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Generalized projection algorithms for nonlinear operators. (English) Zbl 1132.47048
Let $E$ be a reflexive and strictly convex and smooth Banach space, $C$ a closed convex subset of $E$, $T$ a selfmap of $C$ with $F(T)$ denoting the fixed point set of $T$. A point $p$ in $C$ is said to be an asymptotic fixed point of $T$ if $C$ contains a sequence $\{x_n\}$ that converges weakly to $p$ and is such that the strong limit of $(Tx_n - x_n)$ is 0. The set of asymptotic fixed points is denoted by $\widehat{F}(T)$. $T$ is called relatively nonexpansive if $F(T) = \widehat{F}(T)$ and $\varphi(p, Tx) \leq \varphi(p, x)$ for each $p \in F(T)$ and $x \in C$, where $\varphi(x, y) := \Vert x\Vert ^2 - 2\langle x, j(y)\rangle + \Vert y\Vert ^2$. $T$ is called relatively asymptotically nonexpansive if $F(T) = \widehat{F}(T)$ and $\varphi(x, T^nx) \leq k^n\varphi(p, x)$ for each $x \in C$, $p\in F(T)$. With $E$ a uniformly convex and uniformly smooth Banach space, and using a complicated iteration scheme involving duality maps and their inverses, the authors obtain the strong convergence to a fixed point of $T$ for $T$ either strictly relatively nonexpansive or relatively asymptotically nonexpansive, provided that $F(T) \neq \emptyset$.

47J25Iterative procedures (nonlinear operator equations)
47H05Monotone operators (with respect to duality) and generalizations
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
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