Abstract
Recently, W. A. Kirk proved an asymptotic fixed point theorem for nonlinear contractions by using ultrafilter methods. In this paper, we prove his theorem under weaker assumptions. Furthermore, our proof does not use ultrafilter methods.
Highlights
There are many papers in the literature that discuss the asymptotic fixed point theory, in which the existence of the fixed points is deduced from the assumption on the iterates of an operator (e.g., [1, 6] and the references therein)
Let T : M → M be such that d Tnx, Tn y ≤ φn d(x, y) for all x, y ∈ M, where φn : [0, ∞] → [0, ∞] and limn→∞ φn = φ uniformly on any bounded interval [0,b]
Let T : M → M be such that d Tnx, Tn y ≤ φn d(x, y) for all x, y ∈ M, where φn : [0, ∞] → [0, ∞] and limn→∞ φn = φ uniformly on any bounded interval [0, b]
Summary
There are many papers in the literature that discuss the asymptotic fixed point theory, in which the existence of the fixed points is deduced from the assumption on the iterates of an operator (e.g., [1, 6] and the references therein). Let T : M → M be a continuous mapping such that d Tnx, Tn y ≤ φn d(x, y) for all x, y ∈ M, where φn : [0, ∞] → [0, ∞] and limn→∞ φn = φ uniformly on the range of d. Let φ : R+ → R+ be upper semicontinuous, that is, lim supt→t0 φ(t) ≤ φ(t0) for all t0 ∈ R+, and φ(t) < t for t > 0.
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