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