| Abstrakt: |
LetX be a closed subset of a topological spaceF; leta(·) be a continuous map fromX intoX; let {xi} be a sequence generated iteratively bya(·) fromx0 inX, i.e.,xi+1 =a(xi),i=0, 1, 2, ...; and letQ(x0) be the cluster point set of {xi}. In this paper, we prove that, if there exists a pointz inQ(x0) such that (i)z is isolated with respect toQ(x0), (ii)z is a periodic point ofa(·) of periodp, and (iii)z possesses a sequentially compact neighborhood, then (iv)Q(x0) containsp points, (v) the sequence {xi} is contained in a sequentially compact set, and (vi) every point inQ(x0) possesses properties (i) and (ii). The application of the preceding results to the caseF=En leads to the following: (vii) ifQ(x0) contains one and only one point, then {xi} converges; (viii) ifQ(x0) contains a finite number of points, then {xi} is bounded; and (ix) ifQ(x0) containsp points, then every point inQ(x0) is a periodic point ofa(·) of periodp. |
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