Fixed point results for generalized mappings
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Fixed point results for generalized mappings Farhan Golkarmanesh1* , Abdullah E Al-Mazrooei2 , Vahid Parvaneh3 and Abdul Latif2 * Correspondence: [email protected] 1 Department of Mathematics, Sanandaj Branch, Islamic Azad University, Sanandaj, Iran Full list of author information is available at the end of the article
Abstract In this paper, first we establish fixed point results for weak asymptotic pointwise contraction type mappings in metric spaces. Then we study the existence of fixed points for weak asymptotic pointwise nonexpansive type mappings in CAT(0) spaces. Our results improve and extend some corresponding known results in the literature. MSC: 47H09; 47H10; 54H25 Keywords: asymptotic center; asymptotic pointwise contraction type; convexity structure; T-stable; weak asymptotic pointwise nonexpansive type; CAT(0) space
1 Introduction The notion of asymptotic pointwise contraction was introduced by Kirk [] as follows. Let (M, d) be a metric space. A mapping T : M → M is called an asymptotic pointwise contraction if there exists a function α : M → [, ) such that, for each integer n ≥ , d T n x, T n y ≤ αn (x)d(x, y) for each x, y ∈ M, where αn → α pointwise on M. Moreover, Kirk and Xu [] proved that if C be a weakly compact convex subset of a Banach space E and T : C → C an asymptotic pointwise contraction, then T has a unique fixed point v ∈ C and for each x ∈ C the sequence of Picard iterates {T n x} converges in norm to v. Rakotch [] proved that if M be a complete metric space and f : M → M satisfies d(f (x), f (y)) ≤ α(d(x, y))d(x, y), for all x, y ∈ M, where α : [, ∞) → [, ) is monotonically decreasing, then f has a unique fixed point z and {f n (x)} converges to z, for each x ∈ M. Boyd and Wong [] proved that if M be a complete metric space and f : M → M satisfies d(f (x), f (y)) ≤ ψ(d(x, y))d(x, y), for all x, y ∈ M, where ψ : [, ∞) → [, ∞) is upper semicontinuous from the right and satisfies ≤ ψ(t) < t for t > , then f has a unique fixed point z and {f n (x)} converges to z, for each x ∈ M. Using the diameter of an orbit, Walter [] obtained a result that may be stated as follows: Let (M, d) be a complete metric space and let T : M → M be a mapping with bounded orbits. If there exists a continuous, increasing function ϕ : R+ → R+ for which ϕ(r) < r for every r > and d(Tx, Ty) ≤ ϕ diam OT (x, y)
for every x, y ∈ M,
where OT (x, y) = {T n x} ∪ {T n y}, then T has a unique fixed point x . Moreover, {T n x} converges to x , for each x ∈ M. ©2014 Golkarmanesh et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Golkarmanesh et al. Fixed Point Theory and Applications 2014, 2014:217 http://www.fixedpointtheoryandapplications.com/content/2014/1/217
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In [], the authors proved coincidence results fo
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