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Generalized $I$-nonexpansive maps and best approximations in Banach spaces. (English) Zbl 1095.41017
Let $E=(E,\Vert \cdot\Vert )$ be a Banach space and $C$ a subset of $E$. Let $T,I:E\rightarrow E$. $T$ is called $I$-nonexpansive on $C$ if $\Vert Tx-Ty\Vert \leq \Vert Ix-Iy\Vert $ for all $x,y\in C$. The set of fixed points of $T$ (resp. $I$) is denoted by $F(T)$ (resp. $F(I)$). The set $P_C(\hat{x})=\{y\in C:\Vert y-\tilde{x}\Vert =\text{dist}(\hat{x},C)\}$ is called the set of best approximants to $\hat{x}\in X$ from $C$. In the case that $T$ and $I$ are commuting ($ITx=TIx$) on $P_C(\hat{x})$, {\it G. Jungck} and {\it S. Sessa} [Math. Jap. 42, No. 2, 249--252 (1995; Zbl 0834.54026)] proved that, under certain conditions, $P_C(\hat{x})\cap F(T)\cap F(I)\neq \emptyset$. Later, {\it N. Shahzad} [Tamkang J. Math. 32, No. 1, 51--53 (2001; Zbl 0978.41020)] extended this result to a class of noncommuting maps. In this paper it is proved the validity of this result for generalized $I$-nonexpansive maps.

41A50Best approximation, Chebyshev systems
47H10Fixed-point theorems for nonlinear operators on topological linear spaces