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The Farthest Orthogonality, Best Proximity Points and Remotest Points in Banach Spaces

  • R. Rahmani JafarbeigiEmail author
  • H. Mazaheri
Research Paper
  • 12 Downloads
Part of the following topical collections:
  1. Mathematics

Abstract

In this paper, the concept of farthest orthogonality, distance orthogonality and \(*\)-farthest orthogonality in Banach spaces is introduced and the relation between these concepts with the dual space is found. Also, the weakly \(\phi \)-contraction and farthest continuous maps and their relationship are studied. Then, some best proximity and farthest point theorems are proved in Banach spaces. Some examples are given to illustrate the results.

Keywords

Farthest points Farthest orthogonality *-Farthest orthogonality Weakly \(\phi \)-non-orthogonality Farthest continues map 

Mathematics Subject Classification

41A65 41A52 46N10 

References

  1. Alonso J, Benítez C (1989) Orthogonality in normed linear spaces: a survey. II. Relations between main orthogonalities. Extracta Math 4:121–131MathSciNetGoogle Scholar
  2. Alonso J, Martini H, Wu S (2012) On Birkhoff orthogonality and isosceles Orthogonality in normed linear spaces. Aequationes Math 83:153–189MathSciNetCrossRefGoogle Scholar
  3. AL-Thagafi MA, Shahzad N (2009) Convergence and existence result for best proximity points. Nonlinear Anal Theory 70:3665–3671MathSciNetCrossRefGoogle Scholar
  4. Birkhoff G (1935) Orthogonality in linear metric spaces. Duke Math J 1:169–172MathSciNetCrossRefGoogle Scholar
  5. Chmieliński J, Wójcik P (2010) Isosceles orthogonality preserving property and its stability. Nonlinear Anal 72:1445–1453MathSciNetCrossRefGoogle Scholar
  6. Elumalai S, Vijayaragavan R (2006) Farthest points in normed linear spaces. Gen Math 14:9–22MathSciNetzbMATHGoogle Scholar
  7. Franchetti C, Furi M (1972) Some characteristic properties of real Hilbert spaces. Rev Roumaine Math Pures Appl 17:1045–1048MathSciNetzbMATHGoogle Scholar
  8. Franchetti C, Singer I (1979) Deviation and farthest points in normed linear spaces. Rev Roum Math Pures et appl 24:373–381MathSciNetzbMATHGoogle Scholar
  9. Hadzic O (1992) A theorem on best approximations and applications. Zbornik Radova. Prirodno-Matematichkog Fakulteta. Serija za Matematiku 22:47–55MathSciNetzbMATHGoogle Scholar
  10. James RC (1945) Orthogonality in normed linear spaces. Duke Math J 12:291–302MathSciNetCrossRefGoogle Scholar
  11. Jessen B (1940) Two Theorems on Convex Point Sets (in Danish). M t Tidsskr 13:66–70Google Scholar
  12. Kadelburg Z, Radenoić S (2018) Notes on some recent papers concerning $F$-contraction in $b$-metric spaces. Constr Math Anal 1:108–112Google Scholar
  13. Karapinar E (2018) short Survay on the recent fixed point results on $b$-metric spaces. Constr Math Anal 1:15–44Google Scholar
  14. Khalil R, Al-Sharif Sh (2006) Remotal sets in vector valued function spaces. Scientiae Mathematicae Japonicae 3:433–442MathSciNetzbMATHGoogle Scholar
  15. Khalil R, Alkhawalda A (2012) New types of orthogonalities in Banach Spaces. AAMA 7:419–430Google Scholar
  16. Kirk WA, Srinivasan PS, Veeramani P (2003) Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theor Appl 4:79–89MathSciNetzbMATHGoogle Scholar
  17. Machado HV (1973) A characterization of convex subsets of normed spaces. Kodai Math Sem Rep 25:307–320MathSciNetCrossRefGoogle Scholar
  18. Mazaheri H, Narang TD, Khademzadeh HR (2015) Nearest and farthest points in normed spaces. Yazd University Press, YazdGoogle Scholar
  19. Mojškerc B, Turnšek A (2010) Mapping approximately preserving orthogonality in normed spaces. Nonlinear Anal 73:3821–3831MathSciNetCrossRefGoogle Scholar
  20. Roberts BD (1934) On the geometry of abstract vector spaces. Tohoku Math J 39:42–59zbMATHGoogle Scholar
  21. Saidi FB (2002a) Characterization of orthogonality in certain Banach spaces. Bull Austral Math Soc 65:93–104MathSciNetCrossRefGoogle Scholar
  22. Saidi FB (2002b) An extension of the notion of orthogonality to Banach spaces. J Math Anal Appl 267:29–47MathSciNetCrossRefGoogle Scholar
  23. Sangeeta, Narang TD (2005) A note on farthest points in metric spaces. Aligarh Bull Math 24:81–85Google Scholar
  24. Sangeeta, Narang TD (2014) On the farthest points in convex metric spaces and linear metric spaces. Publications de l Institut Mathematique 95:229–238MathSciNetCrossRefGoogle Scholar
  25. Singer I (1970) Best approximation in normed linear spaces by elements of linear subspaces. Springer, New YorkCrossRefGoogle Scholar

Copyright information

© Shiraz University 2020

Authors and Affiliations

  1. 1.Faculty of MathematicsYazd UniversityYazdIran

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