Limit behaviour of trajectories involving subgradients of convex functions. (English) Zbl 0623.49009

Pelant, Jan; Poljak, Svatopluk; Turzík, Daniel

Commentat. Math. Univ. Carol. 28, 457-466 (1987).

We investigate trajectories \(\{y_ i\}^{\infty}_{i=1}\) of mappings \(h=f\circ g\) such that \(y_{i+1}=fg(y_ i,...,y_{i-q+1})\) where \(q\geq 1\), \(f: {\mathbb{R}}^ m\to R^ m\) is cyclically monotone and g is one of the following (a) \(g=g(y_ t,...,y_{t-q+1})=\sum^{q}_{k=1}A_ ky_{t-k+1}\) where \(q\geq 1\) and \(A_{k-q+1}=A^ T_ k\) (the transposed matrix) for \(k=1,...,q.\)
(b) \(g=g(y_ t)\) where g is cyclically monotone (for \(q=1).\)
We show that there is an integer r such that (*) \(\lim_{i\to \infty}\| y_{i+r}-y_ i\| =0\) provided the trajectory is bounded. (Namely, it is \(r=q+1\) in case (a) and \(r=1\) in case (b).
The paper is motivated by the study of cellular automata.

MSC:

49J45	Methods involving semicontinuity and convergence; relaxation
26B25	Convexity of real functions of several variables, generalizations
49J52	Nonsmooth analysis
47H05	Monotone operators and generalizations
68Q80	Cellular automata (computational aspects)

Keywords:

convex function; subgradient; trajectory; cellular automata