Invariants and related Liapunov functions for difference equations
Date of Original Version
Consider the difference equation xn+1 = f(xn), 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. © 2000 Elsevier Science Ltd. All rights reserved.
Applied Mathematics Letters
Kulenović, M. R.S.. "Invariants and related Liapunov functions for difference equations." Applied Mathematics Letters 13, 7 (2000): 1-8. doi:10.1016/S0893-9659(00)00068-9.