On measurable multifunctions in countably separated spaces

1993 ◽  
Vol 42 (1) ◽  
pp. 82-92 ◽  
Author(s):  
Andrzej Spakowski ◽  
Piotr Urbaniec
Keyword(s):  
Measurable Multifunctions

Measurable Multifunctions

1997 ◽  
pp. 138-297
Author(s):  
Shouchuan Hu ◽  
Nikolas S. Papageorgiou
Keyword(s):  
Measurable Multifunctions

scholarly journals Convergence theorems for Banach space valued integrable multifunctions

1987 ◽  
Vol 10 (3) ◽  
pp. 433-442 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Convergent Sequence ◽  
Mosco Convergence ◽  
Convergence Theorems ◽  
Aumann Integral ◽  
Convergence Results ◽  
Measurable Multifunctions ◽  
Dominated Convergence ◽  
Convergence Of Sets

In this work we generalize a result of Kato on the pointwise behavior of a weakly convergent sequence in the Lebesgue-Bochner spacesLXP(Ω) (1≤p≤∞). Then we use that result to prove Fatou's type lemmata and dominated convergence theorems for the Aumann integral of Banach space valued measurable multifunctions. Analogous convergence results are also proved for the sets of integrable selectors of those multifunctions. In the process of proving those convergence theorems we make some useful observations concerning the Kuratowski-Mosco convergence of sets.

scholarly journals Nonlinear random operator equations and inequalities in Banach spaces

1992 ◽  
Vol 15 (1) ◽  
pp. 111-118 ◽  
Author(s):  
Antonios Karamolegos ◽  
Dimitrios Kravvaritis
Keyword(s):  
Banach Spaces ◽  
Existence Theorems ◽  
Random Operator ◽  
Measurable Selection ◽  
Hammerstein Integral Equation ◽  
Operator Equations ◽  
Deterministic Theory ◽  
Measurable Multifunctions ◽  
Selection Theorems ◽  
Monotone Type

In this paper we give some new existence theorems for nonlinear random equations and inequalities involving operators of monotone type in Banach spaces. A random Hammerstein integral equation is also studied. In order to obtain random solutions we use some results from the existing deterministic theory as well as from the theory of measurable multifunctions and, in particular, the measurable selection theorems of Kuratowski/Ryll-Nardzewski and of Saint-Beuve.

scholarly journals Measurable multifunctions and their applications to convex integral functionals

1989 ◽  
Vol 12 (1) ◽  
pp. 175-191 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Control Systems ◽  
Banach Spaces ◽  
Time Dependent ◽  
Linear Control ◽  
Linear Control Systems ◽  
Integral Functionals ◽  
Control Constraints ◽  
Infinite Dimensional ◽  
Measurable Multifunctions

The purpose of this paper is to establish some new properties of set valued measurable functions and of their sets of Integrable selectors and to use them to study convex integral functionals defined on Lebesgue-Bochner spaces. In this process we also obtain a characterization of separable dual Banach spaces using multifunctions and we present some generalizations of the classical “bang-bang” principle to infinite dimensional linear control systems with time dependent control constraints.

scholarly journals Strong convergence of selections implied by weak

1989 ◽  
Vol 39 (2) ◽  
pp. 201-214 ◽  
Author(s):  
Tadeusz Rzezuchowski
Keyword(s):  
Weak Convergence ◽  
Strong Convergence ◽  
Differential Inclusions ◽  
Control Functions ◽  
Measurable Multifunctions ◽  
Derivatives Of ◽  
Convergent Sequences ◽  
Convex Subsets

In some situations weak convergence in L1, implies strong convergence. Let P, L: T → C∘(ℝd) be measurable multifunctions (C∘(ℝd) being the set of closed, convex subsets of ℝd) the values L(t) affine sets and W(t) = P(t) ∩ L(t) extremal faces of P(t). Let pk be integrable selections of P, the projection of pk,(t) on L(t) and pk(t) on W(t). We prove that if converges weakly to zero then pk − k converges to zero in measure. We give also some extensions of this theorem. As applications to differential inclusions we investigate convergence of derivatives of convergent sequences of solutions and we describe solutions which are in some sense isolated. Finally we discuss what can be said about control functions u when the corresponding trajectories of ẋ = f(t, x, u) are convergent to some trajectory.

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