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An implicit iteration process for nonexpansive semigroups. (English) Zbl 1225.47093
Summary: Let $C$ be a closed convex subset of a Banach space $E$. Let $\{T(t):t \ge 0\}$ be a strongly continuous semigroup of nonexpansive mappings on $C$ such that $\bigcap_{t\ge 0} F(T(t))\ne\emptyset$. Let $\{\alpha_n\}$ and $\{t_n\}$ be sequences of real numbers satisfying appropriate conditions, then for arbitrary $x_0\in C$, the Mann type implicit iteration process $\{x_n\}$ given by $x_n=\alpha_nx_{n-1}+(1-\alpha_n)T(t_n)x_n$, $n\ge 0$, weakly (strongly) converges to an element of $\bigcap_{t\ge 0} F(T(t))$.

47J25Iterative procedures (nonlinear operator equations)
47H20Semigroups of nonlinear operators
47H09Mappings defined by “shrinking” properties
Full Text: DOI
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[2] Browder, F. E.: Nonexpansive nonlinear operators in Banach space, Proc. nat. Acad. sci. USA 54, 1041-1044 (1965) · Zbl 0128.35801 · doi:10.1073/pnas.54.4.1041
[3] Shioji, N.; Takahashi, W.: Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces, Nonlinear anal. 34, 87-99 (1998) · Zbl 0935.47039 · doi:10.1016/S0362-546X(97)00682-2
[4] Suzuki, T.: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert space, Proc. amer. Math. soc. 131, 2133-2136 (2003) · Zbl 1031.47038 · doi:10.1090/S0002-9939-02-06844-2
[5] Xu, H. -K.: A strong convergence theorem for contradiction semigroups in Banach spaces, Bull. austral. Math. soc. 72, 371-379 (2005) · Zbl 1095.47016 · doi:10.1017/S000497270003519X
[6] Saejung, S.: Strong convergence theorem for nonexpansive semigroups without Bochner integrals, Fixed point theory appl. 2008, 7 papers (2008) · Zbl 1203.47077 · doi:10.1155/2008/745010