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On the continuity properties of the Lp balls

Huseyin Nesir, Huseyin Anar

Journal of Applied Analysis

2023

Vol. 29, nr 1

151--159

In this paper the right upper semicontinuity at p = 1 and continuity at p = ∞ of the set-valued map p → BΩ,X,p(r), p ∈ [1, ∞], are studied 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 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.

Upper and lower almost cl-supercontinuous multifunctions

Kohli J. K., Arya C. P.

Demonstratio Mathematica

2011

Vol. 44, nr 2

407--421

The notion of almost cl-supercontinuity (≡ almost clopen continuity) of functions (Acta Math. Hungar. 107 (2005), 193–206; Applied Gen. Topology 10 (1) (2009), 1–12) is extended to the realm of multifunctions. Basic properties of upper (lower) almost cl-supercontinuous multifunctions are studied and their place in the hierarchy of strong variants of continuity of multifunctions is discussed. Examples are included to reflect upon the distinctiveness of upper (lower) almost cl-supercontinuity of multifunctions from that of other strong variants of continuity of multifunctions which already exist in the literature.

Continuity of real valued subquadratic functions

Kominek Z., Troczka-Pawelec K.

Commentationes Mathematicae

2011

Vol. 51, [Z] 1

71-75

The aim of this paper is to prove a regularity theorem for real valued subquadratic mappings that are solutions of the inequality [formula], where X = (X, +) is a topological group.

On semicontinuity in impulsive dynamical systems

Ciesielski K.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 1

71-80

In the important paper on impulsive systems [Kl] several notions are introduced and several properties of these systems are shown. In particular, the function [phi] which describes "the time of reaching impulse points" is considered; this function has many important applications. In [Kl] the continuity of this function is investigated. However, contrary to the theorem stated there, the function [phi] need not be continuous under the assumptions given in the theorem. Suitable examples are shown in this paper. We characterize the function [phi] from the point of view of its semicontinuity. Also, we show the analogous properties for impulsive systems given by semidynamical systems. In the last section we investigate the continuity properties of the escape time function in impulsive systems.