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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)
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