  We obtain a new quantitative deformation lemma, and then gain a new mountain
pass theorem. More precisely, the new mountain pass theorem is independent of
the functional value on the boundary of the mountain, which improves the well
known results (\cite{AR,PS1,PS2,Qi,Wil}). Moreover, by our new mountain pass
theorem, new existence of nontrivial periodic solutions for some nonlinear
second-order discrete systems is obtained, which greatly improves the result in
\cite{Z04}.
