On sign-definite pairs of functions

This paper introduces three different sign-definite pairs of functions. Roughly speaking, when a dynamic system is considered, sign-definite pairs of functions can provide sign characteristics of the second function in the zero locus of the first function when time approaches to infinity. These new definitions can be used to facilitate the analysis of uniform global asymptotic stability of dynamic systems. Some useful properties are explored in order to simplify the verification of these pairs. When applied to a time-varying dynamic system, these pairs can be employed to generalize the well-known Matrosov-type stability results. Several examples are presented to illustrate the obtained results.