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An application of fixed point theorems in best approximation theory. (English) Zbl 0909.47044
The result stated below, extends earlier works. Let \(X\) be a Banach space. Let \(T,I:X\to X\) be operators and \(C\) a subset of \(X\) such that \(T:\partial C\to C\) and \(\overline x\in F(T)\cap F(I)\). Further, suppose that \(T\) and \(I\) satisfy \[ \| Tx- Ty\|\leq a\| Ix- Iy\|+ (1- a)\max\{\| Tx- Ix\|,\| Ty- Iy\|\}, \] for all \(x\), \(y\) in \(D_a'= D_a\cup \{\overline x\}\cup E\), where \(E= \{q\in X: Ix_n, Tx_n\to q,\{x_n\}\subset D_a\}\), \(0< a< 1\), \(I\) is linear, continuous on \(D_a\), and \(T\) and \(I\) are compatible in \(D_a\). If \(D_a\) is nonempty, compact, and convex and \(I(D_a)= D_a\), then \(D_a\cap RF(T)\cap F(I)\neq \emptyset\). \([F(T)(F(I))\) stands for fixed of \(T(I)\), and the set \(D_a\) of best \((C,a)\)-approximants to \(\overline x\) consists of the points \(y\) in \(C\) such that \(a\| y-\overline x\|= \inf\{\| z-\overline x\|: z\in C\}]\).

47H10 Fixed-point theorems
41A50 Best approximation, Chebyshev systems
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