Convergence theorem of common fixed points for Lipschitzian pseudo-contraction semi-groups in Banach spaces
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.
Lipschitzian pseudo-contractive semigroup
scientific article; zbMATH DE number 5550582
scientific article