Iterative Sequences for Asymptotically Quasi-nonexpansive Mappings

Qihou, Liu

doi:10.1006/jmaa.2000.6980

Cited by 73 publications

(37 citation statements)

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“…Remark 1.5 It can be clearly seen that the class of total asymptotically quasi-nonexpansive mappings includes the classes of total asymptotically nonexpansive, nearly asymptotically nonexpansive, asymptotically quasinonexpansive, asymptotically nonexpansive, quasinonexpansive and nonexpansive mappings. But the converse of each may not be true (see [8], [18] and [24]). Let C be a nonempty subset of a Banach space X and T, S : C  C be two mappings.…”

Section: Lemma13 [9]mentioning

confidence: 99%

Some Strong and Δ - Convergence Results in Hyperbolic Spaces

Malik¹

2017

IJCA

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Section: Lemma13 [9]mentioning

confidence: 99%

Some Strong and Δ - Convergence Results in Hyperbolic Spaces

Malik¹

2017

IJCA

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“…Finally, combining the notions of asymptotically nonexpansive mappings and quasi-nonexpansive mappings we obtain the concept of asymptotically quasi-nonexpansive mappings first studied in [29] and [20] and more recently in [22,23,24]: In the context of asymptotically (quasi-)nonexpansive mappings f : C → C the so-called Krasnoselski-Mann iteration is defined as follows…”

Section: Definition 13 F : C → C Is Quasi-nonexpansive Ifmentioning

confidence: 99%

Bounds on Iterations of Asymptotically Quasi-Nonexpansive Mappings

Kohlenbach¹,

Lambov²

2003

BRICS

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This paper establishes explicit quantitative bounds on the computation of approximate fixed points of asymptotically (quasi-) nonexpansive mappings f by means of iterative processes. Here f : C → C is a selfmapping of a convex subset C ⊆ X of a uniformly convex normed space X. We consider general Krasnoselski-Mann iterations with and without error terms. As a consequence of our quantitative analysis we also get new qualitative results which show that the assumption on the existence of fixed points of f can be replaced by the existence of approximate fixed points only. We explain how the existence of effective uniform bounds in this context can be inferred already a-priorily by a logical metatheorem recently proved by the first author. Our bounds were in fact found with the help of the general logical machinery behind the proof of this metatheorem. The proofs we present here are, however, completely selfcontained and do not require any tools from logic.

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“…(4) asymptotically quasi-nonexpansive [20] if there exists a real sequence {k n } ⊂ [1, ∞) with k n → 1 as n → ∞ such that T n u − p ≤ k n u − p for all u ∈ K and p ∈ F (T );…”

Section: Introductionmentioning

confidence: 99%

“…In 1972, the class of asymptotically nonexpansive mappings was introduced as a generalization of the class of nonexpansive mappings by Goebel and Kirk [13]. In 2001, the class of asymptotically quasi-nonexpansive mapping was introduced as a generalization of the class of asymptotically nonexpansive mappings by Qihou [20]. Furthermore, it is easy to observe that, if F (T ) = ∅, then a nonexpansive mapping must be quasi-nonexpansive and an asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive mapping.…”

Section: Introductionmentioning

confidence: 99%

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Approximation of a common minimum-norm fixed point of a finite family of σ -asymptotically quasi-nonexpansive mappings with applications

Pathak¹,

Sahu²,

Cho³

2016

J. Nonlinear Sci. Appl.

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In this paper, we use the iterative method proposed by Zegeye and Shahzad [H. Zegeye, N. Shahzed, Fixed Point Theory Appl., 2013 (2013), 12 pages] which converges strongly to the common minimum-norm fixed point of a finite family of σ-asymptotically quasi-nonexpansive mappings. As consequence, convergence results to a common minimum-norm fixed point of a finite family of asymptotically nonexpansive mappings is proved. Our result generalize and improve a recent result of Zegeye and Shahzad [H. Zegeye, N. Shahzed, Fixed Point Theory Appl., 2013 (2013), 12 pages]. In the sequel, we apply our main result to find solution of minimizer of a continuously Frechet-differentiable convex functional which has the minimum norm in Hilbert spaces.

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Iterative Sequences for Asymptotically Quasi-nonexpansive Mappings

Cited by 73 publications

References 4 publications

Some Strong and Δ - Convergence Results in Hyperbolic Spaces

Some Strong and Δ - Convergence Results in Hyperbolic Spaces

Bounds on Iterations of Asymptotically Quasi-Nonexpansive Mappings

Approximation of a common minimum-norm fixed point of a finite family of σ -asymptotically quasi-nonexpansive mappings with applications

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