Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]