×

A result on best approximation. (English) Zbl 0914.41013

Tamkang J. Math. 29, No. 3, 223-226 (1998); correction ibid. 30, No. 2, 165 (1999).
Let \(E\) be a normed linear space. A subset \(C\) of \(E\) is said to be starshaped with respect to a point \(p\in C\) if, for each \(x\in C\), the segment joining \(x\) to \(p\) is contained in \(C\). A mapping \(T:E\to E\) is said to be nonexpansive on \(C\) if \(\| Tx-Ty\|\leq\| x-y\|\) for all \(x,y\in C\). An element \(y\in C\) is called an element of best approximation of \(\widehat{x}\in E\) if \(\|\widehat{x}-y\|= \inf\{\| \widehat{x}-z\|:z\in C\}\). Self mappings \(T\) and \(I\) of \(E\) are called \(R\)-weakly commuting on \(E\) if there exists some positive real number \(R\) such that \[ \| TIx-ITx\|\leq R\| Tx-Ix\| \] for each \(x\in E\). Suppose \(F(T)\) denotes the set of fixed points of \(T\), \(P_c(\widehat{x})\) denotes the set of best approximations to \(\widehat{x}\) in \(C\) and \(\partial C\) denotes the boundary of \(C\). Using a common fixed point theorem for noncommuting mappings of R. P. Pant [J. Math. Anal. Appl. 188, No. 2, 436-440 (1994; Zbl 0830.54031)], the following result on best approximation, which improves and extends a result of S. A. Sahab, M. S. Khan and S. Sessa [J. Approximation Theory 55, No. 3, 349-351 (1988; Zbl 0676.41031)] has been proved in this paper:
Theorem. Let \(T,I:E\to E\) be operators, \(C\) a subset of \(E\) such that \(T:\partial C\to C\), and \(\widehat{x}\in F(T)\cap F(I)\). Further \(T\) and \(I\) satisfy \(\| Tx-Ty\|\leq\| Ix-Iy\|\) on \(P_c(\widehat{x})\cup \{\widehat{x})\) and let \(I\) be linear on \(P_c(\widehat{x})\) and \(T,I\) be \(R\)-weakly commuting on \(P_c(\widehat{x})\).If \(P_c(\widehat{x})\) is non-empty, compact and starshaped with respect to \(p\in F(I)\), if \(I(P_c(\widehat{x}))= P_c(\widehat{x})\), and if either \(T\) or \(I\) is continuous, then \(P_c(\widehat{x})\cap F(T)\cap F(I)\neq \emptyset\). Remark: “This paper contains two errors. The first one occurs in the statement of Theorem 5 due to Pant and the second one occurs in the argument on page 225 (Theorem 6) wherein the author concludes that \(T_n\) and \(I\) are \(R\)-weakly commuting on \(Pc(\widehat x)\) for each \(n\), which is not true. Both these errors have been corrected by the author and the corrections will appear in a forthcoming issue of the same journal”.

MSC:

41A50 Best approximation, Chebyshev systems
PDFBibTeX XMLCite