Some reduction principles of equations independing or depending on a parameter and concerning Lipschitz continuous mappings
are introduced and then are applied to consider the existence of nontrivial solutions and nontrivial periodic solutions with a small norm and the existence of bifurcation and Hopf bifurcation points of equations concerning Lipschitz continuous mappings in Banach spaces, investigating the definition and nonvanishing of the topological degree of mappings or the existence of regular nonzero solutions of algebraic equations in a finite-dimensional space. Some well-known results of other authors are generalized.