Fixed point theorems for mappings with a contractive iterate at a point
Author:
Janusz Matkowski
Journal:
Proc. Amer. Math. Soc. 62 (1977), 344348
MSC:
Primary 54H25
MathSciNet review:
0436113
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Abstract: Let (X,d) be a complete metric space, , and be nondecreasing with respect to each variable. Suppose that for the function , the sequence of iterates tends to 0 in and . Furthermore, suppose that for each there exists a positive integer such that for all , Under these assumptions our main result states that T has a unique fixed point. This generalizes an earlier result of V. M. Sehgal and some recent results of L. Khazanchi and K. Iseki.
 [1]
L. F. Guseman, Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc. 26 (1970), 615618. MR 42 #919. MR 0266010 (42:919)
 [2]
K. Iseki, A generalization of SehgalKhazanchi's fixed point theorems, Math. Seminar Notes Kobe Univ. 2 (1974), 19. MR 0436108 (55:9058)
 [3]
L. Khazanchi, Results on fixed points in complete metric space, Math. Japonicae (to appear). MR 0400197 (53:4032)
 [4]
V. M. Sehgal, A fixed point theorem for mappings with a contractive iterate, Proc. Amer. Math. Soc. 23 (1969), 631634. MR 40 #3531. MR 0250292 (40:3531)
 [1]
 L. F. Guseman, Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc. 26 (1970), 615618. MR 42 #919. MR 0266010 (42:919)
 [2]
 K. Iseki, A generalization of SehgalKhazanchi's fixed point theorems, Math. Seminar Notes Kobe Univ. 2 (1974), 19. MR 0436108 (55:9058)
 [3]
 L. Khazanchi, Results on fixed points in complete metric space, Math. Japonicae (to appear). MR 0400197 (53:4032)
 [4]
 V. M. Sehgal, A fixed point theorem for mappings with a contractive iterate, Proc. Amer. Math. Soc. 23 (1969), 631634. MR 40 #3531. MR 0250292 (40:3531)
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DOI:
http://dx.doi.org/10.1090/S00029939197704361135
PII:
S 00029939(1977)04361135
Keywords:
Fixed point,
orbit,
complete metric,
Cauchy sequence
Article copyright:
© Copyright 1977
American Mathematical Society
