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Weakly absolutely continuous functions without weak, but fractional weak derivatives

  • Hussein A. H. SalemEmail author
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Abstract

Let E be an infinite-dimensional Banach space and I be a compact interval of the real line. The aim of this paper is two-fold: On the one hand, we construct an example of a weakly absolutely continuous function taking its values in E that is nowhere weakly differentiable on I, but has weakly continuous fractional weak derivatives of some critical orders less than one. This also holds for (nearly) all orders less than one if E failing cotype. We believe that this results are of independent interest and discuss it in a rather general setting. On the other hand, we establish some examples of weakly continuous functions taking its values in Gauge space fail to be pseudo differentiable on I, but have fractional-pseudo derivatives of “all” order less than one. An application will be given.

Keywords

Fractional calculus Orlicz spaces Pettis integrals 

Mathematics Subject Classification

26A33 34G20 

Notes

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Faculty of SciencesAlexandria UniversityAlexandriaEgypt

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