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

Matkowski, Janusz

doi:10.1090/S0002-9939-1977-0436113-5

Fixed point theorems for mappings with a contractive iterate at a point
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Proc. Amer. Math. Soc. 62 (1977), 344-348 Request permission

Abstract:

Let (X,d) be a complete metric space, $T:X \to X$, and $\alpha :[0,\infty )^5 \to [0,\infty )$ be nondecreasing with respect to each variable. Suppose that for the function $\gamma (t) = \alpha (t,t,t,2t,2t)$, the sequence of iterates ${\gamma ^n}$ tends to 0 in $[0,\infty )$ and ${\lim _{t \to \infty }}(t - \gamma (t)) = \infty$. Furthermore, suppose that for each $x \in X$ there exists a positive integer $n = n(x)$ such that for all $y \in X$, \[ d({T^n}x,{T^n}y) \leqslant \alpha (d(x,{T^n}x),d(x,{T^n}y),d(x,y),d({T^n}x,y),d({T^n}y,y)).\] 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.

References

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 266010, DOI 10.1090/S0002-9939-1970-0266010-3
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 436108
Lalita Khazanchi, Results on fixed points in complete metric space, Math. Japon. 19 (1974), no. 3, 283–289. MR 400197
V. M. Sehgal, A fixed point theorem for mappings with a contractive iterate, Proc. Amer. Math. Soc. 23 (1969), 631–634. MR 250292, DOI 10.1090/S0002-9939-1969-0250292-X