Weak convergence theorems for asymptotically nonexpansive mappings and semigroups
The authors consider weak convergence in mean ergodic theorems for asymptotically nonexpansive semigroups and the weak convergence of the modified Mann-Ishikawa iteration processes for asymptotically nonexpansive mappings. Their results are new even for nonexpansive mappings and semigroups. The paper consists of nine sections. The first one is an introduction. Section 2 contains some preliminary results. In Section 3 the main Lemma is formulated: Let \(X\) be a uniformly convex Banach space whose dual \(X^*\) has the KK-property (Kadec-Klee). If \(\{x_n\}\) is a bounded sequence such that \(\lim_{n\to\infty} \|\alpha x_n+ (1-\alpha)f_1- f_2\|\) exists for all \(\alpha\in [0,1]\) and \(f_1,f_2\in \omega_w(\{x_n\})\), then \(\omega_w\{x_n\}\) is a singleton. Section 4 deals with asymptotically nonexpansive semigroups. The main result here is: Let \(X\) and \(X^*\) be as in the main Lemma and let \(C\subset X\) be bounded closed convex set. If \({\mathcal F}\) is an asymptotically nonexpansive semigroup acting on \(C\) and \(u\) is an almost orbit of \({\mathcal F}\), then the following conditions are equivalent: (1) \(\omega_w(u)\subset F({\mathcal F})\); (2) \(\text{w-}\lim_{t\to\infty} u(t)= x\in F({\mathcal F})\); (3) \(\text{w-}\lim_{t\to\infty}(u(t)- u(t+ s))= 0\) for each \(s\in [0,\infty)\). In Section 5 the authors consider the weak convergence of almost orbits of asymptotically nonexpansive mappings. The basic theorem here is: If \(T:C\to C\) is asymptotically nonexpansive in the intermediate sense and \(\{x_n\}\) is an almost orbit of \(T\), then \(\{x_n\}\) is weakly convergent to a fixed point of \(T\) iff \(w\)-\(\lim_{n\to\infty}(x_n- x_{n+1})= 0\). Section 6 is devoted to the investigations on modified Mann-Ishikawa iteration processes defined by \[ x_{i+1}= \alpha_i T^{n_i} y_i+ (1-\alpha_i)x_i\quad\text{and}\quad y_i= \beta_i T^{n_i}x_i+ (1- \beta_i) x_i, \] where \(\{\alpha_i\}\) and \(\{\beta_i\}\) are chosen so that \(\alpha_i\in [a,b]\) and \(\beta_i\in [0,b]\text{ ord}_i\in [a,1]\) and \(\beta_i\in [a,b]\) for some \(a\), \(b\) with \(0< a\leq b< 1\), and \(\{n_i\}\) is a sequence of nonnegative integers. In Section 7 unrestricted iterations of nonexpansive mappings are considered, while in Section 8 other applications of the main Lemma are given. Finally in Section 9, using a recent result due to Kirk and Sim, the authors generalize a theorem of Reich-Nevanlinna. Let \(X\) be a reflexive locally uniformly convex Banach space and let \(C\) \(X\) be a closed set. If \(S\) is a nonexpansive semigroup on \(C\) with \(\text{int}(F(S))\neq\emptyset\), then for each almost orbit \(u\), \(u(t)\) converges in norm to a point of \(F(S)\) as \(t\to\infty\).
modified Mann-Ishikawa iteration processes
scientific article; zbMATH DE number 1556814
scientific article