Common fixed points approximation for asymptotically nonexpansive semigroup in Banach spaces
Summary: Let \(E\) be a real Banach space satisfying local uniform Opial's condition, whose duality map is weakly sequentially continuous. Let \(\mathcal J := \{T(t):t \geq 0\}\) be a uniformly asymptotically regular family of asymptotically nonexpansive semigroups of \(E\) with function \(k : [0, \infty) \rightarrow [0, \infty)\). Let \(\mathcal F := \bigcap_{t{\geq}0}F(T(t)) \neq \emptyset\) and \(f : E \rightarrow E\) be weakly contractive map. Let \(G : E \rightarrow E\) be \(\delta\)-strongly accretive and \(\lambda\)-strictly pseudocontractive map with \(\delta + \lambda > 1\). Let \(\{t_n\}\) be an increasing sequence in \([0, \infty)\) and let \(\{\alpha_n\}\) and \(\{\beta_n\}\) be sequences in \([0, 1]\) satisfying some conditions. For some positive real number \(\gamma\) appropriately chosen, let \(\{x_n\}\) be a sequence defined by \(x_0 \in E\), \(x_{n+1} = \beta_nx_n + (1 - \beta_n)y_n\), \(y_n = (I - \alpha_nG)T(t_n)x_n + \alpha_n \gamma f(x_n)\), \(n \geq 0\). It is proved that \(\{x_n\}\) converges strongly to a common fixed point \(q\) of the family \(\mathcal J\) which is also the unique solution of the variational inequality \(\langle(G - \gamma f)q, j(q - x)\rangle \geq 0\) for all \(x \in \mathcal F\).
scientific article; zbMATH DE number 5964498
scientific article