Common Fixed Points with Applications to Best Simultaneous Approximations (Analysis in Theory and Applications, 2012)
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.