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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 $$a\| y-\overline x\|= \inf\{\| z-\overline x\|: z\in C\}]$$.

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