17. Set-valued maps. Upper semicontinuous and lower semicontinuous set-valued maps

In this paper we continue the study of interconnections between separately continuous function which was started by V. K. Maslyuchenko. A pair (g, h) of functions on a topological space is called a pair of Hahn if g ≤ h, g is an upper semicontinuous function and h is a lower semicontinuous function. We say that a pair of Hahn (g, h) is generated by a function f, which depends on two variables, if the inﬁmum of f and the supremum of f with respect to the second variable equals g and h respectively. We prove that for any perfectly normal space X and non-pseudocompact space Y every pair of Hahn on X is generated by a continuous function on X x Y . We also obtain that for any perfectly normal space X and for any space Y having non-scattered compactiﬁcation any pair of Hahn on X is generated by a separately continuous function on X x Y .

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Maximal Classes For Lower And Upper Semicontinuous Strong Świątkowski Functions

Demonstratio Mathematica ◽

10.2478/dema-2014-0004 ◽

2014 ◽

Vol 47 (1) ◽

Author(s):

Paulina Szczuka

Keyword(s):

Lower Semicontinuous ◽

Maximal Class ◽

Upper Semicontinuous

AbstractIn this paper, we characterize the maximal additive and multiplicative classes for lower and upper semicontinuous strong Świątkowsk functions and lower and upper semicontinuous extra strong Świątkowski functions. Moreover, we characterize the maximal class with respect to maximums for lower semicontinuous strong Świątkowski functions and lower and upper semicontinuous extra strong Świątkowski functions.

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Double-dual types over the Banach spaceC(K)

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms.2005.2533 ◽

2005 ◽

Vol 2005 (16) ◽

pp. 2533-2545

Author(s):

Markus Pomper

Keyword(s):

Banach Space ◽

Compact Hausdorff Space ◽

Hausdorff Space ◽

Lower Semicontinuous ◽

Continuous Functions ◽

Upper Semicontinuous ◽

Isolated Points

LetKbe a compact Hausdorff space andC(K)the Banach space of all real-valued continuous functions onK, with the sup-norm. Types overC(K)(in the sense of Krivine and Maurey) can be uniquely represented by pairs(ℓ,u)of bounded real-valued functions onK, whereℓis lower semicontinuous,uis upper semicontinuous,ℓ≤u, andℓ(x)=u(x)for all isolated pointsxofK. A condition that characterizes the pairs(ℓ,u)that represent double-dual types overC(K)is given.

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Types over c(k) spaces

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700010120 ◽

2004 ◽

Vol 77 (1) ◽

pp. 17-28

Author(s):

Markus Pomper

Keyword(s):

Banach Space ◽

Compact Hausdorff Space ◽

Hausdorff Space ◽

Lower Semicontinuous ◽

Continuous Functions ◽

Upper Semicontinuous ◽

Bounded Functions ◽

Isolated Point

AbstractLet K be a compact Hausdorff space and C(K) the Banach space of all real-valued continuous functions on K, with the sup norm. Types over C(K) (in the sense of Krivine and Maurey) are represented here by pairs (l, u) of bounded real-valued functions on K, where l is lower semicontinuous and u is upper semicontinuous, l ≤ u and l(x) = u(x) for every isolated point x of K. For each pair the corresponding type is defined by the equation τ(g) = max{║l + g║∞, ║u + g║∞} for all g ∈ C(K), where ║·║∞ is the sup norm on bounded functions. The correspondence between types and pairs (l, u) is bijective.

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Distance to Spaces of Semicontinuous and Continuous Functions

Journal of Function Spaces ◽

10.1155/2019/9602504 ◽

2019 ◽

Vol 2019 ◽

pp. 1-7

Author(s):

Carlos Angosto

Keyword(s):

Topological Space ◽

Lower Semicontinuous ◽

Continuous Functions ◽

Upper Semicontinuous ◽

Lower Semicontinuous Functions ◽

Upper Semicontinuous Functions ◽

Semicontinuous Functions

Given a topological spaceX, we establish formulas to compute the distance from a functionf∈RXto the spaces of upper semicontinuous functions and lower semicontinuous functions. For this, we introduce an index of upper semioscillation and lower semioscillation. We also establish new formulas about distances to some subspaces of continuous functions that generalize some classical results.

