Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 51-65

A SUFFICIENT CONDITION FOR A FINITE FAMILY OF CONTINUOUS FUNCTIONS TO BE TRANSFORMED INTO ψ-CONTRACTIONS

Radu Miculescu and Alexandru Mihail

Bucharest University, Faculty of Mathematics and Computer Science
Str. Academiei 14, 010014 Bucharest, Romania; miculesc 'at' yahoo.com

University Politehnica of Bucharest, 313 Splaiul Independentei, Bucharest, 060042, Romania
and Bucharest University, Faculty of Mathematics and Computer Science
Str. Academiei 14, 010014 Bucharest, Romania; mihail_alex 'at' yahoo.com

Abstract. Given a metric space (X,d) and a finite set of continuous functions f₁,f₂,...,f_N : X → X, we provide a sufficient condition to find a metric δ on X, equivalent with d, and a comparison function ψ such that the functions f_i : (X,δ) → (X,δ) are ψ-contractions. If the metric space (X,d) is complete, the same condition assures the existence of a unique fixed point of the function F : K(X) → K(X) given by F(C) = ∪_i=1^N f_i(C) for each C ∈ K(X), where K(X) denotes the family of non-empty and compact subsets of X.

2010 Mathematics Subject Classification: Primary 54E35, 47H09, 54E40.

Key words: Metric spaces, comparison function, φ-contraction.

Reference to this article: R. Miculescu and A. Mihail: A sufficient condition for a finite family of continuous functions to be transformed into ψ-contractions. Ann. Acad. Sci. Fenn. Math. 41 (2016), 51-65.

Full document as PDF file

doi:10.5186/aasfm.2016.4103