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On the continuity properties of the Lp balls

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Veröffentlicht/Copyright: 11. November 2022
Journal of Applied Analysis
Aus der Zeitschrift Journal of Applied Analysis Band 29 Heft 1

Abstract

In this paper the right upper semicontinuity at p = 1 and continuity at p = ∞ of the set-valued map p → B Ω , 𝒳 , p ⁢ ( r ) , p ∈ [ 1 , ∞ ] , are studied where B Ω , 𝒳 , p ⁢ ( r ) is the closed ball of the space L p ⁢ ( 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 right upper semicontinuous at p = 1 and continuous at p = ∞ . An application of the obtained results to the set of integrable outputs of the input-output system described by the Urysohn-type integral operator is discussed.

Keywords: Continuity; semicontinuity; set-valued map; Urysohn integral operator; input-output system
MSC 2010: 26E25; 46T20; 93C35

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Received: 2021-12-27
Revised: 2022-08-17
Accepted: 2022-09-21
Published Online: 2022-11-11
Published in Print: 2023-06-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Heaviside function as an activation function
  3. Krylov solvability under perturbations of abstract inverse linear problems
  4. Existence of solutions for a three-point Hadamard fractional resonant boundary value problem
  5. Positive solutions to mixed fractional p-Laplacian boundary value problems
  6. Maximum number of limit cycles for generalized Kukles differential system
  7. Stabilization of a 1-D transmission problem for the Rayleigh beam and string with localized frictional damping
  8. Certain multiplier results on Bp spaces
  9. Remarks on the Belitskii–Lyubich and discrete Markus–Yamabe conjectures
  10. On deferred statistical convergence of complex uncertain sequences
  11. Local existence and uniqueness of regular solutions to a Landau–Lifshitz–Bloch equation with applied current
  12. Converse Ohlin’s lemma for convex and strongly convex functions
  13. Approximate controllability results in α-norm for some partial functional integrodifferential equations with nonlocal initial conditions in Banach spaces
  14. Direct proofs of intrinsic properties of prox-regular sets in Hilbert spaces
  15. On the continuity properties of the Lp balls
  16. On L 𝔮 convergence of the Hamiltonian Monte Carlo
  17. Radii of starlikeness and convexity of generalized k-Bessel functions
  18. An iterative technique for solving split equality monotone variational inclusion and fixed point problems
https://doi.org/10.1515/jaa-2022-1008
Schlagwörter für diesen Artikel
Continuity; semicontinuity; set-valued map; Urysohn integral operator; input-output system

Artikel in diesem Heft

  1. Frontmatter
  2. Heaviside function as an activation function
  3. Krylov solvability under perturbations of abstract inverse linear problems
  4. Existence of solutions for a three-point Hadamard fractional resonant boundary value problem
  5. Positive solutions to mixed fractional p-Laplacian boundary value problems
  6. Maximum number of limit cycles for generalized Kukles differential system
  7. Stabilization of a 1-D transmission problem for the Rayleigh beam and string with localized frictional damping
  8. Certain multiplier results on Bp spaces
  9. Remarks on the Belitskii–Lyubich and discrete Markus–Yamabe conjectures
  10. On deferred statistical convergence of complex uncertain sequences
  11. Local existence and uniqueness of regular solutions to a Landau–Lifshitz–Bloch equation with applied current
  12. Converse Ohlin’s lemma for convex and strongly convex functions
  13. Approximate controllability results in α-norm for some partial functional integrodifferential equations with nonlocal initial conditions in Banach spaces
  14. Direct proofs of intrinsic properties of prox-regular sets in Hilbert spaces
  15. On the continuity properties of the Lp balls
  16. On L 𝔮 convergence of the Hamiltonian Monte Carlo
  17. Radii of starlikeness and convexity of generalized k-Bessel functions
  18. An iterative technique for solving split equality monotone variational inclusion and fixed point problems
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