Summary
Letf be a self-map on a metric space (X, d). We give necessary and sufficient conditions for the sequences {f n x} (x ∈ X) to be equivalent Cauchy. As a typical application we get the following result. Letf be continuous and (X, d) be complete. If, for anyx, y ∈ X d(f n x, f n y) → 0 and for somec > 0, this convergence is uniform for allx, y inX withd(x, y) ≤c thenf has a unique fixed pointp, andf n x →p, for eachx inX.
This theorem includes among others results of Angelov, Browder, Edelstein, Hicks and Matkowski.
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Jachymski, J. An iff fixed point criterion for continuous self-mappings on a complete metric space. Aeq. Math. 48, 163–170 (1994). https://doi.org/10.1007/BF01832983
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DOI: https://doi.org/10.1007/BF01832983