A singular initial value problem for some functional differential equations (Q1769361)

Summary: For the initial value problem \(t^rx'(t)=at+b_1x(t)+ b_2 x(q_1t)+b_3t^rx' (q_2t)+\varphi(t,x(t)\), \(x(q_1t),x'(t),x'(q_2t))\), \(x(0) =0\), where \(r>1\), \(0<q_i \leq 1\), \(i\in\{1,2\}\), we find a nonempty set of continuously differentiable solutions \(x:(0,\rho]\to \mathbb{R}\), each of which possesses nice asymptotic properties when \(t\to+0\).