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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.

Set-valued minimax fractional programming problems under -cone arcwise connectedness

Das Koushik

Control and Cybernetics

2022

Vol. 51, No. 1

43--69

In this paper, we consider a set-valued minimax fractional programming problem (MFP), where the objective as well as constraint maps are set-valued. We introduce the notion of ρ- cone arcwise connectedness of set-valued maps as a generalization of cone arcwise connected set-valued maps. We establish the sufficient Karush-Kuhn-Tucker (KKT) conditions for the existence of minimizers of the problem (MFP) under ρ-cone arcwise connectedness assumption. Further, we study the Mond-Weir (MWD), Wolfe (WD), and mixed (MD) types of duality models and prove the corresponding weak, strong, and converse duality theorems between the primal (MFP) and the corresponding dual problems under ρ-cone arcwise connectedness assumption.

Graph convergence of set-valued maps and its relationship to other convergences

Del Prete I., Di Iorio M., Holá L.

Journal of Applied Analysis

2000

Vol. 6, nr 2

213-226

The notion of even-outer-semicontinuity for set-valued maps is introduced and compared with related ones from [4] and [11]. The coincidence of these notions provides a new characterization of compactness and of local compactness. The following result is proved: Let X be a topological space, Y a uniform space, {Fσ : σ ∈ ∑} be a net of set-valued maps from X to Y and F be a set valued map from X to Y. Then any two of the following conditions imply the third: (1) the net {Fσ : σ ∈ ∑} is evenly-outer semicontinuous; (2) the net {{Fσ : σ ∈ ∑} is graph convergent to F; (3) the net {Fσ : σ ∈ ∑} is pointwise convergent to F. This theorem generalizes some results from [4] and [11].

A fixed point theorem for composed set-valued maps

Balaj M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

1998

[Z] 38

21-27

A fixed point theorem for the composition product of two set-valued maps is established. As an application of this result a generalization of Fan's minimax inequality is obtained.