Abstract
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 > 0. Let /α. It is proved that the sequence {xn} generated by the iterative method 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).
Highlights
Introduction and PreliminariesLet H be a real Hilbert space and T be a nonlinear mapping with the domain D T
We denote by F S the set of common fixed points of S, that is, F S 0≤s
We show that the sequence {xt} generated by above continuous scheme strongly converges to a common fixed point x∗ ∈ F S, which is the unique point in F S solving the variational inequality γf − A x∗, x − x∗ ≤ 0 for all x ∈ F S
Summary
Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces. 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. 1− FS, solves the variational inequality γf − A x∗, x − x∗ ≤ 0 for all x ∈ F S
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