Invariants and related Liapunov functions for difference equations Original Research Article

Consider the difference equation View the MathML source , where xn is in Rk and View the MathML source is continuous where D ⊂ Rk. Suppose that I : Rk → R is a continuous invariant, that is, View the MathML source for every View the MathML source. We will show that if I attains an isolated minimum or maximum value at the equilibrium (fixed) point p of this system, then there exists a Liapunov function, namely View the MathML source and so the equilibrium p is stable. This result is then applied to some difference equations appearing in different fields of applications.