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. First, 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 (SHS-Lemma). 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 subsystems admits a Lyapunov function with homogeneous derivative (LFHD) if and only if it has a convex combination of the vector fields of its two subsystems 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 subsystems, and gives a necessary and sufficient condition under which the conclusion holds for a general switched nonlinear system with two subsystems. It is also shown that for a switched nonlinear system with three or more subsystems, the "if" part holds, but the "only if" part may not. Lastly, 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 (HS-Lemma) and the nonhomogeneous S-Lemma (NHS-Lemma), respectively. In addition, some examples and counterexamples are given to illustrate the main results.