TY - JOUR
AU - A. E. Zernov
AU - O. R. Chaichuk
PY - 2005/10/25
Y2 - 2024/09/12
TI - Qualitative investigation of a singular Cauchy problem for a functional differential equation
JF - Ukrains’kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 57
IS - 10
SE - Research articles
DO -
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/3690
AB - 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.
ER -