ISSN:
1573-2878
Keywords:
Cluster points
;
bounded sequences
;
iterative processes
;
sequentially compact spaces
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract LetX be a closed subset of a topological spaceF; leta(·) be a continuous map fromX intoX; let {x i} be a sequence generated iteratively bya(·) fromx 0 inX, i.e.,x i+1 =a(x i),i=0, 1, 2, ...; and letQ(x 0) be the cluster point set of {x i}. In this paper, we prove that, if there exists a pointz inQ(x 0) such that (i)z is isolated with respect toQ(x 0), (ii)z is a periodic point ofa(·) of periodp, and (iii)z possesses a sequentially compact neighborhood, then (iv)Q(x 0) containsp points, (v) the sequence {x i} is contained in a sequentially compact set, and (vi) every point inQ(x 0) possesses properties (i) and (ii). The application of the preceding results to the caseF=E n leads to the following: (vii) ifQ(x 0) contains one and only one point, then {x i} converges; (viii) ifQ(x 0) contains a finite number of points, then {x i} is bounded; and (ix) ifQ(x 0) containsp points, then every point inQ(x 0) is a periodic point ofa(·) of periodp.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00933379
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