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1. 10. 2000.
Invariants and related Liapunov functions for difference equations
Abstract Consider the difference equation x n+1 = f(x n , where xn is in Rk and f : D → D is continuous where D ⊂ Rk. Suppose that I : Rk → R is a continuous invariant, that is, I( f(x) ) = I( x ) for every x ∈ D . 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 ±(I( x ) − I( p )) and so the equilibrium p is stable. This result is then applied to some difference equations appearing in different fields of applications.