# On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity

• A. E. Zernov

### Abstract

We prove the existence of continuously differentiable solutions with required asymptotic properties as t → +0 and determine the number of solutions of the following Cauchy problem for a functional differential equation: $$\alpha \left( t \right)x\prime \left( t \right) = at + b_1 x\left( t \right) + b_2 x\left( {g\left( t \right)} \right) + \phi \left( {t,x\left( t \right),x\left( {g\left( t \right)} \right),x\prime \left( {h\left( t \right)} \right)} \right),\quad x\left( 0 \right) = 0,$$ where α: (0, τ) → (0, +∞), g: (0, τ) → (0, +∞), and h: (0, τ) → (0, +∞) are continuous functions, 0 < g(t) ≤ t, 0 < h(t) ≤ t, t ∈ (0, τ), $\begin{gathered} \alpha \left( t \right)x\prime \left( t \right) = at + b_1 x\left( t \right) + b_2 x\left( {g\left( t \right)} \right) + \phi \left( {t,x\left( t \right),x\left( {g\left( t \right)} \right),x\prime \left( {h\left( t \right)} \right)} \right),\quad x\left( 0 \right) = 0, \\ \mathop {\lim }\limits_{t \to + 0} \alpha \left( t \right) = 0 \\ \end{gathered}$ , and the function ϕ is continuous in a certain domain.
Published
25.04.2001
How to Cite
ZernovA. E. “On the Solvability and Asymptotics of Solutions of One Functional Differential Equation With Singularity”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 53, no. 4, Apr. 2001, pp. 455-6, https://umj.imath.kiev.ua/index.php/umj/article/view/4268.
Issue
Section
Research articles