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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.
MSC:
47H09Mappings defined by “shrinking” properties
47H30Particular nonlinear operators
54E35Metric spaces, metrizability
54E52Baire category, Baire spaces
65K99Numerical methods for mathematical programming and optimization