On fixed points of one class of
operators

Yu. Korobeinik

Let $T\colon Q\to Q$ be a selfmapping of some subset $Q$ of a Banach space $E$. The operator $T$ does not increase the distance between $Q$ and the origin (the null element $E$) if $$\forall x\in Q\quad\|Tx\|\le\|x\|$$ (such operator is called a NID operator). In this paper some sufficient conditions for the existence of a non-trivial fixed point of a NID operator are obtained. These conditions enable in turn to prove some new results on fixed points of nonexpanding operators.

1. . 10. F.E. Browder Nonexpansive nonlinear operators in Banach spaces // Proc. Nat. Acad. Sci. USA V.54 –1965 p.1041–1044

2. W.A. Kirk A fixed point theorem for mappings which do not increase distance // Amer. Math. Monthly –1965 V.72 p.1004–1006

3. J.P. Gossez and E. Lami Dozo Some geometrical properties related to the fixed point theory for nonexpanding mappings // Pacific J. Math. V.40 –1972 p.565–573

4. L.A. Karlovitz On nonexpansive mappings // Proc. Amer. Math. Soc. V.35 –1975 p.321–325

5. S. Reich The fixed point property for nonexpansive mappings // Amer. Math. Monthly V. 87 –1980 p.292–294

6. P.K. Lin Unconditional bases and fixed points of nonexpansive mappings // Pacific J. of Math. V.116 –1985 p.69–76

7. N.L. Carothers, S.J. Delworth, C.J. Lennard and D.A. Trautman A fixed point property for the Lorentz space $L_{p,1}(\mu)$ // Indiana Univ.\ Math.\ J. V.40--1991 p.345--352

8. D. Van Dulst and B. Sims Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK) in: Banach Space theory and Its Applications info Lecture Notes in Math. 1991 , Springer,addr Berlin–1983 p.35–43

9. J. Garcia-Falset, A. Jiménez-Melado, and E.Lloréns-Fuster Measures of noncompactness and normal structure in Banach spaces // Studia Mathematica–1994 V.110 N.1 p.1–8

10. J. Diestel Geometry of Banach spacesinfo Lecture Notes in Mathematics, Springer-Verlag,addr Berlin-Heilselberg-New York– 1975

11. N. Dunford and J.T. Schwartz Linear Operators. Part I: General Theory , Interscience publishers–1958,addr New York, London

12. N.J. Akhiezer Lectures on the theory of approximation (Lekzii po teorii approximazii), GITTL– 1947,addr Moscow, Leningrad Russian

13. Dale E. Alspach A fixed point free nonexpansive map // Proc. of the Amer. Mathem. Soc. –1981 V.82 N.3 p.423–424

	On fixed points of one class of operators
Author	Yu. Korobeinik kor@ns.unird.ac.ru Rostov State University, Department of Mechanics and Mathematics,Rostov on Don, Zorge 5, 344104, Russia
Abstract	Let $T\colon Q\to Q$ be a selfmapping of some subset $Q$ of a Banach space $E$. The operator $T$ does not increase the distance between $Q$ and the origin (the null element $E$) if $$\forall x\in Q\quad\\|Tx\\|\le\\|x\\|$$ (such operator is called a NID operator). In this paper some sufficient conditions for the existence of a non-trivial fixed point of a NID operator are obtained. These conditions enable in turn to prove some new results on fixed points of nonexpanding operators.
Keywords	selfmapping, Banach space, distance non-increasing, non-trivial fixed point, nonexpanding operators
DOI	doi:10.30970/ms.7.2.187-192
Reference	1. . 10. F.E. Browder Nonexpansive nonlinear operators in Banach spaces // Proc. Nat. Acad. Sci. USA V.54 –1965 p.1041–1044 2. W.A. Kirk A fixed point theorem for mappings which do not increase distance // Amer. Math. Monthly –1965 V.72 p.1004–1006 3. J.P. Gossez and E. Lami Dozo Some geometrical properties related to the fixed point theory for nonexpanding mappings // Pacific J. Math. V.40 –1972 p.565–573 4. L.A. Karlovitz On nonexpansive mappings // Proc. Amer. Math. Soc. V.35 –1975 p.321–325 5. S. Reich The fixed point property for nonexpansive mappings // Amer. Math. Monthly V. 87 –1980 p.292–294 6. P.K. Lin Unconditional bases and fixed points of nonexpansive mappings // Pacific J. of Math. V.116 –1985 p.69–76 7. N.L. Carothers, S.J. Delworth, C.J. Lennard and D.A. Trautman A fixed point property for the Lorentz space $L_{p,1}(\mu)$ // Indiana Univ.\ Math.\ J. V.40--1991 p.345--352 8. D. Van Dulst and B. Sims Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK) in: Banach Space theory and Its Applications info Lecture Notes in Math. 1991 , Springer,addr Berlin–1983 p.35–43 9. J. Garcia-Falset, A. Jiménez-Melado, and E.Lloréns-Fuster Measures of noncompactness and normal structure in Banach spaces // Studia Mathematica–1994 V.110 N.1 p.1–8 10. J. Diestel Geometry of Banach spacesinfo Lecture Notes in Mathematics, Springer-Verlag,addr Berlin-Heilselberg-New York– 1975 11. N. Dunford and J.T. Schwartz Linear Operators. Part I: General Theory , Interscience publishers–1958,addr New York, London 12. N.J. Akhiezer Lectures on the theory of approximation (Lekzii po teorii approximazii), GITTL– 1947,addr Moscow, Leningrad Russian 13. Dale E. Alspach A fixed point free nonexpansive map // Proc. of the Amer. Mathem. Soc. –1981 V.82 N.3 p.423–424
Pages	187-192
Volume	7
Issue	2
Year	1997
Journal	Matematychni Studii
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Table of content of issue	html

On fixed points of one class of operators