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1
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Topological and approximation methods of degree theory of set-valued maps

100%
Kryszewski W.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS    Introduction................................................................................................................5    Preliminaries...............................................................................................................7 A. Elements of homology theory.....................................................................................8    1. Products.................................................................................................................8    2. Orientation of manifolds........................................................................................10 I. Topology of morphisms..............................................................................................12    1. Set-valued maps....................................................................................................12    2. Vietoris maps.........................................................................................................14    3. Category of morphisms..........................................................................................18    4. Operations in the category of morphisms..............................................................21    5. Homotopy and extension properties of morphisms................................................23    6. Essentiality of morphisms.......................................................................................28    7. Concluding remarks................................................................................................32 II. The topological degree theory of morphisms............................................................33    1. Cohomological properties of morphisms.................................................................34    2. The fundamental cohomology class........................................................................36    3. The topological degree of morphisms.....................................................................38    4. The degree of morphisms of spheres and open subsets of Euclidean space..........43    5. Borsuk type theorems..............................................................................................48    6. Applications.............................................................................................................56 III. The class of approximation-admissible morphisms......................................................59    1. Filtrations.................................................................................................................60    2. Approximation-admissible morphisms and maps......................................................63    3. Approximation of A-maps.........................................................................................68 IV. Approximation degree theory for A-morphisms...........................................................73    1. The degree of A-morphisms.....................................................................................73    2. Properties of the degree of A-morphisms................................................................75    3. Further properties of the degree. Applications.......................................................78 V. Other classes of set-valued maps..............................................................................83    1. Single-valued approximations..................................................................................83    2. Linear filtrations. AP-maps of Petryshyn...................................................................91    References..................................................................................................................97
2
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Continuous selection theorems

98%
Kisielewicz M.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2005
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tom 25
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nr 1
159-163
EN
Continuous approximation selection theorems are given. Hence, in some special cases continuous versions of Fillipov's selection theorem follow.
3
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Properties of generalized set-valued stochastic integrals

98%
Kisielewicz M.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2014
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tom 34
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nr 1
131-147
EN
The paper is devoted to properties of generalized set-valued stochastic integrals defined in [10]. These integrals generalize set-valued stochastic integrals defined by E.J. Jung and J.H. Kim in the paper [4]. Up to now we were not able to construct any example of set-valued stochastic processes, different on a singleton, having integrably bounded set-valued integrals defined in [4]. It was shown by M. Michta (see [11]) that in the general case set-valued stochastic integrals defined by E.J. Jung and J.H. Kim, are not integrably bounded. Generalized set-valued stochastic integrals, considered in the paper, are in some non-trivial cases square integrably bounded and can be applied in the theory of stochastic differential equations with set-valued solutions.
4
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A Common Fixed Point Theorem for Set-valued Contraction Mappings in Menger Space

86%
Pant B. D., Chauhan S., Kumar S.
Commentationes Mathematicae
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2012
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tom 52
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nr 1
EN
The aim of this paper is to prove a common fixed point theorem for even number of single-valued and two set-valued mappings in complete Menger space using implicit relation. Our result improves and extends the result of Chen and Chang [Common fixed point theorems in Menger spaces, Int. J. Math. Math. Sci. 2006, Art. ID 75931, 15 pp].
5

Set-valued random differential equations in Banach space

86%
Michta M.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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1995
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tom 15
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nr 2
191-200
EN
We consider the problem of the existence of solutions of the random set-valued equation: (I) $D_HX_t = F(t,X_t)P.1$, t ∈ [0,T] -a.e.; X₀ = U p.1 where F and U are given random set-valued mappings with values in the space $K_c(E)$, of all nonempty, compact and convex subsets of the separable Banach space E. Under certain restrictions on F we obtain existence of solutions of the problem (I). The connections between solutions of (I) and solutions of random differential inclusions are investigated.
6
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Ball intersection model for Fejér zones of convex closed sets

74%
Schott D.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2001
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tom 21
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nr 1
51-79
EN
Strongly Fejér monotone mappings are widely used to solve convex problems by corresponding iterative methods. Here the maximal of such mappings with respect to set inclusion of the images are investigated. These mappings supply restriction zones for the successors of Fejér monotone iterative methods. The basic tool is the representation of the images by intersection of certain balls.
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