Year: 2012
Analysis in Theory and Applications, Vol. 28 (2012), Iss. 1 : pp. 1–12
Abstract
For a subset $K$ of a metric space $(X,d)$ and $x ∈ X$,$$P_K(x)=\Bigg\{y\in K : d(x, y)= d(x, K) ≡\text{inf}\{d(x, k) : k \in K\}\Bigg\}$$ is called the set of best $K$-approximant to $x$. An element $g◦\in K$ is said to be a best simultaneous approximation of the pair $y_1, y_2 \in X$ if $$max\Bigg\{d(y_1, g◦), d(y_2, g◦)\Bigg\}= \inf\limits_{g\in K} max\Bigg\{d(y_1, g), d(y_2, g)\Bigg\}.$$ In this paper, some results on the existence of common fixed points for Banach operator pairs in the framework of convex metric spaces have been proved. For self mappings $T$ and $S$ on $K$, results are proved on both $T$- and $S$- invariant points for a set of best simultaneous approximation. Some results on best $K$-approximant are also deduced. The results proved generalize and extend some results of I. Beg and M. Abbas[1], S. Chandok and T. D. Narang[2], T. D. Narang and S. Chandok[11], S. A. Sahab, M. S. Khan and S. Sessa[14], P. Vijayaraju[20] and P. Vijayaraju and M. Marudai[21].
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.2012.v28.n1.1
Analysis in Theory and Applications, Vol. 28 (2012), Iss. 1 : pp. 1–12
Published online: 2012-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 12
Keywords: Banach operator pair best approximation demicompact fixed point star-shaped nonexpansive asymptotically nonexpansive and uniformly asymptotically regular maps.
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Best approximation and fixed points for rational-type contraction mappings
Chandok, Sumit
Journal of Applied Analysis, Vol. 25 (2019), Iss. 2 P.205
https://doi.org/10.1515/jaa-2019-0021 [Citations: 0]