Approximating fixed points of infinite nonexpansive mappings (Q2782938)

Let \(E\) be a Banach space and let \(I\) be the identity operator in \(E\). Let \(C\) be a nonempty set of \(E\). Then a mapping \(T:C\to C\) is said to be nonexpansive if \(\|Tx-Ty\|\leq\|x-y\|\) for every \(x,y\in C\). Firstly, in this paper the authors introduce an iteration scheme given by infinite nonexpansive mappings which generalizes Das and Debata's scheme from [\textit{G. Das} and \textit{J. P. Debata}, Indian J. Pure Appl. Math. 17, 1263-1269 (1986; Zbl 0605.47054)] in the following way: Let \(C\) be a nonempty closed convex subset of a Banach space \(E\). Let \(T_1,T_2,\dots\) be mappings of \(C\) into itself and \(a_1,a_2,\dots\) be real numbers such that \(0\leq a_i\leq 1\) for \(i=1,2, \dots\). Then for any \(n\in\mathbb{N}\) \(U_{n,k}= a_kT_kU_{n,k+1} +(1-a_k)I\), where \(U_{n,n+1}=I\) and \(W_n=U_{n,1}= a_1T_1U_{n,2}+ (1-a_1)I\). Such a mapping \(W_n\) is called the \(W\)-mapping generated by \(T_n,T_{n-1}\), \(T_1\) and \(a_n,a_{n-1}, \dots,a_1\).NEWLINENEWLINENEWLINEUsing this scheme, the authors prove the following weak convergence theorem: Let \(E\) be a Hilbert space, let \(C\) be a nonempty closed convex subset of \(E\) and let \(T_1,T_2,\dots\) be infinite nonexpansive mappings of \(C\) into itself such that \(\cap^\infty_{i=1} F(T_i)\) is nonempty. Let \(a,b\) be real numbers with \(0<a\leq b<1\) and \(\alpha_1\), \(\alpha_2,\dots\) be real numbers such that \(a\leq\alpha_i\leq b\) for every \(i=1,2,\dots\). Let \(W_n\) \((n= 1,2,\dots)\) be \(W\)-mappings generated by \(T_n\), \(T_{n-1}, \dots,T_1\) and \(\alpha_n\), \(\alpha_{n-1}, \dots,\alpha_1\) and \(W:C\to C\) with \(Wx=\lim W_nX= \lim U_{n,1}x\) for every \(x\in C\), and suppose \(x\in C\) and \(\{x_n\}\) is given by \(x_{n+1}= W_nx_n\) for \(n\geq 1\). Then \(F(W)=\bigcap^\infty_{i=1} F(T_i)\) and \(\{x_n\}\) converges weakly to a common fixed point \(z\) fo \(T_1,T_2,\dots\). Further, \(z=\lim Px_n\), where \(P\) is the metric projection of \(C\) onto \(F(W)= \bigcap^\infty_{i=1} F(T_i)\). Using this result, the authors prove a weak convergence theoren which is connected with the feasibility problem in a Hilbert space setting. In the last part, the authors study the situation where the constraints are inconsistent.