Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces (Q865314)

Let \(K\) be a (nonempty) closed convex subset of a real uniformly convex Banach space \(E\), which is also a nonexpansive retract with retraction \(P\). Let \((T_i:K\to E\), \(i=1,\dots,m)\) be asymptotically nonexpansive maps with associate sequences \(((k_{in})_{n\geq 1}\), \(i=1,\dots,m)\), and \(((\alpha_{in})_{n\geq 1}\), \(i=1,\dots,m)\) be sequences in \([\varepsilon, 1-\varepsilon]\), \(\varepsilon\in (0,1)\). For \(m\geq 2\), define the iterative process \((x_n)\) as \[ \begin{aligned} x_1 &\in K;\\ x_{n+1}&= [(1-\alpha_{1n})x_n+ \alpha_{1n}T_1(PT_1)^{n-1} y_{n+m-2}];\\ y_{n+m-2}&= [(1-\alpha_{2n})x_n+\alpha_{2n} T_2(PT_2)^{n-1} y_{n+m-3}];\\ &\;\vdots\\ y_{n}&= P[(1-\alpha_{mn}) x_n+\alpha_{mn} T_m(PT_m)^{n-1}x_{n}]. \end{aligned} \] Strong and weak convergence of this sequence to an element of \(F(T_1,\dots,T_m):=\bigcap \{F(T_i)\), \(i=1,\dots,m\}\) is discussed.