New results for the best proximity pair in cone Riesz space

Document Type : Research Paper

Authors

1 Computer Geometry and Dynamical Systems Laboratory, Yazd University, Yazd, Iran

2 Faculty of Mathematics, Yazd University, Yazd, Iran

3 Department of Mathematics, Technical and Vocational University (TVU), Tehran, Iran

Abstract

In this paper, the best proximity pair problem is considered with a cone metric.  the conditions for the existence and uniqueness of the best proximity pair problem is discussed by using interesting relationships in Riesz spaces. This problem is studied for $T$-absolutely direct sets. Also, given the conditions considered for this problem, it is shown for the cone cyclic contraction maps, the best proximity pair problem is uniquely solvable.

Keywords

[1] D. Aliprantis and O. Burkinshaw, Positive operators, Springer, Dordrecht, 2006.
[2] J. Anuradha and P. Veeramani, Proximal pointwise contraction, Topology Appl. 156 (2009), no. 18, 2942–2948.
[3] G. Birkhoff, Lattice theory, American Mathematical Society, Providence, R.I., 1979.
[4] Shu Tao Chen, Xin He, and H. Hudzik, Monotonicity and best approximation in Banach lattices, Acta Math. Sin.
(Engl. Ser.) 25 (2009), no. 5, 785–794.
[5] A. A. Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323
(2006), no. 2, 1001–1006.
[6] P. Foralewski, H. Hudzik, W. Kowalewski, and M. Wis l a, Monotonicity properties of Banach lattices and their
applications: A survey, Ordered structures and applications, Trends Math., Birkh¨auser/Springer, Cham, 2016,
pp. 203–232.
[7] H. Hudzik and W. Kurc, Monotonicity properties of Musielak-Orlicz spaces and dominated best approximation in
Banach lattices, J. Approx. Theory 95 (1998), no. 3, 353–368.
[8] A. F. Kalton and J. Nigel, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer,
New York, 2006.
[9] H. R. Khademzadeh and H. Mazaheri, Monotonicity and the dominated farthest points problem in Banach lattice,
Abstr. Appl. Anal. (2014), Art. ID 616989, 7.[10] N. S. Kukushkin, Increasing selections from increasing multifunctions, Order 30 (2013), no. 2, 541–555.
[11] W. Kurc, Strictly and uniformly monotone Musielak-Orlicz spaces and applications to best approximation, J.
Approx. Theory 69 (1992), no. 2, 173–187.
[12] P. Meyer-Nieberg, Banach lattices., Springer-Verlag, Berlin, 1991.
[13] V. Sankar Raj and P. Veeramani, Best proximity pair theorems for relatively nonexpansive mappings, Appl. Gen.
Topol. 10 (2009), no. 1, 21–28.
[14] A. C. Zaanen, Introduction to operator theory in Riesz spaces, Springer-Verlag, Berlin, 1997.
Volume 13, Issue 2
July 2022
Pages 2037-2041
  • Receive Date: 03 April 2019
  • Revise Date: 28 April 2020
  • Accept Date: 08 May 2020