Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

Abstract EN

Let H be a real Hilbert space.

Consider on H a nonexpansive semigroup S={T(s):0≤s<∞} with a common fixed point, a contraction f with the coefficient 0<α<1, and a strongly positive linear bounded self-adjoint operator A with the coefficient γ¯> 0.

Let 0<γ<γ¯/α.

It is proved that the sequence {xn} generated by the iterative method x0∈H, xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)(1/sn)∫0snT(s)xnds, n≥0 converges strongly to a common fixed point x*∈F(S), where F(S) denotes the common fixed point of the nonexpansive semigroup.

The point x* solves the variational inequality 〈(γf-A)x*,x-x*〉≤0 for all x∈F(S).