Qualitative investigation of a singular Cauchy problem for a functional differential equation - Zernov, A. E.; Chaichuk, O. R. (Ukrains’kyi Matematychnyi Zhurnal, 2005)
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.