Qualitative analysis of an implicit singular Cauchy problem (Q2754813)

This paper presents the following singular Cauchy problemNEWLINE\[NEWLINE\alpha(t)\dot x=at+bx+\varphi(t,x,\dot x),\quad x(+0)=0.\tag{1}NEWLINE\]NEWLINE Here, \(\alpha(t)=\sigma t+\omega(t), |\omega(t)|\leq K_1t^2\), the function \(\varphi \) is continuous in the domain \(D:=\{(t,x,y):t\in (0,\tau),|x|<rt, |y|<r\}\) and satisfies the inequality \(|\varphi |\leq K_2t^2\); \(a,b,\sigma,r,K_1,K_2\) are positive numbers, \(b\neq\sigma,2\sigma \). The author introduces the set \(U(\rho,M)\subset C^1((0,\rho ] \mapsto \mathbb{R})\) which consists of functions \(x(t)\) satisfying the inequalities \(|x(t)-ct|\leq Mt^2, |\dot x(t)-c|\leq qMt, c:=a/(b-\sigma), q:=1+\bigl(|b|+|b-2\sigma |\bigr)/\sigma,\) and finds additional conditions on the function \(\varphi \) and constants \(a,b,\sigma,\rho,M\) which guarantee the existence of a unique solution to problem (1) in the case \(b<2\sigma \) and of infinitely many solutions to this problem in the case \(b>2\sigma \).