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Zbl 1071.47052
Butnariu, Dan; Reich, Simeon; Zaslavski, Alexander J.
Weak convergence of orbits of nonlinear operators in reflexive Banach spaces.
(English)
[J] Numer. Funct. Anal. Optimization 24, No. 5-6, 489-508 (2003). ISSN 0163-0563; ISSN 1532-2467/e

Consider a proper convex function $f:X \to (-\infty,+\infty]$ on a reflexive Banach space $X$ and a closed convex subset $K$ of the interior of the domain $D=\{x \in X;\ f(x) < +\infty \}$. An operator $T:K \to K$ is called relatively nonexpansive with respect to the function $f$ if there is $z \in K$ such that $D_f(z,Tx) \le D_f(z,x)$ for all $x \in K$, where $D_f(y,x)=f(y)-f(x)+f^o(x,x-y)$, $f^o(x,y-x)=\lim_{t\to 0+}t^{-1}[f(ty+(1-t)x)-f(x)]$. In this case, $z$ is a fixed point of $T$. A basic question discussed is whether for any $x \in K$, the orbits $\{T^k x\}_{k=1}^\infty$ converge weakly to a fixed point. It is shown that this is in a certain sense a generic property for large classes of operators $T:K \to K$, which are relatively nonexpansive with respect to a function $f$. The function $f$ is supposed to be strictly convex on $K$ and such that the convergence structure induced on $K$ by the function $Df$ is stronger than that induced by the norm of $X$.
[Milan Kučera (Praha)]
MSC 2000:
*47H09 Mappings defined by "shrinking" properties
47H30 Particular nonlinear operators
54E35 Metric spaces, metrizability
54E52 Baire category, Baire spaces
65K99 Numerical methods for mathematical programming and optimization

Keywords: Bergman distance; fixed point; nonexpansiveness with respect to a convex function; orbit; reflexive Banach space; total convexity; uniform convexity; weak convergence

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