Geodesic
On Sehgal's Maps with a Contractive Iterate at a Point
Publications de l'Institut Mathématique, _N_S_33 (1983) no. 47, p. 59 .

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

Let $(X,d)$ be a complete metric space and $T$ a mapping of $X$ into itself. Suppose that for each $x\in X$ there exists a positive integer $n=n(x)$ such that for all $y\in X$, $ d(T^nx,T^ny)\leq \alpha\max\{d(x,y), d(x,Ty), d(x,T^2y),\dots, d(x,T^ny), d(x,T^nx)\}, $ holds tor some $\alpha1$. With these assumptions our main result states that $T$ has a unique fixed point. This generalizes an earlier result of V. M. Sehgal and a recent result of the author.
Classification : 54H25 47H10
Keywords: Mappings with contractive iteration at a point, fixed points, convergence of iterations 
@article{PIM_1983_N_S_33_47_a7,
     author = {Ljubomir \'Ciri\'c},
     title = {On {Sehgal's} {Maps} with a {Contractive} {Iterate} at a {Point}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {59 },
     publisher = {mathdoc},
     volume = {_N_S_33},
     number = {47},
     year = {1983},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1983_N_S_33_47_a7/}
}
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Ljubomir Ćirić. On Sehgal's Maps with a Contractive Iterate at a Point. Publications de l'Institut Mathématique, _N_S_33 (1983) no. 47, p. 59 . http://geodesic.mathdoc.fr/item/PIM_1983_N_S_33_47_a7/