Strongly Generated Banach Spaces and Measures of Noncompactness

Kunze, Markus; Schlüchtermann, Georg

doi:10.1002/mana.19981910110

Cited by 23 publications

(19 citation statements)

References 19 publications

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“…The following result extends [22,Theorem 4.5]. Its proof uses some ideas from [27, Theorem 3.2] and [26].…”

Section: Remark 25 In General Subspaces Of Swcg/sag/scwcg Spaces Nementioning

confidence: 85%

“…In this paper we shall focus on SAG and SCWCG spaces. The basic properties of such spaces were discussed in [22]. Clearly, every Asplund space is SAG and every Banach space not containing 1 is SCWCG.…”

Section: Definition 12 a Banach Space Z Is Calledmentioning

confidence: 99%

“…These classes of spaces were introduced by Kunze and Schlüchtermann [22], inspired by the strongly weakly compactly generated Banach spaces of Schlüchtermann and Wheeler [27]. To recall the definition we need some terminology.…”

Section: Introductionmentioning

confidence: 99%

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Strongly Asplund generated and strongly conditionally weakly compactly generated Banach spaces

Lajara

Rodríguez

2015

Monatsh Math

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We study strongly Asplund generated (SAG) and strongly conditionally weakly compactly generated (SCWCG) Banach spaces. These spaces are defined like the strongly weakly compactly generated (SWCG) Banach spaces of Schlüchtermann and Wheeler, but replacing weakly compact sets by Asplund sets and conditionally weakly compact sets, respectively. We show that every SAG space is SCWCG and that a Banach space is SWCG if and only if it is SAG/SCWCG and weakly sequentially complete. We also prove that the notions of SAG and SCWCG space coincide for Banach lattices. Some related results on Lebesgue-Bochner spaces are also given. We prove that if the norm of the Banach space X is weakly uniformly rotund (WUR) and μ is any probability measure, then L 1 (μ, X ) admits an equivalent norm which is WUR when restricted to any Asplund subspace of L 1 (μ, X ).

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“…The following result extends [22,Theorem 4.5]. Its proof uses some ideas from [27, Theorem 3.2] and [26].…”

Section: Remark 25 In General Subspaces Of Swcg/sag/scwcg Spaces Nementioning

confidence: 85%

Section: Definition 12 a Banach Space Z Is Calledmentioning

confidence: 99%

See 1 more Smart Citation

Strongly Asplund generated and strongly conditionally weakly compactly generated Banach spaces

Lajara

Rodríguez

2015

Monatsh Math

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“…As we have already shown in the proof of Theorem 11, there must be a subsequence (x l kn ) ∞ n=1 of the sequence (x k n ) ∞ n=1 such that x l kn Let K : L q (I, E) → L p (I, E) be given by (18). Observe that solving the equation u = K( f (u)) means to find a solution of the Hammerstein integral inclusion (15).…”

Section: Proof Letmentioning

confidence: 97%

Solvability of inclusions of Hammerstein type

Pietkun¹

2019

Journal of Mathematical Analysis and Applications

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We establish a universal rule for solving operator inclusions of Hammerstein type in Lebesgue-Bochner spaces with the aid of some recently proven continuation theorem of Leray-Schauder type for the class of so-called admissible multimaps. Examples illustrating the legitimacy of this approach include the initial value problem for perturbation of m-accretive mutivalued differential equation, the nonlocal Cauchy problem for semilinear differential inclusion, abstract integral inclusion of Fredholm and Volterra type and the two-point boundary value problem for nonlinear evolution inclusion.2010 Mathematics Subject Classification. 34A60, 34G25, 45N05, 47H10, 47H30. Key words and phrases. acyclic set, admissible map, boundary value problem, evolution inclusion, fixed point, integral inclusion, m-accretive, operator inclusion, p-integrable solution.

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“…we refer to References [5,6] for the details and also for more developments in this direction. We observe that (5) implies that if T z ∈ K(X; Y ) for all z ∈ U thenT ∈ K(X; Y ).…”

mentioning

confidence: 99%