Generalized nonexpansive mappings and a proximal-type algorithm in Banach spaces

Takahashi, Wataru; Ibaraki, Takanori

doi:10.1090/conm/513/10082

Cited by 13 publications

(6 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently Ibaraki and Takahashi [15] also obtained the following result concerning the set of fixed points of a generalized nonexpansive mapping. [15]).…”

Section: Lemma 24 ([26])mentioning

confidence: 97%

“…Recently Ibaraki and Takahashi [15] also obtained the following result concerning the set of fixed points of a generalized nonexpansive mapping. [15]). Let E be a smooth, strictly convex and reflexive Banach space and let T be a generalized nonexpansive mapping from E into itself.…”

Section: Lemma 24 ([26])mentioning

confidence: 97%

“…The following is a direct consequence of Lemmas 2.7 and 2.9. [15]). Let E be a smooth, strictly convex and reflexive Banach space and let T be a generalized nonexpansive mapping from E into itself.…”

Section: Lemma 29 (Ibaraki and Takahashimentioning

confidence: 99%

See 2 more Smart Citations

Attractive Point and Mean Convergence Theorems for New Generalized Nonspreading Mappings in Banach Spaces

Takahashi¹,

Wong²,

Yao³

2015

Contemporary Mathematics

Self Cite

View full text Add to dashboard Cite

In this paper, we first introduce a class of nonlinear mappings called generic generalized nonspreading which contains the class of generalized nonspreading mappings in a Banach space and then prove an attractive point theorem for such mappings in a Banach space. Furthermore, we prove a mean convergence theorem of Baillon's type and a weak convergence theorem of Mann's type for such nonlinear mappings in a Banach space. These results generalize attractive point, mean convergence and weak convergence theorems proved by Lin and Takahashi [26], and Kocourek, Takahashi and Yao [21] in a Banach space.

show abstract

“…Recently Ibaraki and Takahashi [15] also obtained the following result concerning the set of fixed points of a generalized nonexpansive mapping. [15]).…”

Section: Lemma 24 ([26])mentioning

confidence: 97%

Section: Lemma 24 ([26])mentioning

confidence: 97%

See 1 more Smart Citation

Attractive Point and Mean Convergence Theorems for New Generalized Nonspreading Mappings in Banach Spaces

Takahashi¹,

Wong²,

Yao³

2015

Contemporary Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Lemma 2.13. (Ibaraki and Takahashi [15]). Let E be a smooth, strictly convex and reflexive Banach space and let T be a generalized nonexpansive mapping of E into itself.…”

Section: Preliminariesmentioning

confidence: 97%

Weak and Strong Convergence Theorems for Positively Homogenuous Nonexpansive Mappings in Banach Spaces

Takahashi¹,

Yao²

2011

Taiwanese J. Math.

Self Cite

View full text Add to dashboard Cite

Our purpose in this paper is first to prove a weak convergence theorem by Mann's iteration for positively homogeneous nonexpansive mappings in a Banach space. Further, using the shrinking projection method defined by Takahashi, Takeuchi and Kubota, we prove a strong convergence theorem for such mappings. From two results, we obtain weak and strong convergence theorems for linear contractive mappings in a Banach space. These results are new even if the mappings are linear and contractive.

show abstract

“…Inthakon et al [15] obtained the following result concerning the set of fixed points of a generalized nonexpansive mapping in a Banach space; see also Ibaraki and Takahashi [13,14]. Lemma 2.12 (Inthakon, Dhompongsa and Takahashi [15]).…”

Section: Lemma 23 (Bruck [6]) Let E Be a Uniformly Convex Banach Spmentioning

confidence: 99%

Nonlinear Ergodic Theorem for Positively Homogeneous Nonexpansive Mappings in Banach Spaces

Takahashi

Wong

Yao

2013

Numerical Functional Analysis and Optimization

Self Cite

View full text Add to dashboard Cite

Recently, two retractions (projections), which are different from the metric projection and the sunny nonexpansive retraction in a Banach space, were found. In this article, using nonlinear analytic methods and new retractions, we prove a nonlinear ergodic theorem for positively homogeneous and nonexpansive mappings in a uniformly convex Banach space. The limit points are characterized by using new retractions.

show abstract

Generalized nonexpansive mappings and a proximal-type algorithm in Banach spaces

Cited by 13 publications

References 0 publications

Attractive Point and Mean Convergence Theorems for New Generalized Nonspreading Mappings in Banach Spaces

Attractive Point and Mean Convergence Theorems for New Generalized Nonspreading Mappings in Banach Spaces

Weak and Strong Convergence Theorems for Positively Homogenuous Nonexpansive Mappings in Banach Spaces

Nonlinear Ergodic Theorem for Positively Homogeneous Nonexpansive Mappings in Banach Spaces

Contact Info

Product

Resources

About