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Approximation of Coincidence Points and Common Fixed Points of a Collection of Mappings of Metric Spaces
T. N. Fomenko M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics
Abstract:
On a complete metric space
$X$, we solve the problem of constructing an algorithm (in general, nonunique) of successive approximations from any point in space to a given closed subset
$A$. We give an estimate of the distance from an arbitrary initial point to the corresponding limit points. We consider three versions of the subset
$A$: (1)
$A$ is the complete preimage of a closed subspace
$H$ under a mapping from
$X$ into the metric space
$Y$; (2)
$A$ is the set of coincidence points of
$n$ (
$n>1$) mappings from
$X$ into
$Y$; (3)
$A$ is the set of common fixed points of
$n$ mappings of
$X$ into itself (
$n=1,2,\dots$). The problems under consideration are stated conveniently in terms of a multicascade, i.e., of a generalized discrete dynamical system with phase space
$X$, translation semigroup equal to the additive semigroup of nonnegative integers, and the limit set
$A$. In particular, in case (2) for
$n=2$, we obtain a generalization of Arutyunov's theorem on the coincidences of two mappings. In case (3) for
$n=1$, we obtain a generalization of the contraction mapping principle.
Keywords:
metric space, successive approximations, coincidence point, fixed point, discrete dynamical system, translation semigroup, contraction mapping principle.
UDC:
515.124,
515.126.4,
517.938.5 Received: 21.05.2008
DOI:
10.4213/mzm5179
Fulltext:
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