International Journal of Nonlinear Analysis and Applications
Existence of farthest points in Hilbert spaces
Document Type : Research Paper
Authors
1 Faculty of Mathematics, Yazd University, Yazd, Iran
2 Faculty of Mathematics , Yazd University , Yazd , Iran
Abstract
Let $H$ be a real Hilbert space with the inner product $<.,.>$ and the norm $\|\cdot\|$. In this paper, we introduce hybrid algorithms for generating cyclic, non-expansive mapping of H. Also, we discuss about necessary and sufficient conditions on subsets of Hilbert space to be remotal or uniquely remotal. Moreover, we give the basic concepts and theorems of farthest points of Bounded subsets of H. In the end, we will provide examples to illustrate our results.
Keywords
[2] P. Amiri, Sh. Rezapour and N. Shahzad, Fixed points of generalized α−ψ− contractions, Springer-Verlag Italia. 108 (2013) 519–526.
[3] A. Anthony Eldreed and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006) 1001–1006.
[4] M. A. AL-Thagafi and N. Shahzad. Convergence and existence result for best proximity points, Nonlinear. Anal. Theory. 70 (2009) 3665–3671.
[5] M. Baronti and P. L. Papini, Remotal sets revisted, Taiwanese J. Math. 5 (2001) 367–373.
[6] D. W. Boyd and S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969) 458–464.
[7] S . Elumalai and R . Vijayaragavan, Farthest points in normed linear spaces, General Math. 14 (2006) 9–22.
[8] C . Franchetti and I. Singer, Deviation and farthest points in normed linear spaces, Rev. Roum Math. Pures Appl. 24 (1979) 373–381 .
[9] G. Jacob, M. Postolache, M. Marudai and V. Raja, Norm convergence iterations for best proximity points of non-self non-expansive mappings, U.P.B. Sci. Bull, 79 (2017) 1223–7027.
[10] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer-Verlag, New York-Berlin, 1970.
[11] B. Jessen, Two theorems on convex point sets (in Danish), M t. Tidsskr. 13 (1940) 66–70.
[12] E. Karapinar, Best proximity points of cyclic mapping, Appl. Math. Lett. 25 (2012) 1761–1766.
[13] W. A. Kirk, S. Reich and P. Veeramani, Proximinal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim. 24 (2003) 851–862.
[14] C. Martinez-Yanes and H.K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006) 2400–2411.
[15] W. Nilsrakoo and S. Saejung, Weak and strong convergence theorems for countable Lipschitzian mappings and its applications, Nonlinear Anal. 69 (2008) 2695–2708.
[16] W. A. Kirk, P. S. Srinivasan and P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory. 4 (2003) 79–89.
[17] Reny George1, Cihangir Alaca and K. P. Reshma, Best proximity points in b-metric space, J. Nonlinear Anal. Appl. 1 (2015) 45–56.
[18] B. S. Thakur and A. Sharma, Existence and convergence of best proximity points for semi-cyclic contraction pairs, Int. J. Anal. Appl. 5 (2014) 33–44.
[19] M. R. Yadav, B. S. Thakur and A. K. Sharma, Best proximity points for generalized proximity contraction in compelete metric spaces, Adv. Fixed Point Theory, 3 (2013) 392–405.
[20] R. B. Holmes, A Course on Optimization and Best Approximation, lecture Notes 257, Springer-Verlag, NewYork, 1972.
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Volume 12, Issue 2
November 2021Pages 437-446
- Receive Date: 27 February 2019
- Revise Date: 12 November 2019
- Accept Date: 22 November 2019