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Control Systems Described by a Class of Fractional Semilinear Evolution Equations and Their Relaxation Property

Abstract and Applied Analysis ◽

10.1155/2012/850529 ◽

2012 ◽

Vol 2012 ◽

pp. 1-20

Author(s):

Xiaoyou Liu ◽

Xi Fu

Keyword(s):

Control Systems ◽

Evolution Equations ◽

Reflexive Banach Space ◽

Lower Semicontinuous ◽

Original System ◽

Relaxation Property ◽

Solution Sets ◽

State Variable ◽

Upper Semicontinuous ◽

Semilinear Evolution Equations

We consider a control system described by a class of fractional semilinear evolution equations in a separable reflexive Banach space. The constraint on the control is a multivalued map with nonconvex values which is lower semicontinuous with respect to the state variable. Along with the original system we also consider the system in which the constraint on the control is the upper semicontinuous convex-valued regularization of the original constraint. We obtain the existence results for the control systems and the relaxation property between the solution sets of these systems.

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Spaces of lower semicontinuous set-valued maps I

Mathematica Slovaca ◽

10.2478/s12175-010-0030-x ◽

2010 ◽

Vol 60 (4) ◽

Cited By ~ 1

Author(s):

R. McCoy

Keyword(s):

Lower Semicontinuous ◽

Extension Theorem ◽

Continuous Functions ◽

Factorization Theorem ◽

Compact Interval ◽

Unique Factorization ◽

Second Main Theorem ◽

Upper Semicontinuous ◽

Nonempty Compact ◽

The Second Main Theorem

AbstractWe introduce a lower semicontinuous analog, L −(X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L −(X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L −(X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L −(X) and L −(Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L −(X) and L −(Y) can be characterized by a unique factorization.

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On the Borel Classes of Set-Valued Maps of Two Variables

Annales Mathematicae Silesianae ◽

10.2478/amsil-2020-0018 ◽

2020 ◽

Vol 34 (1) ◽

pp. 81-95

Author(s):

Ľubica Holá ◽

Grażyna Kwiecińska

Keyword(s):

Lower Semicontinuous ◽

Topological Spaces ◽

Upper Semicontinuous ◽

Class 1 ◽

Borel Class ◽

Perfectly Normal

AbstractUsing the Borel classification of set-valued maps, we present here some new results on set-valued maps which are similar to some of the well known theorems on functions due to Lebesgue and Kuratowski. We consider set-valued maps of two variables in perfectly normal topological spaces. It was proved in [11] that a set-valued map lower semicontinuous (i.e. of lower Borel class 0) in the first and upper semicontinuous (i.e. of upper Borel class 0) in the second variable is of upper Borel class 1 and also (with stronger assumptions) of lower Borel class 1. This result cannot be generalized into higher Borel classes. In this paper we show that a set-valued map of the upper (resp. lower) Borel class α in the first and lower semicontinuous and upper quasicontinuous (upper semicontinuous and lower quasicontinuous) in the second variable is of the lower (resp. upper) Borel class α + 1. Also other cases are considered.

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2-Strict Convexity and Continuity of Set-Valued Metric Generalized Inverse in Banach Spaces

Abstract and Applied Analysis ◽

10.1155/2014/384639 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8 ◽

Cited By ~ 2

Author(s):

Shaoqiang Shang ◽

Yunan Cui

Keyword(s):

Banach Spaces ◽

Generalized Inverse ◽

Lower Semicontinuous ◽

Continuous Mapping ◽

Linear Operators ◽

Generalized Inverses ◽

Bounded Linear Operators ◽

Bounded Linear ◽

Upper Semicontinuous ◽

Set Valued Mapping

Authors investigate the metric generalized inverses of linear operators in Banach spaces. Authors prove by the methods of geometry of Banach spaces that, ifXis approximately compact andXis 2-strictly convex, then metric generalized inverses of bounded linear operators inXare upper semicontinuous. Moreover, authors also give criteria for metric generalized inverses of bounded linear operators to be lower semicontinuous. Finally, a sufficient condition for set-valued mappingT∂to be continuous mapping is given.

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Regular and singular perturbations of upper semicontinuous differential inclusion

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171297000951 ◽

1997 ◽

Vol 20 (4) ◽

pp. 699-706 ◽

Cited By ~ 4

Author(s):

Tzanko Donchev ◽

Vasil Angelov

Keyword(s):

Differential Inclusion ◽

Singular Perturbations ◽

Differential Inclusions ◽

Existence Result ◽

Lower Semicontinuous ◽

Solution Set ◽

Singularly Perturbed ◽

Solution Sets ◽

Upper Semicontinuous ◽

Continuity Properties

In the paper we study the continuity properties of the solution set of upper semicontinuous differential inclusions in both regularly and singularly perturbed case. Using a kind of dissipative type of conditions introduced in [1] we obtain lower semicontinuous dependence of the solution sets. Moreover new existence result for lower semicontinuous differential inclusions is proved.

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