Difference equations of Volterra type and extension of Lyapunov's method
The author studies the nonlinear difference equation of Volterra type (1) \(\Delta x(n)=f(n,x(n),\sum^{n-1}_{s=n_ 0}G(n,s,x(s))),\) \(x(n_ 0)=n_ 0, n_ 0\in {\mathbb{N}}^+_{n_ 0},\) where \(f,G:{\mathbb{N}}^+_{n_ 0}\times {\mathbb{R}}^ d\times {\mathbb{R}}^ d\to {\mathbb{R}}^ d.\) With the help of an extension of Lyapunov's method and choosing a minimal class of suitable functions the author obtains conditions for the stability properties of (1). Another approach used by author is an estimation of the variation of Lyapunov's function by means of some functional.
nonlinear difference equation of Volterra type
scientific article; zbMATH DE number 4210732