# A note on a common fixed point theorem of b. fisher  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 ).
Our theorem solves also the problem posed in . 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  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 , 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  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) ...