The brouwer fixed point theorem and tetragon with all vertexes in a surface |
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| Authors: | Jiehua Mai |
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| Institution: | (1) Institute of Mathematics, Shantou University, 515063 Shantou, China |
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| Abstract: | LetD be a disc with radiusr in the Euclidean plane ℝ2, and letF be a Lipschitz continuous real valued function onD. SupposeA
1
A
21
A
3
A
4 is an isosceles trapezoid with lengths of edges not greater thanr, and ∠A
1
A
21
A
3 = α≤π/2 By means of the Brouwer fixed point theorem, it is proved that ifF has a Lipschitz constant λ≤min{1, tgα}, then there exist four coplanar points in the surfaceM = {(x, y, F(x, y))∈ℝ3:(x, y)ℝ} which span a tetragon congruent toA
1
A
21
A
3
A
4. In addition, some further problems are discussed.
Project supported by the National Natural Science Foundation of China (Grant No. 19231201). |
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| Keywords: | surface Lipschitz constant continuous functional homotopy mapping degree Brouwer fixed point theorem |
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