On the Vietoris semicontinuity property of the Lp balls at p= 1 and an application

HÜSEYİN A.

Archiv der Mathematik, cilt.121, sa.2, ss.171-182, 2023 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 121 Sayı: 2
  • Basım Tarihi: 2023
  • Doi Numarası: 10.1007/s00013-023-01881-y
  • Dergi Adı: Archiv der Mathematik
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, DIALNET
  • Sayfa Sayıları: ss.171-182
  • Anahtar Kelimeler: Input–output system, Set valued map, Urysohn integral operator, Vietoris semicontinuity
  • Sivas Cumhuriyet Üniversitesi Adresli: Evet

Özet

In this paper, the Vietoris right lower semicontinuity at p= 1 of the set valued map p→ BΩ,X,p(r) , p∈ [1 , ∞] , is discussed where BΩ,X,p(r) is the closed ball of the space Lp(Ω , Σ , μ; X) centered at the origin with radius r, (Ω , Σ , μ) is a finite and positive measure space, X is a separable Banach space. It is proved that the considered set valued map is Vietoris right lower semicontinuous at p= 1 . Introducing additional geometric constraints on the functions from the ball BΩ,X,1(r) , a property, which is close to the Hausdorff right lower semicontinuity, is derived. An application of the obtained result to the set of integrable outputs of the input–output system described by a Urysohn type integral operator is studied.