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Fixed point properties for Lorentz sequence spaces \(\ell _{\rho ,\infty }^0\) and \(\ell _{\rho ,1}\)

  • Veysel NezirEmail author
Article
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Abstract

In this paper, we work on Lorentz sequence spaces and explore the fixed point property for \(\ell _{\rho ,\infty }^0\) and \(\ell _{\rho ,1}\) space. We prove that both spaces enjoy the weak fixed point property for nonexpansive mappings. To prove the weak fixed point property for \(\ell _{\rho ,1}\), we provide alternative methods such that one gives exact value of its Riesz angle. We show that both spaces fail the fixed point property for nonexpansive mappings since \(\ell _{\rho ,1}\) contains an asymptotically isometric copy of \(\ell ^1\) and \(\ell _{\rho ,\infty }^0\) contains an asymptotically isometric copy of \(c_0\). We also show that there exists a non-empty, closed, bounded and convex subset C of \(\ell _{\rho ,1}\) and a fixed point-free affine, nonexpansive mapping \(T:C\longrightarrow C\) and so \(\ell _{\rho ,1}\) fails to have the fixed point property for affine nonexpansive mappings. Contrary to this, in the final section, we get a Goebel and Kuczumow analogy for \(\ell _{\rho ,1}\) by proving that there exists a large class of non-weak* compact, closed, bounded and convex subsets of \(\ell _{\rho ,1}\) with the fixed point property for affine nonexpansive mappings.

Keywords

Nonexpansive mapping Banach lattice fixed point property weak fixed point property Lorentz spaces 

Mathematics Subject Classification

Primary 46B45 47H09 Secondary 46B42 46B10 

Notes

Acknowledgements

The author is grateful to Chris Lennard for his valuable comments and helpful discussions on the subject. We also would like to thank the anonymous reviewers for their helpful and constructive comments that greatly contributed to improving the final version of the paper.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and LettersKafkas UniversityKarsTurkey

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