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К теории неподвижных точек на плоскости.

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2014. ? 7. ??????-?????????????? ?????. ???????????. 178-181
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Alexejhohlov@yandex.ru, SHALA@utmn.ru
??? 512.12
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On the theory of plane fixed points
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Summary. The article presents a new proof of the case showing that any connected
compact separating the plane has a fixed point. The provided proof does not use the
fundamental results of the fixed point theory or functional analysis.
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Key words. Plane, connected compact, fixed point.
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?????? ?? ??????? ???????, ?? ??????????? ?????????, ????? ???????????
?????? ??? ???????? ???? ??????? ????? ?????????? ? ?????????? ???????
????????? ?? ?????????. ?? ??????????? ???? ??? ???????????? ??????? ??????????, ??? ??????? ???????, ?? ??????????? ?????????, ???????? ?????????
??????????? ?????.
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f : X ? X, ???? f(x) = x. ???????, ??? ????????? X ???????? ????????? ??????????? ?????, ???? ?????? ??????????? ??????????? f : X ? X ????? ????????? ? ???? ????? ??????????? ?????.
????? ??????, ??? ?????????? ?? ???????? ????????? ??????????? ?????:
??? ????????, ????????, ?? ???? ?/2 ??? ????? ?? ?????????. ?? ??????????
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?? ???????????? ???????? ??????????? ?????? ??????????? ????? ? ??????????????? ???????, ?????????? ? ????????? ???????.
© ????? ??? Тюменск?? государственн?? университет
? ?????? ??????????? ????? ?? ????????? ...
179
?? (0, 1]. ?????????????, ??? ??????? ???????????? ? ???? ?????????????, ????
??? p ???????????? ?? ???????? ???????????? (0, 1] (? R ??????? ? ????????).
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???? A ? ??????? ????????? ? ? ??????, ???
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,
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A ? TT21 ????
?? diam A > 2.
2) ??????? ???????: f : X ? X, ??? f ? ??????????, X, Y ? ???????????
????????????, X ? ???????, ?? f ?????????? ??????????.
3) ?? ?????????? ???????????? ??????????? ? : [0,1] ? X ??????, ??? ?(0) ? ?,
?(1) ? ?.
??????????? ?????????. ??? ? = 1 ???????? ? > 0 ?????, ??? diam ? (B) < 1,
???? diam B < ? (? ???? ??????????? ?????????????). ???????? ??????? [0,1]
??????? 0 < x1 < x2 < ... < xn < xn +1 ?? ????? ? ?xi <<?.d ????? diam xi ?1 , xi < 1 .
? ???? 1) ??????? ???????????? ? ? ? [0, xi] ??????, ??? p (?[0, xi]) ??????????
???? T1, ???? T2, ?? ?? ??? ??????. ?????????? ????? ???? ? ?[x1, x2], ? ?.?. ???????, ??? ???? ??????? [0,1] ????? ? ?, ?.?. ????????????.
????? ???????, ??? ??????? ???????????? ??????????? ? : [0,1] ? X ?????
????????? ???? ?([0,1]) ? ?, ???? ?(
p[0,1])
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]
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????? g : X?X ? ??????????? ???????????. ???? g ({0}Ч [? 1, 1]) ? {0}Ч [? 1, 1],
?? ?? ??????? ??????? ??????? ??????????? ?????.
????? g ({}
0 Ч [? 1, 1]) ? {}
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? ??? ??????? ???????, ?????????????, p(g(X)) = [?, ?] ? (0,1]. ?????
.
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??????-?????????????? ?????. ???????????
180
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?????? ??????????
1. Dugundji, J., Granas, A. Fixed point theory. NY.: Springer-Verlag, 2003. 690 p.
2. Hagopian, Ch.L. Fixed-point Problem in Continuum Theory //Contemporary Math.
1991. Vol. 117. Pp. 79?86.
3. Klee, V., Wagon, S. Old and New Unsolved problems in Plane Geometry and Number
Theory // Dolciani Mathematical Expositions. Washington, DS: Math. Assoc. Amer., 1991.
Vol. 11. 340 p.
4. Akis, V.N. On the plane fixed point problem // Topology Proc. 1999. Vol. 24. Pp.
15?31.
5. Bell, H. On fixed point properties of plane Continua. // Tranc. Amer. Math. Soc.
1967. Vol. 128. Pp. 539?548.
6. Bellamy, D.P. A tree-like Continuum without the fixed point property. // Nonstom
J. Math. 1979. Vol. 6. Pp. 1?13.
7. Minc, P. A fixed point theorem for weakly chainable plane continua. // Tranc. Amer.
Math. Soc. 1990. Vol. 317. Pp. 303?312.
8. Mill, Y., van, Reed, G.M. Open Problems In Topology. Amsterdam: North-Holland
Publ. Co., 1990. Pp. 354-357.
9. Kiang, T. Theory of Fixed Point. Berlin: Classes, Springer, 1989. 424 p.
10. ???????? ?.?. ???????? ????????????? ? ???????????????? ?????????.
?.: ?????, 2004. 352 ?.
11. ?????????? ?. ????? ?????????. ???. ? ????. ?.: ???, 1986. 752 ?.
references
1. Dugundji, J., Granas, A. Fixed point theory. NY.: Springer-Verlag, 2003. 690 p.
2. Hagopian, Ch.L. Fixed-point Problem in Continuum Theory. Contemporary Math.
1991. Vol. 117. Pp. 79?86.
3. Klee, V., Wagon, S. Old and New Unsolved problems in Plane Geometry and Number
Theory // Dolciani Mathematical Expositions. Washington, DS: Math. Assoc. Amer., 1991.
Vol. 11. 340 p.
4. Akis, V.N. On the plane fixed point problem. Topology Proc. 1999. Vol. 24. Pp. 15?
31.
5. Bell, H. On fixed point properties of plane Continua. Tranc. Amer. Math. Soc. 1967.
Vol. 128. Pp. 539?548.
6. Bellamy, D.P. A tree-like Continuum without the fixed point property. Nonstom J.
Math. 1979. Vol. 6. Pp. 1?13.
7. Minc, P. A fixed point theorem for weakly chainable plane continua. Tranc. Amer.
Math. Soc. 1990. Vol. 317. Pp. 303?312.
8. Mill, Y., van, Reed, G.M. Open Problems In Topology. Amsterdam: North-Holland
Publ. Co., 1990. Pp. 354-357.
9. Kiang, T. Theory of Fixed Point. Berlin: Classes, Springer, 1989. 424 p.
10. Prasolov, V.V. Elementy kombinatornoi i differentsial'noi topologii [The elements
of combinatorial and differential topology]. Moscow, 2004. 352 p. (in Russian).
11. Engel'king, R. Obshchaia topologiia [General topology] / Transl. fr. Eng. by
M. Ya. Antonovsky, A.V. Arkhangelsky. Moscow, 1986. 752 p. (in Russian).
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? ?????? ??????? ????????? ?????????? ? ???????????? ???? ?????????? ???????????????? ????????????, ???????? ??????-?????????????? ????
Вестник Тюменского государственного университета.? 2014.? №? 7
? ?????? ??????????? ????? ?? ????????? ...
181
Authors of the publication
Aleksey G. Khokhlov ? Cand. Sci. (Phys.-Math.), Associate Professor, Department
of Mathematical Analysis and Theory of Functions, Institute of Mathematics and Computer
Sciences, Tyumen State University
Sergey D. Shalaginov ? Cand. Sci. (Phys.-Math.), Associate Professor, Department
of Mathematical Analysis and Theory of Functions, Institute of Mathematics and Computer
Sciences, Tyumen State University
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