E-methods for fixed point equations f(x)=x

Kaucher, E.; Rump, Siegfried M.

doi:10.1007/bf02237993

Cited by 29 publications

(5 citation statements)

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“…Rädström's embedding theorem and its generalizations also establish the preservation of topological properties. Furthermore, Kaucher [5] proved abstract embedding theorems which ensure the preservation of order properties, and embedding theorems for classes of convex sets in a Hausdorff locally convex vector Space which simultaneously guarantee the preservation of topological and order properties were proven by Hörmander [4] and Schmidt [14]. In view of these remarks, it is a natural question to ask whether or not Fischer's results can be improved.…”

Section: Introductionmentioning

confidence: 94%

“…While the first of these results is a partial extension of Rädström' embedding theorem [12] for the class of all nonempty, compact, convex subsets of anormed vector space, the second one contains embedding theorems for the class of all norm balls of a normed vector space and for the class of all order intervals of a vector lattice. These special cases are of particular interest in interval mathematics [1,5,6,8,9,10,11].…”

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confidence: 98%

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Embedding Theorems for Classes of Convex Sets in a Hypernormed Vector Space

Schmidt¹

1986

Analysis

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As a partial extension of Rädström's einbedding theorem, Fischer recently proved that the class of all nonempty, hypernorm sequentially compact, convex subsets of a hypernormed vector space can be embedded as a convex cone in a vector space, and he also obtained a similar embedding theorem for the class of all hypernorm balls. In the present paper, Fischer's results are improved by taking into account the inclusion of sets as an order relation, and it is also shown that the füll Statement of Rädström's embedding theorem cannot be extended to the general hypernormed case. Furthermore, embedding theorems are proven for the class of all order intervals of a vector lattice. These results are roore precise than Fischer's embedding theorem for hypernorm balls and cannot be extended to the general case. AMS 1980 subject Classification: 52A05, 52A07, 06F05, 06F20, 46A40, 46B30. Brought to you by | Michigan State University Authenticated Download Date | 6/26/15 8:20 PM 58 Schmidt 1. INTRODUCTIONIn a recent paper published in this journal, Fischer [2] proved that the class of all nonempty, hypernorm sequentially compact, convex subsets of a hypernormed vector space can be embedded as a convex cone in a suitable vector space. Under the assumption that the hypernorm is splittable, he also obtained a similar embedding theorem for the class of all hypernorm balls. While the first of these results is a partial extension of Rädström' embedding theorem [12] for the class of all nonempty, compact, convex subsets of anormed vector space, the second one contains embedding theorems for the class of all norm balls of a normed vector space and for the class of all order intervals of a vector lattice. These special cases are of particular interest in interval mathematics [1,5,6,8,9,10,11].The usefulness of such embedding theorems clearly depends on the amount of Information they provide on the structure of the embedding vector space and the properties of the embedding map. While Fischer's results only guarantee the existence of an embedding vector space and an embedding map which is additive and positively homogeneous. Rädström's embedding theorem and its generalizations also establish the preservation of topological properties. Furthermore, Kaucher [5] proved abstract embedding theorems which ensure the preservation of order properties, and embedding theorems for classes of convex sets in a Hausdorff locally convex vector Space which simultaneously guarantee the preservation of topological and order properties were proven by Hörmander [4] and Schmidt [14]. In view of these remarks, it is a natural question to ask whether or not Fischer's results can be improved. This question will be studied in the present paper. Brought to you by | Michigan State University Authenticated Download Date | 6/26/15 8:20 PM Embedding theorems 59This paper is organized as follows:In Sections 2 and 3, we recall some definitions concerning cones and hypernormed vector spaces.In the first part of Section 4, we study the question of whether or not the füll Stat...

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Section: Introductionmentioning

confidence: 94%

mentioning

confidence: 98%

Embedding Theorems for Classes of Convex Sets in a Hypernormed Vector Space

Schmidt¹

1986

Analysis

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“…Various kinds of computational algorithm have been developed with the above technique. 6 To illustrate the technique, a solution for a simultaneous linear equation is explained. The target equation is expressed as…”

Section: Computational Algorithmmentioning

confidence: 99%

High-accuracy optical computing based on interval arithmetic and the fixed-point theorem

1996

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A method for high-accuracy analog optical computing based on interval arithmetic and the fixed-point theorem is considered. Two-variable simultaneous equations are studied to investigate the proposed method. An optical implementation is considered by the use of spatial coding of intervals, affine transformation, and image magnification. Computational simulation verifies the principle of the method.

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“…For the role of fixed-point theorems in interval mathematics, we refer to the papers by Kaucher and Rump [28] and Nickel [35].…”

Section: Subadditive Ergodic Theorymentioning

confidence: 99%

Embedding theorems for classes of convex sets

Schmidt

1986

Acta Appl Math

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R/idstrOm's embedding theorem states that the nonempty compact convex subsets of a normed vector space can be identified with points of another normed vector space such that the embedding map is additive, positively homogeneous, and isometric. In the present paper, extensions of R~dstr6m's embedding theorem are proven which provide additional information on the embedding space. These results include those of H6rmander who proved a similar embedding theorem for the nonempty closed bounded convex subsets ofa Hausdorfflocally convex vector space. In contrast to HOrmander's approach via support function als, all embedding theorems of the present paper are proven by a refinement of Rgtdstr6m's original method which is constructive and does not rely on Zorn's lemma. This paper also includes a brief discussion of some actual or potential applications of embedding theorems for classes of convex sets in probability theory, mathematical economics, interval mathematics, and related areas.

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E-methods for fixed point equations f(x)=x

Cited by 29 publications

References 6 publications

Embedding Theorems for Classes of Convex Sets in a Hypernormed Vector Space

Embedding Theorems for Classes of Convex Sets in a Hypernormed Vector Space

High-accuracy optical computing based on interval arithmetic and the fixed-point theorem

Embedding theorems for classes of convex sets

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