On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity - Zernov, A. E. (Ukrains’kyi Matematychnyi Zhurnal, 2001)
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: α ( t )x′ ( t ) = at + b_1 x( t ) + b_2 x( g( t ) ) + φ ( t,x( t ),x( g( t ) ),x′ ( h( t ) ) ),x( 0 ) = 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), \(gathered α ( t )x′ ( t ) = at + b_1 x( t ) + b_2 x( g( t ) ) + φ ( t,x( t ),x( g( t ) ),x′ ( h( t ) ) ),x( 0 ) = 0, _t → + 0 α ( t ) = 0 gathered\) , and the function ϕ is continuous in a certain domain.