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Journal of Applied Analysis

Editor-in-Chief: Liczberski, Piotr / Ciesielski, Krzysztof

Managing Editor: Gajek, Leslaw

2 Issues per year


SCImago Journal Rank (SJR): 0.752
Source Normalized Impact per Paper (SNIP): 1.104

Mathematical Citation Quotient 2013: 0.13

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Nonlinear Contractions on Semimetric Spaces

J. Jachymski1 / J. Matkowski2 / T. Świa̧tkowski

1Institute of Mathematics, Technical University, Ul. Żwirki 36, 90-924 Lódź, Poland.

2Department of Mathematics, Technical University, Ul. Willowa 2, 43-300 Bielsko–Biala, Poland

Citation Information: Journal of Applied Analysis. Volume 1, Issue 2, Pages 125–133, ISSN (Online) 1869-6082, ISSN (Print) 1425-6908, DOI: 10.1515/JAA.1995.125, June 2010

Publication History

Published Online:
2010-06-03

Abstract

Let (X, d) be a Hausdorff semimetric (d need not satisfy the triangle inequality) and d–Cauchy complete space. Let ƒ be a selfmap on X, for which dx, ƒy) ≤ φ(d(x, y)), (x, yX), where φ is a non– decreasing function from R +, the nonnegative reals, into R + such that φn(t) → 0, for all tR +. We prove that ƒ has a unique fixed point if there exists an r > 0, for which the diameters of all balls in X with radius r are equi-bounded. Such a class of semimetric spaces includes the Frechet spaces with a regular ecart, for which the Contraction Principle was established earlier by M. Cicchese [Boll. Un. Mat. Ital 13–A: 175-179, 1976], however, with some further restrictions on a space and a map involved. We also demonstrate that for maps ƒ satisfying the condition dx, ƒy) ≤ φ(max{d(x, ƒx), d(y, ƒy)}), (x, yX) (the Bianchini [Boll. Un. Mat. Ital. 5: 103–108, 1972] type condition), a fixed point theorem holds under substantially weaker assumptions on a distance function d.

Key words and phrases.: Fixed point; nonlinear contraction; semimetric; symmetric; space with a regular ecart; E–space; d–Cauchy completeness

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