Volume 28, Issue 1
Common Fixed Points with Applications to Best Simultaneous Approximations

Anal. Theory Appl., 28 (2012), pp. 1-12.

Published online: 2012-03

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• 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].

• Keywords

Banach operator pair, best approximation, demicompact, fixed point, star-shaped, nonexpansive, asymptotically nonexpansive and uniformly asymptotically regular maps.

41A50, 41A60, 41A65, 47H10, 54H25

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@Article{ATA-28-1, author = {}, title = {Common Fixed Points with Applications to Best Simultaneous Approximations}, journal = {Analysis in Theory and Applications}, year = {2012}, volume = {28}, number = {1}, pages = {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].

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2012.v28.n1.1}, url = {http://global-sci.org/intro/article_detail/ata/4535.html} }
TY - JOUR T1 - Common Fixed Points with Applications to Best Simultaneous Approximations JO - Analysis in Theory and Applications VL - 1 SP - 1 EP - 12 PY - 2012 DA - 2012/03 SN - 28 DO - http://doi.org/10.4208/ata.2012.v28.n1.1 UR - https://global-sci.org/intro/article_detail/ata/4535.html KW - Banach operator pair, best approximation, demicompact, fixed point, star-shaped, nonexpansive, asymptotically nonexpansive and uniformly asymptotically regular maps. AB -

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].

Sumit Chandok & T. D. Narang. (1970). Common Fixed Points with Applications to Best Simultaneous Approximations. Analysis in Theory and Applications. 28 (1). 1-12. doi:10.4208/ata.2012.v28.n1.1
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