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Strong convergence to common fixed points of families of nonexpansive mappings. (English) Zbl 0883.47075

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. A mapping T of C into itself is said to be nonexpansive, if |Tx-Ty||x-y| for each x,yC.

For a mapping T of C into itself, we denote by F(T) the set of fixed points of T. We also denote by N and R + the set of positive integers and nonnegative real numbers, respectively. A family {S(t)} tR + of mappings of C into itself is called a nonexpansive semigroup of C, if it satisfies the following conditions:

(1) S(t 1 +t 2 )x=S(t 1 )S(t 2 )x for each t 1 ,t 2 R + and xC;

(2) S(0)x=x for each xC;

(3) for each xC, tS(t)x is continuous;

(4) |S(t)x-S(t)y||x-y| for each tR + and x,yC.

Convergence theorem for a finite mapping.

Convergence theorem for two commutative mappings in a Hilbert space.

Theorem 1. Let H be a Hilbert space, and let C be a nonempty closed convex subset of H. Let S and T be nonexpansive mappings of C into itself such that ST=TS and F(S)F(T) is nonempty. Suppose that {α n } n=0 [0,1] satisfies

lim n α n =0,and n=0 α n =·

Then, for an arbitrary xC, the sequence {x n } n=0 generated by x 0 =x and

x n+1 =α n x+(1-α n )2 (n+1)(n+2) k=0 n i+j=k S i T i x n ,n0,

converges strongly to a common fixed point Px of S and T, where P is the metric projection of H onto F(S)F(T).

Convergence theorem for nonexpansive semigroups.

Convergence theorem for a nonexpansive semigroup in a Hilbert space.

Theorem 2. Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let {S(t)} tR + be a nonexpansive semigroup on C such that tR + F(S(T)) is nonempty. Suppose that {β n } n=0 satisfies

lim n β n =0,and n=0 β n =·

Then, for an arbitrary zC, the sequence {z n } n=0 generated by z 0 =z and

z n+1 =β n z+(1-β n )1 t n 0 t n S(u)z n du,n0,

converges strongly to a common fixed point Pz of S(t), tR + , where P is the metric projection of H onto tR + F(S(T)) and {t n } n=0 is a positive real divergent sequence.


MSC:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H20Semigroups of nonlinear operators