EXISTENCE OF FIXED POINT AND BEST PROXIMITY
POINT OF $p$-CYCLIC ORBITAL CONTRACTION
OF BOYD-WONG TYPE
P.K. Hemalatha1, T. Gunasekar2, S. Karpagam3
1,2Department of Mathematics
Vel Tech Rangarajan Dr. Sangunthala R and D Institute
of Science and Technology
Chennai - 600054, INDIA
3Department of Science and Humanities
Saveetha School of Engineering
Saveetha Institute of Medical and Technical Sciences
Chennai - 602105, INDIA

Abstract


Abstract. Let $A_1, A_2,...,A_p$ ( $p\in \mathbb{N}$) be non empty subsets of a metric space $(X,d)$. In this paper, a map $T:\cup _{i=1}^p A_i \rightarrow \cup_{i=1}^ p A_i $, called $p$-cyclic orbital contraction of Boyd-Wong type is introduced. Convergence of a unique fixed point and a best proximity point for this map are obtained in a uniformly convex Banach space settings. Moreover, the obtained best proximity point is the unique periodic point of the map.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 31
Issue: 6
Year: 2018

DOI: 10.12732/ijam.v31i6.9

Download Section



Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

  1. [1] D.W. Boyd and J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20 (1969), 458-464.
  2. [2] W.A. Kirk, P.S. Srinivasan and P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4, No 1 (2003), 79-89.
  3. [3] A.A. Eldred, Existence and Convergence of Best Proximity Points, Ph. D Thesis, 2006, Indian Institute of Technology Madras, Chennai, India.
  4. [4] S. Karpagam, On Fixed Point Theorems and Best Proximity Point Theorems , Ph. D Thesis, 2009, The Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India.
  5. [5] S. Karpagam and S. Agrawal, Best proximity point theorems for cyclic orbital Meir-Keeler contraction maps, Nonlinear Anal., 74 (2011), 1040- 1046.
  6. [6] S. Karpagam and S. Agrawal, Best proximity point for p-cyclic Meir-Keeler contractions, Fixed Point Theory Appl., 2009 (2009), Article ID 197308, 9 pp.
  7. [7] S. Karpagam and S. Agrawal, Existence of best proximity points for pcyclic contractions, Fixed Point Theory, 13 (2012), 99-105.
  8. [8] M. Petric, B. Zlatanov, Best proximity points and Fixed points for psumming maps, Fixed Point Theory Appl., 2012 (2012), Art. 86.
  9. [9] B. Zlatanov, Best proximity points for p-summing cyclic orbital Meir- Keeler contractions, Nonlinear Anal. Model. Control., 20, No 4 (2015), 528-544.
  10. [10] S. Karpagam and B. Zlatanov, Best proximity points of p-cyclic orbital Meir-Keeler contraction maps, Nonlinear Anal. Model. Control, 21, No 6 (2016), 790-806.
  11. [11] S. Karpagam and B. Zlatanov, A note on p-summing orbital Meir-Keeler contraction maps, Int. J. Pure and Applied Math., 107, No 1 (2016), 225-243.