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Weak convergence of orbits of nonlinear operators in reflexive Banach spaces. (English) Zbl 1071.47052
Consider a proper convex function f:X(-,+] on a reflexive Banach space X and a closed convex subset K of the interior of the domain D={xX;f(x)<+}. An operator T:KK is called relatively nonexpansive with respect to the function f if there is zK such that D f (z,Tx)D f (z,x) for all xK, where D f (y,x)=f(y)-f(x)+f o (x,x-y), f o (x,y-x)=lim t0+ 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 xK, the orbits {T k x} k=1 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:KK, 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.
47H09Mappings defined by “shrinking” properties
47H30Particular nonlinear operators
54E35Metric spaces, metrizability
54E52Baire category, Baire spaces
65K99Numerical methods for mathematical programming and optimization