Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

Let ð » be a real Hilbert space. Consider on ð » a nonexpansive semigroup 𠑆 = { 𠑇 ( ð ‘ ) ∶ 0 ≤ ð ‘ < ∞ } with a common fixed point, a contraction ð ‘“ with the coefficient 0 < ð ›¼ < 1 , and a strongly positive linear bounded self-adjoint operator ð ´ with the coefficient ð ›¾ >  0. Let 0 < ð ›¾ < ð ›¾ / ð ›¼ . It is proved that the sequence { ð ‘¥ ð ‘› } generated by the iterative method ð ‘¥ 0 ∈ ð » , ð ‘¥ ð ‘› + 1 = ð ›¼ ð ‘› ð ›¾ ð ‘“ ( ð ‘¥ ð ‘› ) + ð ›½ ð ‘› ð ‘¥ ð ‘› + ( ( 1 − ð ›½ ð ‘› ) ð ¼ âˆ’ ð ›¼ ð ‘› ð ´ ) ( 1 / ð ‘ ð ‘› ) ∫ ð ‘ ð ‘› 0 𠑇 ( ð ‘ ) ð ‘¥ ð ‘› ð ‘‘ ð ‘ , ð ‘› ≥ 0 converges strongly to a common fixed point ð ‘¥ ∗ ∈ ð ¹ ( 𠑆 ) , where ð ¹ ( 𠑆 ) denotes the common fixed point of the nonexpansive semigroup. The point ð ‘¥ ∗ solves the variational inequality ⟨ ( ð ›¾ ð ‘“ − ð ´ ) ð ‘¥ ∗ , ð ‘¥ − ð ‘¥ ∗ ⟩ ≤ 0 for all ð ‘¥ ∈ ð ¹ ( 𠑆 ) .