Fixed point theorems for mappings with a contractive iterate at a point

Author:
Janusz Matkowski

Journal:
Proc. Amer. Math. Soc. **62** (1977), 344-348

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 ,

*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 Jr.,*Fixed point theorems for mappings with a contractive iterate at a point*, Proc. Amer. Math. Soc.**26**(1970), 615–618. MR**0266010**, 10.1090/S0002-9939-1970-0266010-3**[2]**Kiyoshi Iseki,*A generalization of Sehgal-Khazanchi’s fixed point theorems*, Math. Sem. Notes Kobe Univ.**2**(1974), no. 2, paper no. XV, 9. MR**0436108****[3]**Lalita Khazanchi,*Results on fixed points in complete metric space*, Math. Japon.**19**(1974), no. 3, 283–289. MR**0400197****[4]**V. M. Sehgal,*A fixed point theorem for mappings with a contractive iterate*, Proc. Amer. Math. Soc.**23**(1969), 631–634. MR**0250292**, 10.1090/S0002-9939-1969-0250292-X

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DOI:
https://doi.org/10.1090/S0002-9939-1977-0436113-5

Keywords:
Fixed point,
orbit,
complete metric,
Cauchy sequence

Article copyright:
© Copyright 1977
American Mathematical Society