Qualitative investigation of a singular Cauchy problem for a functional differential equation

We consider the singular Cauchy problem , where x: (0, τ) → ℝ, g: (0, τ) → (0, + ∞), h: (0, τ) → (0, + ∞), g(t) ≤ t, and h(t) ≤ t, t ∈ (0, τ), for linear, perturbed linear, and nonlinear equations. In each case, we prove that there exists a nonempty set of continuously differentiable solutions x: (0, ρ] → (ρ is sufficiently small) with required asymptotic properties. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1344–1358, October, 2005.