Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces. (English) Zbl 0935.47039

The authors consider a closed convex subset \(C\) of a Hilbert space \(H\), a given semigroup \(S\) and an asymptotically nonexpansive semigroup \(\mathcal S = \{T_t\); \(t \in S \}\) on \(C\) (i.e. for any \(t \in S\), \(T_t\) is a mapping from \(C\) into itself, \(\|T_tx-T_ty\|\leq k_t \|x-y\|\), \(\sup k_t < \infty\), \(T_{ts}x = T_tT_sx\)). It is supposed that \(F(\mathcal S) \neq \emptyset\) where \(F(\mathcal S)= \cap_{t\in S} \{x\in C\); \(T_tx=x\}\) – the set of common fixed points of \(\mathcal S\). For any \(x \in C\) and \(x, y_0 \in C\), iteration processes defined by \[ x_n = a_n x + (1-a_n)T_{\mu_n}x_n, \quad \text{and}\quad y_{n+1} = b_n x + (1-b_n)T_{\mu_n}y_n \] are studied. Here \(0<a_n\leq 1\), \(\lim a_n =0\), \(0\leq b_n\leq 1\), \(\lim b_n =0\), \(\sum b_n = \infty\), \(\mu_n\) is a sequence of means on a certain subspace \(X\) of \(B(S)\) containing 1 (i.e. \(\mu_n \in X^*\), \(\|\mu_n \|= \mu_n(1)=1\)), \(T_{\mu}x\) is the element of \(C\) for which \(\mu (f_{x,y}) = (T_{\mu}x,y)\) for all \(y \in H\), \(f_{x,y} \in X\), \(f_{x,y}(t)=(T_tx,y)\). It is proved that under some assumptions, for any \(x \in C\) or \(x\), \(y_0 \in C\), the iteration processes mentioned converge strongly to the element of \(F(\mathcal S)\) which is the nearest to \(x\) in \(F(\mathcal S)\). By a special choice of the means, the authors prove extensions of the results of Shimizu and Takahashi concerning the convergence of some iteration processes to the fixed points of asymptotically nonexpansive mappings in a closed convex subset of a Hilbert space.
Reviewer: M.Kučera (Praha)


47H20 Semigroups of nonlinear operators
47H10 Fixed-point theorems
65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
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