A new approach to measures of noncompactness of Banach spaces

Cheng, Lixin; Cheng, Qingjin; Shen, Qinrui; Tu, Kun; Wen, Zhang

doi:10.4064/sm8448-2-2017

Cited by 13 publications

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“…then it becomes a complete normed semigroup with respect to the Hausdorff metric [13]. (See, also, Section 2 in detail.)…”

Section: Introductionmentioning

confidence: 99%

“…We should mention here that a Banach space setting for the study of MNGCs would lose no generality, because that every metric space is isometric to a subset of a Banach space [9, Lemma 1.1]. Besides, the concept is easily to be generalized by Banach space theory, for example, it has been generalized in various ways such as MNWC (see, for instance, [11], [18], [23], [24]); of MNRNP [36]; or, more general, measures of nonproperty D [13].…”

Section: Introductionmentioning

confidence: 99%

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Representation of measures of noncompactness and its applications related to an initial-value problem in Banach spaces

Chen¹,

Cheng²

2021

Preprint

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The purpose of this paper is devoted to studying representation of measures of non generalized compactness, in particular, measures of noncompactness, of non-weak compactness, and of non-super weak compactness, etc, defined on Banach spaces and its applications. With the aid of a three-time order preserving embedding theorem, we show that for every Banach space X, there exist a Banach function space C(K) for some compact Hausdorff space K, and an order-preserving affine mapping T from the super space B of all nonempty bounded subsets of X endowed with the Hausdorff metric to the positive cone C(K) + of C(K) such that for every convex measure, in particular, regular measure, homogeneous measure, sublinear measure of non generalized compactness µ on X, there is a convex function ̥ on the cone V = T(B) which is Lipschitzian on each bounded set of V such thatAs its applications, we show a class of basic integral inequalities related to an initial-value problem in Banach spaces, and prove a solvability result of the initial-value problem, which is an extension of some classical results due to Goebel, Rzymowski, and Banaś.

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“…then it becomes a complete normed semigroup with respect to the Hausdorff metric [13]. (See, also, Section 2 in detail.)…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Representation of measures of noncompactness and its applications related to an initial-value problem in Banach spaces

Chen¹,

Cheng²

2021

Preprint

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“…We use Ω to denote the closed unit ball B X * of the dual X * , and C b (Ω), the Banach space of all real-valued bounded norm-continuous functions on Ω endowed with the sup-norm. Let J :With the symbols as above, we first recall the main results presented in [2], which will be used in the proof. Theorem 1.…”

mentioning

confidence: 99%

“…With the symbols as above, we first recall the main results presented in [2], which will be used in the proof. Theorem 1.…”

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confidence: 99%

Erratum to: Every Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure

Ablet

Cheng

et al. 2019

Sci. China Math.

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1, Theorem 4.4] states that every infinite dimensional Banach space admits a homogenous measure of noncompactness not equivalent to the Hausdorff measure. Howevere, there is a gap in the proof. In fact, we found that [1, Lemma 4.3] is not true. In this erratum, we give a corrected proof of [1, Theorem 4.4].For a Banach space X, let C (X) (resp. B(X), K (X)) be the collection of all non-empty bounded closed convex (resp. nonempty bounded, nonempty convex compact) sets of X endowed with the Hausdorff metric. If there is no confusion, we simply denote them by C , B and K , respectively. We use Ω to denote the closed unit ball B X * of the dual X * , and C b (Ω), the Banach space of all real-valued bounded norm-continuous functions on Ω endowed with the sup-norm. Let J :With the symbols as above, we first recall the main results presented in [2], which will be used in the proof. Theorem 1. (1) (See [2, Theorem 2.3i)]) Given a Banach space X, the collection C consisting of all nonempty closed bounded convex sets of X endowed with the set addition A ⊕ B = A + B, the usual scalar multiplication of sets λC = {λc : c ∈ C}, and the norm ||| · ||| defined by |||C||| = sup c∈C ∥c∥ is a complete normed convex cone. (2) (See [2, Theorem 2.3ii)]) If we endow with the Hausdorff metric d H on C , i.e., d H (A, B) = max { sup a∈A inf b∈B ∥a − b∥, sup b∈B inf a∈A ∥b − a∥ }

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“…On the other hand, in the mentioned Banach tempered sequence space c β 0 we can use the so-called Hausdorff measure of noncompactness being the most convenient measure in applications as far as we know (cf. [1,3,6,9]).…”

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confidence: 99%

On solutions of semilinear upper diagonal infinite systems of differential equations

Banaś¹,

Krajewska²

2019

Discrete &Amp; Continuous Dynamical Systems - S

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The goal of the paper is to investigate the existence of solutions for semilinear upper diagonal infinite systems of differential equations. We will look for solutions of the mentioned infinite systems in a Banach tempered sequence space. In our considerations we utilize the technique associated with the Hausdorff measure of noncompactness and some existence results from the theory of ordinary differential equations in abstract Banach spaces.

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A new approach to measures of noncompactness of Banach spaces

Cited by 13 publications

References 0 publications

Representation of measures of noncompactness and its applications related to an initial-value problem in Banach spaces

Representation of measures of noncompactness and its applications related to an initial-value problem in Banach spaces

Erratum to: Every Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure

On solutions of semilinear upper diagonal infinite systems of differential equations

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