Iterative methods for common fixed points for a countable family of nonexpansive mappings in uniformly convex spaces. (English) Zbl 1222.47095
Summary: Let \(E\) be a uniformly convex real Banach space with a uniformly Gâteaux differentiable norm. Let \(K\) be a closed, convex and nonempty subset of \(E\). Let \(\{T_i\})_{i-1}^\infty\) be a family of nonexpansive self-mappings of \(K\). For arbitrary fixed \(\delta\in (0,1)\), define a family of nonexpansive maps \(\{S_i\}^\infty\) by \(S_i:0(1-\delta)I+\delta T_i\) where \(I\) is the identity map of \(K\). Let \({\mathcal F}:=\bigcap^\infty_{i=1}F(T_i)\neq \emptyset\). It is proved that an iterative sequence \(\{x_n\}\) defined by \(x_0\in K\), \(x_{n+1}=\alpha_nu+\sum_{i\geq 1} \sigma_{i,t_n}S_ix_n\), \(n\geq 0\), converges strongly to a common fixed point of the family \(\{T_i\}^\infty_{i=1}\), where \(\{\alpha_n\}\) and \(\{\sigma_i,t_n\}\) are sequences in \((0,1)\) satisfying appropriate conditions, in each of the following cases: (a) \(E=l_p\), \(1<p<\infty\); and (b) at least one of the \(T_i\)’s is demicompact.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
Keywords:
demiclosed mappings; demicompact operator; nonexpansive mappings; strong convergence; convex real Banach space; uniformly Gâteaux differentiable norm; common fixed pointReferences:
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