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Convergence theorem of common fixed points for Lipschitzian pseudo-contraction semi-groups in Banach spaces. (English) Zbl 1181.47074
The author considers a uniformly convex Banach space $E$ satisfying the Opial condition, a closed, convex subset $C$ of $E$, and a Lipschitzian pseudo-contractive semigroup $\text{T}:=\{T(t):t\geq 0\}:C\to C$ such that $\text{F}:= \bigcap_{t\geq 0}F(T(t))\neq \emptyset$, where $F(T(t))$ denotes the set of fixed points of $T(t)$. Let $(\alpha _n)_n$ be a sequence in $(0,1)$ and $(t_n)_n$ a sequence in $(0,\infty)$ satisfying (i) $\limsup_{n}\alpha_n<1$ and (ii) $\lim_nt_n=\lim_n\frac{\alpha_n}{t_n}=0$. Under these circumstances, he shows that the sequence $x_0\in C$, $x_n=\alpha_nx_{n-1}+(1-\alpha_n)T(t_n)x_n,\ n\geq 1$, is weakly convergent to a common fixed point of the semigroup.

47J25Iterative procedures (nonlinear operator equations)
47H05Monotone operators (with respect to duality) and generalizations
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H20Semigroups of nonlinear operators
Full Text: DOI
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