Borsuk–Dugundji type extension theorems with Busemann convex target spaces

Espínola, Rafa; Castro, Óscar Reynaldo Madiedo; Nicolae, Adriana

doi:10.5186/aasfm.2018.4313

Cited by 9 publications

(3 citation statements)

References 13 publications

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“…It is worth noticing that in a Busemann convex space X the closure of con(A) is convex and so, coincides with con(A) ( [8]).…”

Section: Definition 23 ([1]mentioning

confidence: 99%

Existence and approximating of common best proximity points of relatively nonexpansive mappings via Ishikawa iteration method

Moosa,

Jack T.,

Vladimir

2023

FPT-RO

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In this article, we study the existence of a common best proximity points for a finite class of cyclic relatively nonexpansive mappings in the setting of Busemann convex spaces. In this way, we extend the main results given in Eldred and Raj (2009) [A.A. Eldred, V.S. Raj, On common best proximity pair theorems, Acta Sci. Math. (Szeged), 75, 707-721] for relatively nonexpansive mappings in Banach spaces to more general metric spaces. We then present a strong convergence theorem of a common best proximity point for a pair of cyclic mappings in uniformly convex Banach spaces by using the Ishikawa iterative process.

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“…It is worth noticing that in a Busemann convex space X the closure of con(A) is convex and so, coincides with con(A) ( [8]).…”

Section: Definition 23 ([1]mentioning

confidence: 99%

Existence and approximating of common best proximity points of relatively nonexpansive mappings via Ishikawa iteration method

Moosa,

Jack T.,

Vladimir

2023

FPT-RO

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“…It is remarkable to note that in a Busemann convex space X the closure of con(E) is convex and so, coincides with con(A) (see [10]).…”

Section: Lemma 21 ([5]mentioning

confidence: 99%

Best proximity point (pair) results via MNC in Busemann convex metric spaces

Gabeleh

Patle

2022

Appl. Gen. Topol.

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In this paper, we present a new class of cyclic (noncyclic) α-ψ and β-ψ condensing operators and survey the existence of best proximity points (pairs) as well as coupled best proximity points (pairs) in the setting of reflexive Busemann convex spaces. Then an application of the main existence result to study the existence of an optimal solution for a system of differential equations is demonstrated.

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“…See also [8,Section 3.2]. Other barycentric constructions can be found in [2,34]. For our proofs it would actually suffice to just have a barycenter map β : P f (X) → X defined on the subset P f (X) ⊂ P 1 (X) of all µ ∈ P 1 (X) whose support is finite.…”

Section: Preliminariesmentioning

confidence: 99%

Absolute Lipschitz extendability and linear projection constants

Basso¹

2022

Studia Math.

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In this mostly expository article, we give streamlined proofs of several well-known Lipschitz extension theorems. We pay special attention to obtaining statements with explicit expressions for the Lipschitz extension constants. One of our main results is an explicit version of a very general Lipschitz extension theorem of Lang and Schlichenmaier. A special case of the theorem reads as follows. We prove that if X is a metric space and A ⊂ X satisfies the condition Nagata(n, c), then any 1-Lipschitz map f : A → Y to a Banach space Y admits a Lipschitz extension F : X → Y whose Lipschitz constant is at most 1000 • (c + 1) • log 2 (n + 2). By specifying to doubling metric spaces, this recovers an extension result of Lee and Naor. We also revisit another theorem of Lee and Naor by showing that if A ⊂ X consists of n points, then Lipschitz extensions as above exist with a Lipschitz constant of at most 600 • log n • (log log n) −1 .

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Borsuk–Dugundji type extension theorems with Busemann convex target spaces

Abstract: Abstract. In this work we study continuity properties of convex combinations in Busemann convex geodesic spaces and apply them to obtain two extension results for continuous and Lipschitz mappings with values in a Busemann convex space.

Cited by 9 publications

References 13 publications

Existence and approximating of common best proximity points of relatively nonexpansive mappings via Ishikawa iteration method

Existence and approximating of common best proximity points of relatively nonexpansive mappings via Ishikawa iteration method

Best proximity point (pair) results via MNC in Busemann convex metric spaces

Absolute Lipschitz extendability and linear projection constants

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