Nonlinear Contractions on Semimetric Spaces

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 d(ƒx, ƒy) ≤ φ(d(x, y)), (x, y ∈ X), where φ is a non– decreasing function from R +, the nonnegative reals, into R + such that φn (t) → 0, for all t ∈ R +. 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 d(ƒx, ƒy) ≤ φ(max{d(x, ƒx), d(y, ƒy)}), (x, y ∈ X) (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.