We present mathematical analyses of the evolution of solutions of
the self-consistent equation derived from variational calculations
based on the displaced-oscillator-state and the
displaced-squeezed-state in spin-boson model at a zero temperature
and a finite temperature. It is shown that, for a given spectral
function defined as $J(\omega)=\pi\sum_k c_k^2=\ddfrac{\pi}{2}\alpha
\omega^{ s}\omega_{\rm c}^{ 1-s}$, there exists a universal $s_{\rm
c}$ for both kinds of variational schemes, the localized transition
happens only for $s\le s_{\rm c}$, moreover, the localized
transition is discontinuous for $s
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