# A note on a common fixed point theorem of b. fisher

A note on a common fixed point theorem of b. fisher
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The subject of this note is to establish a common fixed point theorem in complete metric spaces which improves a well known result of B. Fisher see 1. Our theorem solves also the problem posed in 2.

Tipo de documento: Artículo - Article

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

## Teaser

Boletı́n de Matemáticas
Nueva Serie, Volumen VIII No.1 (2001), pp.

26–30
A NOTE ON A COMMON FIXED
POINT THEOREM OF B.

FISHER
Mohamed Akkouchi(*)
Abstract.

The subject of this note is to establish a common fixed point theorem in complete metric spaces which improves a well known result of B.

Fisher
(see [1]).

Our theorem solves also the problem posed in [2].
Key words and phrases: Common fixed points in complete metric spaces.
2000 Mathematics Subject Classification.

47H10, and 54H25.
§ 1 Introduction and statement of the result
The study of common ﬁxed points has started in the year 1936 by the well
known result of Markov and Kakutani.

Since this year, many works were
devoted to Fixed Point Theory.

Many authors have studied the existence of
ﬁxed and common ﬁxed points and now the literature on the subject is very
rich.

B.

Fisher has proved in his paper [1] the following result:
Theorem 1.1.

[B.

Fisher]: Let (M, d) be a complete metric space.

let S, T be
two self-mappings of M such that
(i) S is continuous,
(ii) d(Sx, T Sy) ≤ αd(x, Sy) β [d(x, Sx) d(Sy, T Sy)] γ[d(x, T Sy)
d(Sx, Sy)], for every x, y ∈ M, where α, β, γ ≥ 0 are such that α 2β
2γ 1.
Then S and T have a unique common fixed point.
In this note, we shall prove that the assumption of continuity made on S,
in Theorem 1.1, is superﬂuous and can be removed.

In the paper [2], L.

Nova
has tried to remove the assumption (i) of continuity on S but she replaced it
by another condition.

More precisely the main result of [2] was the following:
(*) Mohamed Akkouchi, University Cadi Ayyad.

Faculty of sciences-semlalia.

Department
of Mathematics.

Av.

Prince My Abdellah, BP.

2390, Marrakech, Morocco.

e-mail:
makkouchi@hotmail.com
26
ON A THEOREM OF B.

FISHER
27
Theorem 1.2.

[L.

Nova]: Let (M, d) be a complete metric space.

Let a, b ≥ 0
such that a 2b 1 and let α, β ≥ 0, such that β 1.

Let S, T be two
self-mappings of M such that
(i) d(Sx, Sy) ≤ ad(x, y) ...