where γ ∈ ( 0, 2 λ ), with λ being the largest eigenvalue of the matrix A t A ( A t stands for matrix transposition) and S and T are quasi-nonexpansive operators.
To solve the SCFP of nonexpansive operators, Censor and Segal [23] proposed and proved, in finite-dimensional spaces, the convergence of the following algorithm: x_{n+1}=U bigl(x_{n}-gamma A^{ast} I-T Ax_{n} I-T Ax_{nd nin N, (1.7) where (gammain(0, frac{2}{lambigr)) with λ being the largest eigenvalue of the matrix (A^{ast}A).
where τ ∈ ( 0, 2 L ), with L being the largest eigenvalue of the matrix A ∗ A. Note that x ∗ solves (1.2) if and only if x ∗ solves the fixed point equation x ∗ = P C ( I − λ A ∗ ( I − P Q ) A ) x ∗. (1.6).
To solve (1.6), Censor and Segal [20] proposed and proved, in finite-dimensional spaces, the convergence of the following algorithm: x k + 1 = U ( x k + γ A t ( T − I ) A x k ), k ∈ N, where γ ∈ ( 0, 2 λ ), with λ being the largest eigenvalue of the matrix A t A ( A t stands for matrix transposition).
A seemingly more popular algorithm that solves the SFP is the CQ algorithm presented by Byrne [2]: x_{n+1}=P_{C} bigl I-gamma A^(I-P_{Q})A bigl I-gammauA^ I-P_{, (1.3) where the initiA^ I-P_{ (x_{0}inmathcal{H}_{1}) and (gammain(0,frac {2}{lambda})), with λ being the largest eigenvalue of the matrix (A^A).
One is Byrne's CQ algorithm [1] x n + 1 = P C ( x n − τ A ∗ ( I − P Q ) A x n ), n ∈ N, where τ ∈ ( 0, 2 L ) with L being the largest eigenvalue of the matrix A ∗ A, I is the unit matrix or operator, and P C and P Q denote the orthogonal projections onto C and Q, respectively.
