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Strong convergence theorem for nonexpansive semigroups in Hilbert space. (English) Zbl 1193.41036

Let H be a real Hilbert space, C a nonempty closed and convex subset of H, T:CC a nonexpansive mapping, F(T) the set of fixed point of T, and + be the set of nonnegative real numbers. Let {T(T(t):t + } be a nonexpansive semigroup on C, i.e.

(1) for each t + , T(t) is a nonexpansive mapping on C;

(2) T(0)x=x for all xC;

(3) T(s+t)=T(s)T(t) for all s,t + ;

(4) for each xC, the mapping T(·) from + into C is continuous.

The purpose of this paper is to prove the strong convergence of a method combining the decent method and the hybrid method in mathematical programming for finding a point pF t0 F(T(t)) i.e. a point in the common fixed set of a semigroup of nonexpansive mappings in Hilbert-space.

41A65Abstract approximation theory
47H20Semigroups of nonlinear operators