A set of time map $U(\alpha,\lambda)$ formulae for the problem $x'' +\lambda f(x) =0$ $\; (i),$ $x(a) = 0, \; x'(a) = \alpha>0\; (ii)$ is provided. We derive conditions for monotonicity and convexity of $U$ in both arguments and apply them a) to obtain some multiplicity results for the Dirichlet problem $(i)$, $x(a)=x(b)=0\; (iii)$ and b) to get assertions about similarity of the Dirichlet (Neumann) nonlinear Fu\v{c}\'{i}k type spectra to the classical Fu\v{c}\'{i}k spectrum.