Abstract
Uniformly convex W-hyperbolic spaces with monotone modulus of uniform convexity are a natural generalization of both uniformly convexnormed spaces and CAT(0) spaces. In this article, we discuss the existence of fixed points and demiclosed principle for mappings of asymptotically non-expansive type in uniformly convex W-hyperbolic spaces with monotone modulus of uniform convexity. We also obtain a Δ-convergence theorem of Krasnoselski-Mann iteration for continuous mappings of asymptotically nonexpansive type in CAT(0) spaces.MSC: 47H09; 47H10; 54E40
Highlights
In 1974, Kirk [1] introduced the mappings of asymptotically nonexpansive type and proved the existence of fixed points in uniformly convex Banach spaces
In 1993, Bruck et al [2] introduced the notion of mappings which are asymptotically nonexpansive in the intermediate sense and obtained the weak convergence theorems of averaging iteration for mappings of asymptotically nonexpansive in the intermediate sense in uniformly convex Banach space with the Opial property
We prove the existence of fixed points and demiclosed principle for mappings of asymptotically nonexpansive type in uniformly convex W-hyperbolic spaces with monotone modulus of uniform convexity
Summary
In 1974, Kirk [1] introduced the mappings of asymptotically nonexpansive type and proved the existence of fixed points in uniformly convex Banach spaces. We prove the existence of fixed points and demiclosed principle for mappings of asymptotically nonexpansive type in uniformly convex W-hyperbolic spaces with monotone modulus of uniform convexity. [[23], Lemma [7]] Let (X, d, W) be a UCW-hyperbolic space with modulus of uniform convexity h. It is known that CAT(0) spaces are UCW-hyperbolic spaces with modulus of uniform convexity h(r, ε) = ε2/8 quadratic in ε (refer to [23] for details). 3. Fixed point theorem for mappings of asymptotically nonexpansive type The First main result of this article is the existence of fixed points for the mappings of asymptotically nonexpansive type in UCW-hyperbolic space with a monotone modulus of uniform convexity. Let ζ >0 and using the Definition of T choose M so that i ≥ M implies sup (d(Tiz, Tix) − d(z, x)) ≤ 1 ζ
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