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Common fixed points of Greguš type multi-valued mappings

R. A. Rashwan, Magdy A. Ahmed (2002)

Archivum Mathematicum

This work is considered as a continuation of [19,20,24]. The concepts of δ -compatibility and sub-compatibility of Li-Shan [19, 20] between a set-valued mapping and a single-valued mapping are used to establish some common fixed point theorems of Greguš type under a φ -type contraction on convex metric spaces. Extensions of known results, especially theorems by Fisher and Sessa [11] (Theorem B below) and Jungck [16] are thereby obtained. An example is given to support our extension.

Commutativity and non-commutativity of topological sequence entropy

Francisco Balibrea, Jose Salvador Cánovas Peña, Víctor Jiménez López (1999)

Annales de l'institut Fourier

In this paper we study the commutativity property for topological sequence entropy. We prove that if X is a compact metric space and f , g : X → X are continuous maps then h A ( f ∘ g ) = h A ( g ∘ f ) for every increasing sequence A if X = [ 0 , 1 ] , and construct a counterexample for the general case. In the interim, we also show that the equality h A ( f ) = h A ( f | ∩ n ≥ 0 f n ( X ) ) is true if X = [ 0 , 1 ] but does not necessarily hold if X is an arbitrary compact metric space.

Commuting contractive families

Luka Milićević (2015)

Fundamenta Mathematicae

A family f₁,..., fₙ of operators on a complete metric space X is called contractive if there exists a positive λ < 1 such that for any x,y in X we have d ( f i ( x ) , f i ( y ) ) ≤ λ d ( x , y ) for some i. Austin conjectured that any commuting contractive family of operators has a common fixed point, and he proved this for the case of two operators. We show that Austin’s conjecture is true for three operators, provided that λ is sufficiently small.

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