  This paper extends some results on the S-Lemma proposed by Yakubovich and
uses the improved results to investigate the asymptotic stability of a class of
switched nonlinear systems.
  Firstly, the strict S-Lemma is extended from quadratic forms to homogeneous
functions with respect to any dilation, where the improved S-Lemma is named the
strict homogeneous S-Lemma (the SHS-Lemma for short). In detail, this paper
indicates that the strict S-Lemma does not necessarily hold for homogeneous
functions that are not quadratic forms, and proposes a necessary and sufficient
condition under which the SHS-Lemma holds.
  It is well known that a switched linear system with two sub-systems admits a
Lyapunov function with homogeneous derivative (LFHD for short), if and only if
it has a convex combination of the vector fields of its two sub-systems that
admits a LFHD. In this paper, it is shown that this conclusion does not
necessarily hold for a general switched nonlinear system with two sub-systems,
and gives a necessary and sufficient condition under which the conclusion holds
for a general switched nonlinear system with two sub-systems. It is also shown
that for a switched nonlinear system with three or more sub-systems, the "if"
part holds, but the "only if" part may not.
  At last, the S-Lemma is extended from quadratic polynomials to polynomials of
degree more than $2$ under some mild conditions, and the improved results are
called the homogeneous S-Lemma (the HS-Lemma for short) and the non-homogeneous
S-Lemma (the NHS-Lemma for short), respectively.
  Besides, some examples and counterexamples are given to illustrate the main
results.
