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Fixed points for mean non-expansive mappings. (English) Zbl 1128.47053
Summary: For a Banach space $X$, {\it J. García--Falset} [Houston J. Math. 20, No. 3, 495--506 (1994; Zbl 0816.47062)] introduced the coefficient $R(X)$ and showed that if $R(X) < 2$, then $X$ has a fixed point. In the present paper, we define a mean nonexpansive mapping $T$ on $X$ in the sense that $\|Tx - Ty\|\leq a\|x - y\| + b\|x - Ty\|$ for any $x, y \in X$, where $a, b \geq 0$, $a + b \leq 1$. We show that if $R{\left(X \right)}<\frac{2}{1 + b}$, then $T$ has a fixed point in $X$.
MSC:
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
46B20Geometry and structure of normed linear spaces
47H05Monotone operators (with respect to duality) and generalizations
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References:
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