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Nonlinear Contractions on Semimetric Spaces

  • J. Jachymski , J. Matkowski and T. Świa̧tkowski

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.

Published Online: 2010-06-03
Published in Print: 1995-December
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