Neumann boundary value problems with singularities in a phase variable

Summary.

The singular Neumann boundary value problem $(g({x}\ifmmode{'}\else$'$\fi){)}\ifmmode{'}\else$'$\fi = f(t,x,{x}\ifmmode{'}\else$'$\fi),{x}\ifmmode{'}\else$'$\fi(0) = {x}\ifmmode{'}\else$'$\fi(T) = 0$ is considered. Here f(t, x, y) satisfies local Carathéodory conditions on $[0,T] \times \mathbb{R} \times (0,\infty )$ and f may be singular at the value 0 of the phase variable y. Conditions guaranteeing the existence of a solution to the above problem with a positive derivative on (0, T) are given. The proofs are based on regularization and sequential techniques and use the topological transversality method.