Difference between revisions of "Condensing operator"
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| + | An operator $ U $, | ||
| + | generally non-linear, defined on the set $ \mathfrak M $ | ||
| + | of all subsets of a set $ M $ | ||
| + | in a normed vector space $ X $, | ||
| + | with values in a normed vector space $ Y $, | ||
| + | such that $ \psi _ {Y} [ U ( A) ] $— | ||
| + | the measure of non-compactness of the set $ U ( A) \subset Y $— | ||
| + | is less than the measure of non-compactness $ \psi _ {X} ( A) $ | ||
| + | for any non-compact set $ A \in \mathfrak M $. | ||
| + | Here, the measures of non-compactness may be the same in both cases or different. For example, as $ \psi _ {X} $ | ||
| + | and $ \psi _ {Y} $ | ||
| + | one may take the Kuratowski measure of non-compactness: $ \alpha ( A) = \inf \{ d > 0, A \textrm{ may be decomposed into finitely many subsets of diameter less than } d \} $. | ||
For a continuous condensing operator many constructions and facts of the theory of completely-continuous operators can be carried over, for instance, the rotation of contracting vector fields, the fixed-point principle of contraction operators, etc. | For a continuous condensing operator many constructions and facts of the theory of completely-continuous operators can be carried over, for instance, the rotation of contracting vector fields, the fixed-point principle of contraction operators, etc. | ||
Latest revision as of 17:46, 4 June 2020
An operator $ U $,
generally non-linear, defined on the set $ \mathfrak M $
of all subsets of a set $ M $
in a normed vector space $ X $,
with values in a normed vector space $ Y $,
such that $ \psi _ {Y} [ U ( A) ] $—
the measure of non-compactness of the set $ U ( A) \subset Y $—
is less than the measure of non-compactness $ \psi _ {X} ( A) $
for any non-compact set $ A \in \mathfrak M $.
Here, the measures of non-compactness may be the same in both cases or different. For example, as $ \psi _ {X} $
and $ \psi _ {Y} $
one may take the Kuratowski measure of non-compactness: $ \alpha ( A) = \inf \{ d > 0, A \textrm{ may be decomposed into finitely many subsets of diameter less than } d \} $.
For a continuous condensing operator many constructions and facts of the theory of completely-continuous operators can be carried over, for instance, the rotation of contracting vector fields, the fixed-point principle of contraction operators, etc.
References
| [1] | B.N. Sadovskii, "Limit-compact and condensing operators" Russian Math. Surveys , 27 (1972) pp. 85–155 Uspekhi Mat. Nauk , 27 : 1 (1972) pp. 81–146 |
| [2] | C. Kuratowski, "Sur les espaces complets" Fund. Math. , 15 (1930) pp. 301–309 |
How to Cite This Entry:
Condensing operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Condensing_operator&oldid=46439
Condensing operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Condensing_operator&oldid=46439
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article