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Weak convergence of orbits of nonlinear operators in reflexive Banach spaces. (English) Zbl 1071.47052
Consider a proper convex function \(f:X \to (-\infty,+\infty]\) 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 \(D_f(z,Tx) \leq D_f(z,x)\) 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 \(X\).

MSC:
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
54E35 Metric spaces, metrizability
54E52 Baire category, Baire spaces
65K99 Numerical methods for mathematical programming, optimization and variational techniques
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