On the asymptotic behavior of solutions to a singular Cauchy problem (Q2784878)

The authors consider the singular Cauchy problem NEWLINE\[NEWLINE\alpha (t) x'=F(t,x,x'), \qquad x(0)=0,NEWLINE\]NEWLINE with \(x:(0,\tau)\to\mathbb{R},\) \(\lim_{t\to {0+}} \alpha(t) = 0, \) \(\lim_{\alpha\to{0+}}\frac{\alpha(t)}{t}= \infty.\) First, the existence of a continuously differentiable solution \(x(t)\) such that \(x'(t)\to 0,\) \(t\to 0,\) is investigated. An asymptotic estimation on the function \(x(t)\) and on the derivative \(x'(t)\) are found. A uniqueness theorem is proved with the help of a fixed-point theorem. The authors also consider the existence of a continuously differentiable solution \(x(t)\) such that \(x'(t)\to+\infty\) as \(t\to 0_+.\) The asymptotic behavior of the solution is investigated and the number of solutions is found.