Wupper semicontinuous multivalued mappings and Kakutani theorem Inese

W-upper semicontinuous multivalued mappings and Kakutani theorem Inese Bula (in collaboration with Oksana Sambure) University of Latvia ibula@lanet. lv 1

Let X and Y be metric spaces. U(x, r) - open ball with center x and radius r. Let. Then is a neighbourhood of the set A. Definition 1. A multivalued mapping w-upper semicontinuous at a point is called if If f is w-upper semicontinuous multivalued mapping for every point of space X, then such a mapping is called w-upper semicontinuous multivalued mapping in space X (or w-u. s. c. ). 2

Every upper semicontinuous multivalued mapping is w-upper semicontinuous multivalued mapping (w>0) but not conversely. Example 1. and y 3 2 1 0 1 2 3 4 x This mapping is not upper semicontinuous multivalued mapping in point 2: But this mapping is 1 -upper semicontinuous multivalued mapping in point 2. It is w-upper semicontinuous multivalued mapping in point 2 for every too. 3

We consider Definition 2. A multivalued mapping is called w-closed at a point x, if for all convergent sequences which satisfy it follows that If f is w-closed mapping for every point of space X, then such a mapping is called w-closed mapping in space X. In Example 1 considered function is 1 -closed in point 2. It is w-closed mapping in point 2 for every too. 4

Let X, Y be normed spaces. We define a sum f + g of multivalued mappings as follows: We prove Theorem 1. If is w 1 -u. s. c. and is w 2 -u. s. c. , then f + g is (w 1+w 2)-u. s. c. Corollary. If is w-u. s. c. and is u. s. c. , then f + g is w-u. s. c. 5

Let X, Y be metric spaces. It is known for u. s. c. : If K is compact subset of X and is compact-valued u. s. c. , then the set If is compact-valued w-u. s. c. , then it is possible that is not compact even if K is compact subset of X. Example 2. Suppose the mapping is y This mapping is compact-valued and 0. 5 -u. s. c. , its domain is compact set [0, 2], but 3 2 2. 5 2. 3 1 0 this set is not compact, only bounded. 1 2 x 6

We prove Theorem 2. Let compact set, then is compact-valued w-u. s. c. If is is bounded set. In Example 1 considered mapping is 1 -u. s. c. , compact-valued and 1 -closed. Is it regularity? We can observe: if mapping is w-closed, then it is possible that there is a point such that the image is not closed set. For example, Theorem 3. If multivalued mapping for every is w-u. s. c. and the image set f(x) is closed, then f is w-closed. 7

Analog of Kakutani theorem Theorem 4. Let K be a compact convex subset of normed space X. Let be a w-u. s. c. multivalued mapping. Assume that for every , the image f(x) is a convex closed subset of K. Then there exists such that , that is B(x, r) - closed ball with center x and radius r. 8

Idea of PROOF. We define mapping This mapping satisfies the assumptions of the Kakutani theorem: If C be a compact convex subset of normed space X and if be a closed and convex-valued multivalued mapping, then there exists at least one fixed point of mapping f. Then It follows (f is w-u. s. c. multivalued mapping!) Therefore 9

In one-valued mapping case we have: Definition 1. A mapping is called w-continuous at a point if If f is w-continuous mapping for every point of space X, then such a mapping is called w-continuous mapping in space X. Corollary. Let K be a compact convex subset of normed space X. Let is w-continuous mapping. Then 10

References I. Bula, Stability of the Bohl-Brouwer-Schauder theorem, Nonlinear Analysis, Theory, Methods & Applications, V. 26, P. 1859 -1868, 1996. M. Burgin, A. Šostak, Towards theory of continuity defect and continuity measure for mappings of metric spaces, Latvijas Universitātes Zinātniskie Raksti, V. 576, P. 45 -62, 1992. M. Burgin, A. Šostak, Fuzzyfication of the Theory of Continuous Functions, Fuzzy Sets and Systems, V. 62, P. 71 -81, 1994. O. Zaytsev, On discontinuous mappings in metric spaces, Proc. of the Latvian Academy of Sciences, Section B, v. 52, 259 -262, 1998. 11

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