Weak and strong convergence for quasi-nonexpansive mappings in Banach spaces (Q2902039)

Let \(E\) be a real uniformly convex Banach space, and let \(C\) be a nonempty closed convex subset of \(E\) which is also a nonexpansive retract of \(E\).NEWLINENEWLINE A mapping \(T: C\to E\) is called quasi-nonexpansive if the set \(F(T)\) of fixed points of \(T\) is nonempty and \(\| Tx-y\|\leq\| x-y\|\) for all \(x\in C\) and \(y\in F(T)\).NEWLINENEWLINE A mapping \(T: C\to E\) with \(F(T)\neq\emptyset\) is said to satisfy Condition A if there exists a nondecreasing function \(f: [0,\infty)\to[0,\infty)\) with \(f(0)= 0\) and \(f(r)> 0\) for all \(r\in(0,\infty)\) such that \(\| x-Tx\|\geq f(\text{inf}_{z\in F(T)}\| x-z\|)\) for all \(x\in C\).NEWLINENEWLINE In the present paper, the author proves, among others, the following:NEWLINENEWLINE Theorem 3. Let \(T: C\to E\) be a quasi-nonexpansive mapping and satisfy Condition A. If for any \(x_1\in C\), NEWLINE\[NEWLINEx_{n+1}= P((1- \alpha_n) x_n+\alpha_n TP[\beta_n Tx_n+(1- \beta_n) x_n]),NEWLINE\]NEWLINE where \(0< a\leq\alpha_n\), \(\beta_n\leq b< 1\) for all \(n\geq 1\), and \(P:E\to C\) is a nonexpansive retraction, then \(\{x_n\}\) converges strongly to a fixed point of \(T\).NEWLINENEWLINE Strong convergence of the Mann iteration process with errors is discussed as well.