Approximating fixed points of non-self asymptotically nonexpansive mappings in Banach spaces
Summary: Suppose that \(K\) is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space \(E\) with \(P\) as a nonexpansive retraction. Let \(T:K\to E\) be an asymptotically nonexpansive mapping with \(\{k_n\}\subset [1,\infty)\) such that \(\sum^\infty_{n=1}(k_n-1)<\infty\) and \(F(T)\) is nonempty, where \(F(T)\) denotes the fixed points set of \(T\). Let \(\{\alpha_n\},\{\alpha_n'\}\), and \(\{\alpha_n''\}\) be real sequences in (0,1) and \(\varepsilon\leq\alpha_n,\alpha_n',\alpha_n''\leq 1-\varepsilon\) for all \(n\in\mathbb N\) and some \(\varepsilon>0\). Starting from arbitrary \(x_1\in K\), define the sequence \(\{x_n\}\) by \(x_1\in K\), \(z_n=P(\alpha_n''T(PT)^{n-1}x_n+(1-\alpha_n'')x_n)\),\ \(y_n=P(\alpha_n'T(PT)_{n-1}z_n+(1-\alpha_n')x_n)\), \(x_{n+1}=P(\alpha_nT(PT)_{n-1}y_n+(1-\alpha_n)x_n)\). (i) If the dual \(E^*\) of \(E\) has the Kadec-Klee property, then \(\{x_n\}\) converges weakly to a fixed point \(p\in F(T)\); (ii) if \(T\) satisfies condition (A), then \(\{x_n\}\) converges strongly to a fixed point \(p\in F(T)\).
scientific article; zbMATH DE number 5312409