On the solvability of the Cauchy problem for a certain class of equations with deviations depending on the unknown function (Q1901908)

The author considers the Cauchy problem (1) \(\dot x(t) = (Fx)(t)\), \(x(0) = \alpha\), where \[ (Fx)(t) = \begin{cases} f \Bigl( x \biggl( h \bigl( x(t) \bigr) \biggr) \Bigr) \quad & \text{for } h \bigl( x(t) \bigr) \in [0,b], \\ 0 \quad & \text{for } h \bigl( x(t) \bigr) \notin [0,b], \end{cases} \] i.e. the deviation \(h\) of the argument depends on the unknown function \(x\). The author obtains three different sufficient conditions which imply that (1) has a solution.