Qualitative investigation of a singular Cauchy problem for a functional differential equation
We consider the singular Cauchy problem $$txprime(t) = f(t,x(t),x(g(t)),xprime(t),xprime(h(t))), x(0) = 0,$$ 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 $x: (0, ρ] → ℝ$ ($ρ$ is sufficiently small) with required asymptotic properties.
English version (Springer): Ukrainian Mathematical Journal 57 (2005), no. 10, pp 1571-1589.
Citation Example: Chaichuk O. R., Zernov A. E. Qualitative investigation of a singular Cauchy problem for a functional differential equation // Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1344–1358.