [τ₀, 0)} < + ∞. Sufficient conditions are obtained for problem (0.1) and (0.2) to have a solution, a unique solution and a unique oscillatory solution.