# On a fixed point theorem by totik

On a fixed point theorem by totik
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In this paper we prove a common fixed point theorem see Theorem 3.1 in metric spaces for two self-mappings satisfying a general implicit relation involving the diameter of nite sets, without requiring continuity. This theorem may be considered as a generalization of a result by Totik 1983. Also, it unies and generalizes some other results obtained by Fisher 1977, Akkouchi 2001 and Nova 1997.

Tipo de documento: Artículo - Article

Source: http://www.bdigital.unal.edu.co

## Teaser

Bol.

Mat.

16(1), 21–32 (2009)
21
On a fixed point theorem by Totik
Mohamed Akkouchi1
Faculté des Sciences–Semlalia
Département de Mathématiques
Université Cadi Ayyad, Avenue du Prince My.

Abdellah, B.P.

2390
Marrakech – Maroc – Morocco
En este artı́culo se demuestra un teorema de punto fijo común (Teorema 3.1) en espacios métricos para dos auto–mapeos que satisfacen una
relación implı́cita general que involucra el diámetro de conjuntos finitos
sin exigir continuidad.

Este teorema se puede considerar como una generalización de un resultado de Totik (1983).

También, unifica y generaliza
algunos otros resultados obtenidos por Fisher (1977), Akkouchi (2001) y
Nova (1997).
Palabras claves: Teoremas de punto fijo común en espacios métricos
In this paper we prove a common fixed point theorem (see Theorem 3.1) in
metric spaces for two self–mappings satisfying a general implicit relation
involving the diameter of finite sets, without requiring continuity.

This
theorem may be considered as a generalization of a result by Totik (1983).
Also, it unifies and generalizes some other results obtained by Fisher
(1977), Akkouchi (2001) and Nova (1997).
Keywords: Common fixed point theorems in metric spaces.
MSC: 54H25, 47H10
1
Introduction
The common fixed point theory has seen a great developpement during
the three last decades.

One can say that its story has started with the
well known result of Markov and Kakutani.
For two self–mappings S and T of a given metric space (X, d), many
kind of contractive (exapnsive or nonexpansive) conditions may be considered.

We list here some examples.
1
akkouchimo@yahoo.fr
22
M.

Akkouchi, On a fixed poitn theorem
d(Sx, Sy) ≤ q d(T x, T y) ,
∀x, y ∈ X .
(1.1)
This condition generalizes the well known Banach contraction principle.

It has been generalized by the following condition
d(Sx, Sy) ≤ q max{d(T x, T y), d(Sx, T x), d(Sy, T y),
d(T x, Sy), d(T y, Sx)} .
(1.2)
A more general c...